We prove that a plane curve
We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, andcircles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.
]]>Let 𝑔(𝑥) ∈ ℝ[𝑥] be a polynomial of degree 𝑛 whose 𝑛 roots are real and distinct. Then 𝑔 has exactly 𝑛 -1 critical points, all of them real, interlacing the roots of 𝑔.
In this paper, we study the two-dimensional version of this phenomenon. We call a bivariate real polynomial 𝐺(𝑥, 𝑦) ∈ ℝ[𝑥, 𝑦] (or the corresponding affine plane curve 𝐶) expressive if the locations of the critical points of 𝐺 are determined by the set of real points 𝐶 ℝ = {(𝑥, 𝑦) ∈ ℝ 2 | 𝐺(𝑥, 𝑦) = 0}, as follows:
• there is precisely one extremum inside each bounded region of ℝ 2 ⧵ 𝐶 ℝ ;
• all other critical points of 𝐺 are the saddles located at hyperbolic nodes of 𝐶. (Recall that a hyperbolic node is an intersection of two smooth real local branches.) In particular, all critical points of an expressive polynomial 𝐺 are real.
An example is shown in Figure 1. For a nonexample, see Figure 2. Our main result (Theorem 7.17) gives an explicit characterization of expressive curves, subject to a mild requirement of "𝐿 ∞ -regularity." (This requirement forbids some exotic behaviour of 𝐶 at infinity.) We prove that a plane algebraic curve 𝐶 with real irreducible components is expressive and 𝐿 ∞ -regular if and only if
• each component of 𝐶 has a trigonometric or polynomial parametrization,
• all singular points of 𝐶 in the affine plane are real hyperbolic nodes, and • the set of real points of 𝐶 in the affine plane is connected.
To illustrate, a union of circles is an expressive curve provided any two of them intersect at two real points, as in Figure 1. On the other hand, the circle and the ellipse in Figure 2 intersect at four points, two of which are complex conjugate. (In the case of a pair of circles, those two points escape to infinity.)
The above characterization allows us to construct numerous examples of expressive plane curves, including arrangements of lines, parabolas, circles, and singular cubics; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more. See Figures 34for an assortment of examples; many more are scattered throughout the paper.
Our main result (Theorem 7.17) gives an explicit characterization of expressive curves, subject to a mild requirement of "L ∞ -regularity." (This requirement forbids some exotic behaviour of C at infinity.) We prove that a plane algebraic curve C with real irreducible components is expressive and L ∞ -regular if and only if • each component of C has a trigonometric or polynomial parametrization, • all singular points of C in the affine plane are real hyperbolic nodes, and • the set of real points of C in the affine plane is connected.
To illustrate, a union of circles is an expressive curve provided any two of them intersect at two real points, as in Figure 1. On the other hand, the circle and the ellipse in Figure 2 intersect at four points, two of which are complex conjugate. (In the case of a pair of circles, those two points escape to infinity.)
The above characterization allows us to construct numerous examples of expressive plane curves, including arrangements of lines, parabolas, circles, and singular cubics; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more. See Figures 34 On the face of it, expressivity is an analytic property of a function G : R 2 → R. This is however an illusion: just like in the univariate case, in order to rule out accidental critical points, we need G to be a polynomial of a certain kind. Thus, expressivity is essentially an algebraic phenomenon. Accordingly, its study requires tools of algebraic geometry and singularity theory. 1 Jun 2022 08:44:25 PDT 201027-Fomin Version 2 -Submitted to Commun. Amer. Math. Soc. On the face of it, expressivity is an analytic property of a function 𝐺 ∶ ℝ 2 → ℝ. This is however an illusion: just like in the univariate case, in order to rule out accidental critical points, we need 𝐺 to be a polynomial of a certain kind. Thus, expressivity is essentially an algebraic phenomenon. Accordingly, its study requires tools of algebraic geometry and singularity theory.
For a real plane algebraic curve 𝐶 to be expressive, one needs #{critical points of 𝐶 in the complex affine plane} = #{double points in 𝐶 ℝ } + #{bounded components of ℝ 2 ⧵ 𝐶 ℝ }. ( * ) Since a generic plane curve of degree 𝑑 has (𝑑 -1) 2 critical points, whereas expression ( * ) is typically smaller than (𝑑 -1) 2 , we need all the remaining critical points to escape to infinity. Our analysis shows that this can only happen if each (real) irreducible component of 𝐶 either has a unique point at infinity or a pair of complex conjugate points; moreover the components must intersect each other in the affine plane at real points, specifically at hyperbolic nodes. The requirement of having one or two points at infinity translates into the condition of having a polynomial or trigonometric parametrization, yielding the expressivity criterion formulated above.
As mentioned earlier, these results are established under the assumption of 𝐿 ∞regularity, which concerns the behaviour of the projective closure of 𝐶 at the line at infinity 𝐿 ∞ . This assumption ensures that the number of critical points accumulated at each point 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ is determined in the expected way by the topology of 𝐶 in the vicinity of 𝑝 together with the intersection multiplicity (𝐶 ⋅ 𝐿 ∞ ) 𝑝 . All polynomial and trigonometric curves are 𝐿 ∞ -regular, as are all expressive curves of degrees ≤ 4.
Section-by-section overview. Sections 1-4 are devoted to algebraic geometry groundwork. Section 1 reviews basic background on plane algebraic curves, intersection numbers, and topological invariants of isolated singularities. The number of critical points escaping to infinity is determined by the intersection multiplicities of polar curves at infinity, which are studied in Section 2. Its main result is Proposition 2.5, which gives a lower bound for such a multiplicity in terms of the Milnor number and the order of tangency between the curve and the line at infinity. When this bound becomes an equality, a plane curve 𝐶 is called 𝐿 ∞ -regular.
In Section 3, we provide several criteria for 𝐿 ∞ -regularity. We also show (see Proposition 3.4) that for an 𝐿 ∞ -regular curve 𝐶 = {𝐺(𝑥, 𝑦) = 0} all of whose singular points in the affine plane are ordinary nodes, the number of critical points of 𝐺 is completely determined by the number of those nodes, the geometric genus of 𝐶, and the number of local branches of 𝐶 at infinity. This statement relies on classical formulas due to H. Hironaka [22] and J. Milnor [29].
The technical material of Sections 2-3 can be safely skipped by the readers who are willing to treat the notion of 𝐿 ∞ -regularity as a "black-box" genericity condition that automatically holds for most, if not all, expressive curves that arise in applications.
Section 4 introduces polynomial and trigonometric curves, the plane curves possessing a parametrization 𝑡 ↦ (𝑋(𝑡), 𝑌 (𝑡)) in which both 𝑋 and 𝑌 are polynomials, resp. trigonometric polynomials. We review a number of examples of such curves, recall the classical result [1] characterizing polynomial curves as those with a single place at infinity, and provide an analogous characterization for trigonometric curves.
Expressive curves are introduced in Section 5. We formulate their basic properties and discuss a large number of examples, which include an inventory of expressive curves of degrees ≤ 4.
In Section 6, we introduce divides and relate them to the notion of expressivity. Section 7 contains our main results. Using the aforementioned bounds and criteria, we show (see Theorem 7.10) that an irreducible real plane algebraic curve is expressive and 𝐿 ∞ -regular if and only if it is either trigonometric or polynomial, and moreover all its singular points in the complex affine plane are (real) hyperbolic nodes. This criterion is then extended (see Theorem 7.17) to general plane curves with real irreducible components. Additional expressivity criteria are given in Section 8.
As a byproduct, we obtain the following elementary statement (see Corollary 7.19): if 𝐶 = {𝐺(𝑥, 𝑦) = 0} is a real polynomial or trigonometric affine plane curve that intersects itself solely at hyperbolic nodes, then all critical points of 𝐺 are real.
Multiple explicit constructions of expressive curves are presented in Sections 9-13, demonstrating the richness and wide applicability of the theory. In Section 9, we describe the procedures of bending, doubling, and unfolding. Each of them can be used to create new (more "complicated") expressive curves from existing ones. Arrangements of lines, parabolas, and circles, discussed in Section 10, provide another set of examples. These examples are generalized in Section 11 to arrangements consisting of shifts, dilations and/or rotations of a given expressive curve. Explicit versions of these constructions for polynomial (resp., trigonometric) curves are presented in Section 12 (resp., Section 13).
In Section 14, we briefly discuss alternative notions of expressivity: a "topological" notion that treats real algebraic curves set-theoretically, and an "analytic" notion that does not require the defining equation of a curve to be algebraic.
The class of divides which can arise from 𝐿 ∞ -regular expressive curves is studied in Section 15. In particular, we show that a simple pseudoline arrangement belongs to this class if and only if it is stretchable. In Section 16, we compare this class to the class of algebraic divides studied in [16].
Motivations and outlook. This work grew out of the desire to develop a global version of the A'Campo-Guseın-Zade theory [3,5,20,21] of morsifications of isolated singularities of plane curves. The defining feature of such morsifications is a local expressivity property, which prescribes the locations (up to real isotopy) of the critical points of a morsified curve in the vicinity of the original singularity. In this paper, expressivity is a global property of a real plane algebraic curve, prescribing the locations of its critical points (again, up to real isotopy) on the entire affine plane.
In a forthcoming follow-up to this paper, we intend to develop a global analoguein the setting of expressive curves-of A'Campo's theory of divides and their links. As shown in [16], this theory has intimate connections to the combinatorics of quivers, cluster mutations, and plabic graphs.
It would be interesting to explore the phenomenon of expressivity in higher dimensions, and in particular find out which results of this paper generalize.
The concept of an expressive curve/hypersurface can be viewed as a generalization of the notion of a line/hyperplane arrangement. (Expressivity of such arrangements in arbitrary dimension can be established by a log-concavity argument.) This opens the possibility of extending the classical theory of hyperplane arrangements [6,13,33] to arrangements of expressive curves/surfaces.
Definition 1.1. Let ℙ 2 denote the complex projective plane. We fix homogeneous coordinates 𝑥, 𝑦, 𝑧 in ℙ 2 . Any homogeneous polynomial 𝐹 ∈ ℂ[𝑥, 𝑦, 𝑧] defines a plane algebraic curve 𝐶 = 𝑍(𝐹) in ℙ 2 given by 𝐶 = 𝑍(𝐹) = {𝐹(𝑥, 𝑦, 𝑧) = 0}.
We understand the notion of a curve (and the notation 𝑍(𝐹)) scheme-theoretically: if the polynomial 𝐹 splits into factors, we count each component of the curve 𝐶 = 𝑍(𝐹) with the multiplicity of the corresponding factor.
For two distinct points 𝑝, 𝑞 ∈ ℙ 2 , we denote by 𝐿 𝑝𝑞 the line passing through 𝑝 and 𝑞. The line at infinity 𝐿 ∞ ⊂ ℙ 2 is defined by 𝐿 ∞ = 𝑍(𝑧).
For 𝐹 a smooth function in 𝑥, 𝑦, 𝑧, we use the shorthand
for the partial derivatives of 𝐹. The following elementary statement is well known, and easy to check.
Lemma 1.2 (Euler's formula). Let 𝐹 = 𝐹(𝑥, 𝑦, 𝑧) be a homogeneous polynomial of degree 𝑑. Then The polar curve 𝐶 (𝑞) associated with a curve 𝐶 = 𝑍(𝐹) and a point 𝑞 ∈ ℙ 2 is defined by 𝐶 (𝑞) = 𝑍(𝐹 (𝑞) ). In particular, for 𝑞 = (1, 0, 0) ∈ 𝐿 ∞ (resp., 𝑞 = (0, 1, 0) ∈ 𝐿 ∞ ), we get the polar curve 𝑍(𝐹 𝑥 ) (resp., 𝑍(𝐹 𝑦 )).
For a point 𝑝 lying on two plane curves 𝐶 and C , we denote by (𝐶⋅ C ) 𝑝 the intersection number of these curves at 𝑝. We will also use this notation for analytic curves, i.e., curves defined by analytic equations in a neighborhood of 𝑝. Definition 1.4. Let 𝐶 = 𝑍(𝐹) be a plane algebraic curve, and 𝑝 an isolated singular point of 𝐶. Let us recall the following topological invariants of the singularity (𝐶, 𝑝):
• the multiplicity mult(𝐶, 𝑝) = (𝐶 ⋅ 𝐿) 𝑝 , where 𝐿 is any line passing through 𝑝 which is not tangent to the germ (𝐶, 𝑝); • the 𝜘-invariant 𝜘(𝐶, 𝑝) = (𝐶 ⋅ 𝐶 (𝑞) ) 𝑝 , where 𝑞 ∈ ℙ 2 ⧵ {𝑝} is such that the line 𝐿 𝑝𝑞 is not tangent to (𝐶, 𝑝); • the number Br(𝐶, 𝑝) of local branches (irreducible components) of the germ (𝐶, 𝑝); • the 𝛿-invariant 𝛿(𝐶, 𝑝), which can be determined from 𝜘(𝐶, 𝑝) = 2𝛿(𝐶, 𝑝) + mult(𝐶, 𝑝) -Br(𝐶, 𝑝);
• the Milnor number 𝜇(𝐶, 𝑝) = (𝐶 (𝑞 ′ ) ⋅ 𝐶 (𝑞 ″ ) ) 𝑝 , where the points 𝑞 ′ , 𝑞 ″ ∈ ℙ 2 are chosen so that 𝑝, 𝑞 ′ , 𝑞 ″ are not collinear. More generally, for any point 𝑝 ∈ 𝐶 (𝑞 ′ ) ∩ 𝐶 (𝑞 ″ ) , not necessarily lying on the curve 𝐶, we can define the Milnor number 𝜇(𝐶, 𝑝) = (𝐶 (𝑞 ′ ) ⋅ 𝐶 (𝑞 ″ ) ) 𝑝 (provided 𝑝, 𝑞 ′ , 𝑞 ″ are not collinear). Note that for 𝑝 ∉ 𝐿 ∞ , we can simply define (1.2) 𝜇(𝐶, 𝑝) = (𝑍(𝐹 𝑥 ) ⋅ 𝑍(𝐹 𝑦 )) 𝑝 .
See [29, §5 and §10] and [18,Sections I.3.2 and I.3.4] for additional details as well as basic properties of these invariants. See also Remark 1.5 and Proposition 1.6. Remark 1.5. All invariants listed in Definition 1.4 depend only on the topological type of the singularity at hand. The Milnor number 𝜇(𝐶, 𝑝) measures the complexity of the singular point 𝑝 viewed as a critical point of 𝐹. It is equal to the maximal number of critical points that a small deformation of 𝐹 may have in the vicinity of 𝑝. The 𝛿invariant is the maximal number of critical points lying on the deformed curve in a small deformation of the germ (𝐶, 𝑝). The 𝜘-invariant is the number of ramification points of a generic projection onto a line of a generic deformation of the germ (𝐶, 𝑝).
It has two points on the line at infinity 𝐿 ∞ , namely 𝑝 1 = (0, 1, 0) and 𝑝 2 = (1, 1, 0). At 𝑝 1 , the cubical component has an elliptic node, and the two line components are the two tangents to the cubic at 𝑝
Note that (1.2) does not hold for 𝑝 = 𝑝 1 ; this is not a contradiction since 𝑝 1 ∈ 𝐿 ∞ .
In this section, we study the properties of intersection numbers of polar curves at their common points located at the line at infinity. Lemma 2.1. Let 𝐹(𝑥, 𝑦, 𝑧) ∈ ℂ[𝑥, 𝑦, 𝑧] be a nonconstant homogeneous polynomial.
(i) The set 𝑍(𝐹 (𝑞 ′ ) ) ∩ 𝑍(𝐹 (𝑞 ″ ) ) ∩ 𝐿 ∞ does not depend on the choice of a pair of distinct points 𝑞 ′ , 𝑞 ″ ∈ 𝐿 ∞ . Moreover this set is contained in 𝐶. (ii) For a point 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ , the intersection multiplicity (𝑍(𝐹 (𝑞 ′ ) ) ⋅ 𝑍(𝐹 (𝑞 ″ ) )) 𝑝 does not depend on the choice of a pair of distinct points 𝑞 ′ , 𝑞 ″ ∈ 𝐿 ∞ .
Proof. Any other pair q ′ , q ″ of distinct points in 𝐿 ∞ satisfies (2.1) q ′ = 𝑎 11 𝑞 ′ + 𝑎 12 𝑞 ″ , q ″ = 𝑎
21 𝑞 ′ + 𝑎 22 𝑞 ″ , with | | | 𝑎 11 𝑎 12 𝑎 21 𝑎 22 | | | ≠ 0.
Consequently 𝐹 ( q ′ ) = 𝑎 11 𝐹 (𝑞 ′ ) + 𝑎 12 𝐹 (𝑞 ″ ) , 𝐹 ( q ″ ) = 𝑎 21 𝐹 (𝑞 ′ ) + 𝑎 22 𝐹 (𝑞 ″ ) and the first claim in (i) follows. To establish the second claim, set 𝑞 ′ = (1, 0, 0) and 𝑞 ″ = (0, 1, 0) (i.e., take the polar curves 𝑍(𝐹 𝑥 ) and 𝑍(𝐹 𝑦 )), and note that by Euler's formula (1.1), 𝐹 vanishes as long as 𝐹 𝑧 , 𝐹 𝑦 and 𝑧 vanish.
To prove (ii), factor the nonsingular 2×2 matrix in (2.1) into the product of an upper triangular and a lower triangular matrix; then use that, for 𝑏𝑐 ≠ 0,
Definition 2.2. Let 𝐶 be a plane projective curve 𝐶 that does not contain the line at infinity 𝐿 ∞ as a component. For a point 𝑝 ∈ 𝐿 ∞ , we denote
Note that by Lemma 2.1(i), if 𝑝 lies on both 𝑍(𝐹 𝑥 ) and 𝑍(𝐹 𝑦 ), then it necessarily lies on 𝐶, so 𝜇(𝐶, 𝑝, 𝐿 ∞ ) can only be nonzero at points 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ .
Remark 2.3. For 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ , the number 𝜇(𝐶, 𝑝, 𝐿 ∞ ) may differ from the Milnor number 𝜇(𝐶, 𝑝) (cf. (1.2)), since the points 𝑝, 𝑞 ′ , 𝑞 ″ lie on the same line 𝐿 ∞ . Moreover, 𝜇(𝐶, 𝑝, 𝐿 ∞ ) is not determined by the topological type of the singularity (𝐶, 𝑝), as it also depends on its "relative position" with respect to the line 𝐿 ∞ . The following example illustrates this phenomenon. Consider the curves 𝑍(𝑥 2 𝑦 -𝑧 3 -𝑥𝑧 2 ) and 𝑍(𝑥 2 𝑦 2 -𝑦𝑧 3 ).
Each of them has an ordinary cusp (type 𝐴 2 ) at the point 𝑝 = (0, 1, 0). On the other hand, we have 𝜇(𝐶, 𝑝, 𝐿 ∞ ) = 4 in the former case versus 𝜇(𝐶, 𝑝, 𝐿 ∞ ) = 3 in the latter. Additional examples can be produced using Proposition 2.5.
Remark 2.4. The intersection number 𝜇(𝐶, 𝑝, 𝐿 ∞ ) is also different from "the Milnor number at infinity" (as defined, for instance, in [8,30]) since 𝜇(𝐶, 𝑝, 𝐿 ∞ ) depends on the choice of a point 𝑝 ∈ 𝐿 ∞ . Moreover, 𝜇(𝐶, 𝑝, 𝐿
∞ ) is not determined by the local topology of the configuration consisting of the germ (𝐶, 𝑝) and the line 𝐿 ∞ . To see this, consider Example 3.8 and Example 5.11 with 𝑝 = 𝑝 1 . In both cases, (𝐶, 𝑝) is an ordinary cusp transversal to 𝐿 ∞ . In Example 3.8, we have 𝜇(𝐶, 𝑝, 𝐿 ∞ ) = 4, whereas in Example 5.11, we get 𝜇(𝐶, 𝑝, 𝐿 ∞ ) = 𝜇(𝐶, 𝑝) + (𝐶 ⋅ 𝐿 ∞ ) 𝑝 -1 = 3 by Proposition 3.6). Proposition 2.5. Let 𝐶 = 𝑍(𝐹) be an algebraic curve in ℙ 2 . Let 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ . Then
The proof of Proposition 2.5 will rely on two lemmas, one of them very simple.
Lemma 2.6. For any 𝑞 ∈ 𝐿 ∞ ⧵ {𝑝}, we have
Proof. Using (1.4) and (1.5), we obtain:
1. □ Lemma 2.7. Let 𝑄 be a local branch (i.e., a reduced, irreducible component) of the germ of the curve 𝑍(𝐹 𝑦 ) at 𝑝. Then (2.4) (𝑍(𝑧𝐹 𝑧 ) ⋅ 𝑄) 𝑝 ≥ (𝑍(𝐹) ⋅ 𝑄) 𝑝 .
Proof. We will prove the inequality (2.4) inductively, by blowing up the point 𝑝. As a preparation step, we will apply a coordinate change intended to reduce the general case to a particular one, in which the blowing-up procedure is easier to describe. Without loss of generality, we assume that 𝑝 = (1, 0, 0). In a neighborhood of 𝑝, we can set 𝑥 = 1 and then work in the affine coordinates 𝑦, 𝑧. Abusing notation, for a homogeneous polynomial 𝐺(𝑥, 𝑦, 𝑧), we will write 𝐺(𝑦, 𝑧) instead of 𝐺(1, 𝑦, 𝑧).
For any curve 𝑍(𝐺), the intersection multiplicity (𝐺⋅𝑄) 𝑝 can be computed as follows. Write 𝑄 = {𝑓(𝑦, 𝑧) = 0}, where 𝑓(𝑦, 𝑧) is an irreducible element of the ring ℂ{𝑦, 𝑧} of germs at 𝑝 of holomorphic functions in the variables 𝑦 and 𝑧. By [18, Proposition I. It follows that the variable change (𝑦, 𝑧) = 𝜏(𝑦 1 , 𝑧 1 ) def = (𝑦 1 , 𝑧 𝑘 1 ) multiplies both sides of (2.4) by 𝑘. For 𝑔 ∈ ℂ{𝑦, 𝑧}, let us denote 𝜏 * 𝑔(𝑦 1 , 𝑧 1 ) = 𝑔 ∘ 𝜏(𝑦 1 , 𝑧 1 )). Then 𝜏 * 𝑄 splits into 𝑘 smooth branches 𝑄 𝑖 = {𝑦 1 -𝜉 𝑖 (𝑧 1 ) = 0}, 𝑖 = 1, . . . , 𝑘, and it suffices to prove the inequality (2.4) with 𝑄 replaced by each of the 𝑄 𝑖 's. Moreover, with respect to 𝑄 𝑖 , the desired inequality is of the same type. Namely, 𝜏 * (𝐹 𝑦 ) = (𝜏 * 𝐹) 𝑦 1 , and hence 𝑄 𝑖 is a local branch of the polar curve 𝑍((𝜏 * 𝐹) 𝑦 1 ) of the curve 𝜏 * 𝐶 = 𝑍(𝜏 * 𝐹). Furthermore,
which implies (𝑍(𝑧 1 (𝜏 * 𝐹) 𝑧 1 ) ⋅ 𝑄 𝑖 ) 𝑝 = (𝑍(𝜏 * (𝑧𝐹 𝑧 )) ⋅ 𝑄 𝑖 ) 𝑝 , 𝑖 = 1, . . . , 𝑘.
We have thus reduced the proof of (2.4) to the case where 𝑄 is a smooth curve germ transversal to the line 𝐿 ∞ . To simplify notation, we henceforth write 𝐶, 𝐹, 𝑦, 𝑧 instead of 𝜑 * 𝐶, 𝜑 * 𝐹, 𝑦 1 , 𝑧 1 , respectively.
We proceed by induction on 𝜇(𝐶, 𝑝). If 𝜇(𝐶, 𝑝) = 0, then (𝐶, 𝑝) is a smooth germ. If 𝐶 intersects 𝐿 ∞ transversally, then (𝐶, 𝑝) is given by 𝐹(𝑦, 𝑧) = 𝑎𝑦 + 𝑏𝑧 + h.o.t., 𝑎 ≠ 0 (hereinafter h.o.t. is a shorthand for "higher order terms"), implying 𝐹 𝑦 (0, 0) = 𝑎 ≠ 0. Thus the polar curve 𝑍(𝐹 𝑦 ) does not pass through 𝑝; consequently both sides of (2.4) vanish. If 𝐶 is tangent to 𝐿 ∞ at 𝑝, then it is transversal to 𝑄 at 𝑝, so we have
Suppose that 𝜇(𝐶, 𝑝) > 0. Then 𝑚 def = mult(𝐶, 𝑝) ≥ 2. If 𝐶 and 𝑄 intersect transversally at 𝑝 (i.e., have no tangent in common), then
On the other hand, mult(𝑍(𝐹 𝑧 ), 𝑝) ≥ 𝑚 -1, and therefore
If 𝐶 and 𝑄 have a common tangent at 𝑝, we apply the blowing-up 𝜋 ∶ P2 → ℙ 2 of the plane at the point 𝑝. For a curve 𝐷 passing through 𝑝, let 𝐷 * denote its strict transform, i.e., the closure of the preimage 𝜋 -1 (𝐷 ⧵ {𝑝}) in P2 . (For more details, see [18,Section I.3.3,p. 185].). Since 𝑄 is smooth, the strict transform 𝑄 * is smooth too, and intersects transversally the exceptional divisor 𝐸 at some point 𝑝 * . More precisely, if 𝑄 = 𝑍(𝑦 -𝜂𝑧 -h.o.t.), then in the coordinates (𝑦 * , 𝑧 * ) given by 𝑦 = 𝑦 * 𝑧 * , 𝑧 = 𝑧 * , we have 𝐸 = 𝑍(𝑧 * ) and 𝑝 * = (𝜂, 0). Since 𝐶 and its polar curve 𝑍(𝐹 𝑦 ) are tangent to the line 𝑍(𝑦-𝜂𝑧), the lowest homogeneous form of 𝐹(𝑦, 𝑧) is divisible by (𝑦-𝜂𝑧) 2 , while the lowest homogeneous form of 𝐹 𝑦 is divisible by 𝑦-𝜂𝑧 and, moreover, mult(𝑍(𝐹 𝑦 ), 𝑝) = 𝑚-1.
We now recall some properties of the blowing-up. For a curve 𝐷 = 𝑍(𝐺(𝑦, 𝑧)) passing through 𝑝, we have [18, In the rest of this section, we provide 𝐿 ∞ -regularity criteria for large classes of plane curves.
As mentioned in the Introduction, the technical material in this section can be skipped if the reader is willing to take it on faith and to view the requirement of 𝐿 ∞regularity as a genericity condition that holds in all "nonpathological" examples arising in common applications. Proposition 3.2. Let 𝐶 = 𝑍(𝐹(𝑥, 𝑦, 𝑧)) ⊂ ℙ 2 be a reduced algebraic curve of degree 𝑑 which does not contain 𝐿 ∞ as a component. Assume that the polynomial 𝐹(𝑥, 𝑦, 1) has 𝜉 < ∞ critical points, counted with multiplicities. Then we have
with equality if and only if 𝐶 is 𝐿 ∞ -regular.
Proof. In view of Proposition 2.5, we have
with equality if and only if 𝐶 is 𝐿 ∞ -regular. Since 𝐹 has finitely many critical points, Bézout's theorem for the polar curves 𝑍(𝐹 𝑥 ) and 𝑍(𝐹 𝑦 ) applies, yielding
The claim follows. □ Example 3.3. As in Example 1.7, consider the quintic curve 𝐶 = 𝑍(𝐹) defined by the polynomial
Set 𝐺(𝑥, 𝑦) = 𝐹(𝑥, 𝑦, 1) = (𝑥 2 + 1)(𝑦𝑥 2 + 𝑦 -𝑥 3 ). Then
and we see that 𝐺 has no critical points in the complex (𝑥, 𝑦)-plane; thus 𝜉 = 0. Using the values of Milnor numbers computed in Example 1.7, we obtain:
It follows by Proposition 3.2 that 𝐶 is 𝐿 ∞ -regular. Alternatively, one can check directly that the equality (3.1) holds at 𝑝 1 and 𝑝 2 . Recall that the geometric genus of a plane curve 𝐶 is defined by (3.5) 𝑔(𝐶) = ∑ 𝐶 ′ ∈𝐂𝐨𝐦𝐩(𝐶) (𝑔(𝐶 ′ ) -1) + 1, where 𝐂𝐨𝐦𝐩(𝐶) is the set of irreducible components of 𝐶, and 𝑔(𝐶 ′ ) denotes the genus of the normalization of a component 𝐶 ′ . Proposition 3.4. Let 𝐶 = 𝑍(𝐹(𝑥, 𝑦, 𝑧)) ⊂ ℙ 2 be a reduced algebraic curve of degree 𝑑. Suppose that • 𝐶 does not contain the line at infinity 𝐿 ∞ as a component; • all singular points of 𝐶 in the affine (𝑥, 𝑦)-plane ℙ ⧵ 𝐿 ∞ are ordinary nodes; • the polynomial 𝐹(𝑥, 𝑦, 1) ∈ ℂ[𝑥, 𝑦] has finitely many critical points. Let 𝜈 denote the number of nodes of 𝐶 in the (𝑥, 𝑦)-plane, and let 𝜉 denote the number of critical points of the polynomial 𝐹(𝑥, 𝑦, 1), counted with multiplicities. Then we have (3.6) 𝜉 ≤ 2𝑔(𝐶) -1 + 2𝜈 + ∑ 𝑝∈𝐶∩𝐿 ∞ Br(𝐶, 𝑝), with equality if and only if 𝐶 is 𝐿 ∞ -regular. Proof. By Hironaka's genus formula [22] (cf. also [19, Chapter II, (2.1.4.6)]), we have (3.7) 𝑔(𝐶) = (𝑑-1)(𝑑-2) 2 -∑ 𝑝∈𝐒𝐢𝐧𝐠(𝐶) 𝛿(𝐶, 𝑝), where 𝐒𝐢𝐧𝐠(𝐶) denotes the set of singular points of 𝐶. Combining this with Milnor's formula (1.3), we obtain:
Br(𝐶, 𝑝).
Therefore
and the claim follows from Proposition 3.2. □
Our next result (Proposition 3.6) shows that equation (3.1) holds under certain rather mild local conditions, To state these conditions, we will need to recall some terminology and notation. Definition 3.5 ([18, Definitions I.2.14-I.2.15]). We denote by Γ(𝐺) the Newton diagram of a bivariate polynomial 𝐺, i.e., the union of the edges of the Newton polygon of 𝐺 which are visible from the origin. The truncation of 𝐺 along an edge 𝑒 of Γ(𝐺) is the sum of all monomials in 𝐺 corresponding to the integer points in 𝑒.
An isolated singularity of an affine plane curve {𝐺(𝑦, 𝑧) = 0} at the origin is called Newton nondegenerate (with respect to the local affine coordinates (𝑦, 𝑧)) if the Newton diagram Γ(𝐺) intersects each of the coordinate axes, and the truncation of 𝐺 along any edge of the Newton diagram is a quasihomogeneous polynomial without critical points in (ℂ * ) 2 . Proposition 3.6. Let 𝐶 = 𝑍(𝐹(𝑥, 𝑦, 𝑧)) ⊂ ℙ 2 be a reduced curve not containing the line at infinity 𝐿 ∞ = 𝑍(𝑧) as a component. Let 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ . Without loss of generality, assume that 𝑝 = (1, 0, 0). Suppose that 𝑝 is either a smooth point of 𝐶 or a singular point of 𝐶 such that the singularity (𝐶, 𝑝) is Newton nondegenerate, in the local coordinates 𝑦, 𝑧; Thus, if conditions (3.8)-(3.9) hold at every point 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ , then 𝐶 is 𝐿 ∞ -regular.
Remark 3.7. Although we did not find Proposition 3.6 in the literature, similar resultsproved using similar tools-appeared before, see for example [8].
Proof. If 𝑝 is a smooth point of 𝐶 with the tangent 𝐿 ≠ 𝐿 ∞ , then
implying that 𝑝 ∉ 𝑍(𝐹 𝑦 ). The 𝐿 ∞ -regularity follows:
(𝑍(𝐹 𝑥 ) ⋅ 𝑍(𝐹 𝑦 )) 𝑝 = 0 = 𝜇(𝐶, 𝑝) + (𝐶 ⋅ 𝐿 ∞ ) 𝑝 -1.
If 𝐶 is smooth at 𝑝 with the tangent line 𝐿 ∞ , then
which implies the Newton nondegeneracy as well as condition (3.9):
(𝑍(𝑦) ⋅ 𝐶) 𝑝 = 1 < 𝑛 ≤ deg 𝐶 .
Thus, this situation can be viewed as a particular case of the general setting where we have a singular point 𝑝 satisfying conditions (3.8)-(3.9). We next turn to the treatment of this setting. We proceed in two steps. We first consider semi-quasihomogeneous singular points, and then move to the general case. We set 𝑥 = 1 in a neighborhood of 𝑝 and write 𝐹(𝑦, 𝑧) as a shorthand for 𝐹(1, 𝑦, 𝑧).
(1) Assume that Γ(𝐹) is a segment with endpoints (𝑚, 0) and (0, 𝑛). By the assumptions of the lemma, 𝑚 ≤ 𝑑 = deg 𝐹 and 𝑛 < 𝑑. The Newton nondegeneracy condition means that the truncation 𝐹 Γ(𝐹) of 𝐹 on Γ(𝐹) is a square-free quasihomogeneous polynomial.
Assuming that 𝑠 = gcd{𝑚, 𝑛}, 𝑚 = 𝑚 1 𝑠, 𝑛 = 𝑛 1 𝑠, we can write
𝑎 𝑘 𝑦 𝑚 1 𝑘 𝑧 𝑛 1 (𝑠-𝑘) , where 𝑎 0 𝑎 𝑠 ≠ 0.
Consider the family of polynomials 𝐹 𝑡 (𝑦, 𝑧) = 𝑡 -𝑚𝑛 𝐹(𝑦𝑡 𝑛 , 𝑧𝑡 𝑚 ) = 𝐹 Γ(𝐹) (𝑦, 𝑧) + ∑ 𝑖𝑛+𝑗𝑚>𝑚𝑛 𝑐 𝑖𝑗 𝑡 𝑖𝑛+𝑗𝑚-𝑚𝑛 𝑦 𝑖 𝑧 𝑗 , 𝑡 ∈ [0, 1].
Note that 𝐹 0 = 𝐹 Γ(𝐹) and the polynomials 𝐹 and 𝐹 𝑡 , 0 < 𝑡 < 1, differ by a linear change of the variables. This together with the lower semicontinuity of the intersection multiplicity implies
Here
Since 𝑎 𝑠 ≠ 0 and 𝑛 1 𝑠 = 𝑛 < 𝑑, these are nonzero polynomials, and moreover 𝐹 Γ(𝐹) 𝑦 splits into 𝑙 1 ≥ 0 factors of type 𝑧 𝑛 1 + 𝛼𝑦 𝑚 1 , 𝛼 ≠ 0 and the factor 𝑦 𝑚-1-𝑙 1 𝑚 1 , while 𝑧𝐹 Γ(𝐹) 𝑧 -𝑑𝐹 Γ(𝐹) splits into 𝑙 2 ≥ 0 factors of type 𝑧 𝑛 1 +𝛽𝑦 𝑚 1 , 𝛽 ≠ 0, and the factor 𝑧 𝑛-𝑙 2 𝑛 1 .
Observe that the polynomials 𝐹 Γ(𝐹) 𝑦 and 𝑧𝐹 Γ(𝐹) 𝑧 -𝐹 Γ(𝐹) are coprime. (Otherwise, they would have a common factor 𝑧 𝑛 1 + 𝛾𝑦 𝑚 1 with 𝛾 ≠ 0, which would also be a divisor of the polynomial
Then 𝐹 Γ(𝐹) and its derivative 𝐹 Γ(𝐹) 𝑦 would have a common factor, contradicting the square-freeness of 𝐹 Γ(𝐹) .) Since (𝑍(𝑦) ⋅ 𝑍(𝑧)) 𝑝 = 1, (𝑍(𝑦) ⋅ 𝑍(𝑧 𝑛 1 + 𝛽𝑦 𝑚 1 )) 𝑝 = 𝑛 1 , (𝑍(𝑧) ⋅ 𝑍(𝑧 𝑛 1 + 𝛼𝑦 𝑚 1 )) 𝑝 = 𝑚 1 as 𝛼 ≠ 0, (𝑍(𝑧 𝑛 1 + 𝛼𝑦 𝑚 1 ) ⋅ 𝑍(𝑧 𝑛 1 + 𝛽𝑦 𝑚 1 )) 𝑝 = 𝑚 1 𝑛 1 as 𝛼 ≠ 𝛽, the right-hand side of (3.11) equals
which together with (2.3) yields the desired equality.
(2) Suppose that Γ(𝐹) consists of 𝑟 ≥ 2 edges 𝜎 (1) , . . . , 𝜎 (𝑟) successively ordered so that 𝜎 (1) touches the axis of exponents of 𝑦 at the point (𝑚, 0), where 𝑚 = (𝐶 ⋅𝐿 ∞ ) 𝑝 , and 𝜎 (𝑟) touches the axis of exponents of 𝑧 at the point (0, 𝑛), where 𝑛 = (𝐶 ⋅ 𝑍(𝑦)) 𝑝 < 𝑑 = deg 𝐹. By the hypotheses of the lemma, for any edge 𝜎 = 𝜎 (𝑖) , the truncation 𝐹 𝜍 (𝑦, 𝑧) is the product of 𝑦 𝑎 𝑧 𝑏 , 𝑎, 𝑏 ≥ 0, and of a quasihomogeneous, square-free polynomial 𝐹 𝜍 0 , whose Newton polygon Δ(𝐹 𝜍 0 ) is the segment 𝜎 0 with endpoints on the coordinate axes, obtained from 𝜎 by translation along the vector (-𝑎, -𝑏).
Note that the minimal exponent of 𝑧 in the polynomial 𝑑𝐹(0, 𝑧) -𝑧𝐹 𝑧 (0, 𝑧) is 𝑛, and hence (3.13) (𝑍(𝐹 𝑦 ) ⋅ 𝑍(𝑑𝐹 -𝑧𝐹 𝑧 )) 𝑝 = (𝑍(𝑦𝐹 𝑦 ) ⋅ 𝑍(𝑑𝐹 -𝑧𝐹 𝑧 )) 𝑝 -𝑛 .
Next, we note that the Newton diagram Γ(𝑦𝐹 𝑦 ) contains entire edges 𝜎 (1) , . . . , 𝜎 (𝑟-1) and some part of the edge 𝜎 (𝑟) , while Γ(𝑑𝐹 -𝑧𝐹 𝑧 ) = Γ(𝐹), since the monomials of 𝑑𝐹 -𝑧𝐹 𝑧 and of 𝐹 on the Newton diagram Γ(𝐹) are in bijective correspondence, and the corresponding monomials differ by a nonzero constant factor, cf. (3.12). By [18, Proposition I. for 𝑐 the minimal exponent of 𝑧 in (𝑦𝐹 𝑦 ) 𝜍 (𝑟) . Thus,
3.14) (𝑍(𝑦𝐹 𝑦 ) ⋅ 𝑍(𝑑𝐹 -𝑧𝐹 𝑧 )) 𝑝 = ∑ 1≤𝑖,𝑗≤𝑟 (𝑍(𝜓 𝑖 ) ⋅ 𝑍(𝜃 𝑗 )) 𝑝 . We claim that (3.15) (𝑍(𝜓 𝑖 ) ⋅ 𝑍(𝜃 𝑗 )) 𝑝 = (𝑍(𝜑 𝑖 ) ⋅ 𝑍(𝜃 𝑗 )) 𝑝 for all 1 ≤ 𝑖, 𝑗 ≤ 𝑟 . Having this claim proven, we derive from (3.14) that (𝑍(𝑦𝐹 𝑦 ) ⋅ 𝑍(𝑑𝐹 -𝑧𝐹 𝑧 )) 𝑝 = ∑ 1≤𝑖,𝑗≤𝑟 (𝑍(𝜑 𝑖 ) ⋅ 𝑍(𝜃 𝑗 )) 𝑝 = (𝑍(𝑦𝐹 𝑦 ) ⋅ 𝑍(𝐹)) 𝑝 = (𝑍(𝑦) ⋅ 𝑍(𝐹)) 𝑝 + (𝑍(𝐹 𝑦 ) ⋅ 𝑍(𝐹)) 𝑝 = 𝑛 + 𝜘(𝐶, 𝑝) + (𝐶 ⋅ 𝐿 ∞ ) 𝑝 -mult(𝐶, 𝑝) = 𝑛 + (𝜇(𝐶, 𝑝) + mult(𝐶, 𝑝) -1) + (𝐶 ⋅ 𝐿 ∞ ) 𝑝 -mult(𝐶, 𝑝) = 𝑛 + 𝜇(𝐶, 𝑝) + (𝐶 ⋅ 𝐿 ∞ ) 𝑝 -1 , which completes the proof in view of (3.13). The equality (3.15) follows from the fact that both sides of the relation depend only on the geometry of the segments 𝜎 (𝑖) 0 and 𝜎 (𝑗)
0 . Suppose that 1 ≤ 𝑖 < 𝑗 ≤ 𝑟. We have 𝜎 (𝑖) = [(𝑚 ′ , 0), (0, 𝑛 ′ )], 𝜎 (𝑗) = [(𝑚 ″ , 0), (0, 𝑛 ″ )] with
𝑛 ″ . By the Newton-Puiseux algorithm [18, pp. 165-170], the function 𝜑 𝑖 (𝑦, 𝑧) (or 𝜓 𝑖 (𝑦, 𝑧)) splits into the product of 𝑢(𝑦, 𝑧) ∈ ℂ{𝑦, 𝑧}, 𝑢(0, 0) ≠ 0, and 𝑛 ′ factors of the form 𝑧 -𝛼𝑦 𝑚 ′ /𝑛 ′ + h.o.t., 𝛼 ≠ 0, and then by (2.5) we get (𝑍(𝜑 𝑖 ) ⋅ 𝑍(𝜃 𝑗 )) 𝑝 = (𝑍(𝜓 𝑖 ) ⋅ 𝑍(𝜃 𝑖 )) 𝑝 = 𝑚 ″ 𝑛 ′ . This holds even for 𝑗 = 𝑟 since the only monomial of 𝜃 𝑟 (𝑦, 𝑧) that comes into play is 𝛽𝑦 𝑚 ″ , 𝛽 ≠ 0. The case 1 ≤ 𝑗 < 𝑖 ≤ 𝑟 is settled analogously. Now suppose 1 ≤ 𝑖 = 𝑗 ≤ 𝑟. As we observed in the first part of the proof, the pairs of quasihomogeneous polynomials 𝜑 For another (more complicated) instance of this phenomenon, see Example 5.11.
Corollary 3.9. Let 𝐶 = 𝑍(𝐹(𝑥, 𝑦, 𝑧)) ⊂ ℙ 2 be a reduced curve not containing the line at infinity 𝐿 ∞ = 𝑍(𝑧) as a component. Let 𝐺(𝑥, 𝑦) = 𝐹(𝑥, 𝑦, 1). Assume that the Newton polygon Δ(𝐺) intersects each of the coordinate axes in points different from the origin, and the truncation of 𝐺 along any edge of the boundary 𝜕Δ(𝐺) not visible from the origin is a square-free polynomial, except possibly for factors of the form 𝑥 𝑖 or 𝑦 𝑗 . Then 𝐶 is 𝐿 ∞ -regular.
Proof. The intersection 𝐶 ∩ 𝐿 ∞ is determined by the top degree form of 𝐺(𝑥, 𝑦), which has the form 𝑥 𝑖 𝑦 𝑗 𝑓(𝑥, 𝑦), with 𝑓(𝑥, 𝑦) a square-free homogeneous polynomial. Hence 𝐶 ∩ 𝐿 ∞ consists of the points (0, 1, 0) (if 𝑖 > 0), (1, 0, 0) (if 𝑗 > 0), and deg(𝑓) other points, at which 𝐶 is smooth and 𝐿 ∞ -regular (see Proposition 3.6). The Newton diagram of each of the points (0, 1, 0) and (1, 0, 0) consists of some edges of 𝜕Δ(𝐺) mentioned in the lemma, and therefore these points (if they lie on 𝐶) satisfy the requirements of Proposition 3.6. □ Remark 3.10. A reducible plane curve 𝐶 = 𝑍(𝐹) with 𝐿 ∞ -regular components does not have to be 𝐿 ∞ -regular. For example, take 𝐹 = (𝑥𝑦-1)(𝑥𝑦-2): each factor is 𝐿 ∞ -regular, but the product is not, since 𝐹 𝑥 and 𝐹 𝑦 have a common divisor 2𝑥𝑦 -3.
Conversely, a curve may be 𝐿 ∞ -regular even when one of its components is not. For example, let 𝐹 = 𝑥 2 𝑦 + 𝑥 3 + 𝑥 2 𝑧 + 𝑥𝑧 2 + 𝑧 3 . Then 𝑍((𝑥 -𝑦)𝐹) is 𝐿 ∞ -regular by Proposition 3.6. On the other hand, 𝐿 ∞ -regularity of 𝑍(𝐹) fails at 𝑝 = (0, 1, 0): direct computation yields (𝑍(𝐹 𝑥 ) ⋅ 𝑍(𝐹 𝑦 )) 𝑝 = 4, whereas 𝜇(𝑍(𝐹), 𝑝) + (𝑍(𝐹) ⋅ 𝐿 ∞ ) 𝑝 -1 = 3.
In Sections 4 and 5, we introduce several classes of affine (rather than projective) plane curves. Before we begin, let us clarify what we mean by a real affine plane curve. There are two different notions here: an algebraic and a topological one: Definition 4.1. As usual, a reduced real algebraic curve 𝐶 in the complex affine plane 𝔸 2 ≅ ℂ 2 is the vanishing set
of a squarefree bivariate polynomial 𝐺(𝑥, 𝑦) ∈ ℝ[𝑥, 𝑦] ⊂ ℂ[𝑥, 𝑦]. We view 𝐶 as a subset of 𝔸 2 , and implicitly identify it with the polynomial 𝐺(𝑥, 𝑦) (viewed up to a constant nonzero factor), or with the principal ideal generated by it, the ideal of polynomials vanishing on 𝐶.
Alternatively, one can consider a "topological curve" 𝐶 ℝ in the real affine (𝑥, 𝑦)plane ℝ 2 , defined as the set of real points of an algebraic curve 𝐶 as above:
In contrast to the real algebraic curve 𝑉(𝐺) ⊂ ℂ 2 , the real algebraic set 𝑉 ℝ (𝐺)-even when it is one-dimensional-does not determine the polynomial 𝐺(𝑥, 𝑦) up to a scalar factor. In other words, an algebraic curve 𝐶 is not determined by the set of its real points 𝐶 ℝ , even when 𝐶 ℝ "looks like" an algebraic curve. (Roughly speaking, this is because 𝐶 can have "invisible" components which either have no real points at all or all such points are isolated in ℝ 2 .) There is however a canonical choice, provided 𝐶 ℝ is nonempty and without isolated points: we can let 𝐶 be the Zariski closure of 𝐶 ℝ , or equivalently let 𝐺(𝑥, 𝑦) be the minimal polynomial of 𝐶 ℝ , a real polynomial of the smallest possible degree satisfying 𝑉 ℝ (𝐺) = 𝐶 ℝ . (The minimal polynomial is defined up to a nonzero real factor.) Throughout this paper, we switch back and forth between a projective curve 𝑍(𝐹(𝑥, 𝑦, 𝑧)) and its affine counterpart 𝑉(𝐺(𝑥, 𝑦)), where 𝐺(𝑥, 𝑦) = 𝐹(𝑥, 𝑦, 1). (Remember that the line at infinity 𝐿 ∞ = 𝑍(𝑧) = ℙ 2 ⧵ 𝔸 2 is fixed throughout.) For example, we say that 𝑉(𝐺) is 𝐿 ∞ -regular if and only if 𝑍(𝐹) is 𝐿 ∞ -regular, cf. Definition 3.1. Definition 4.2. Let 𝐶 be a complex curve in the affine (𝑥, 𝑦)-plane. We say that 𝐶 is a polynomial curve if it has a polynomial parametrization, i.e., if there exist polynomials 𝑋(𝑡), 𝑌 (𝑡) ∈ ℂ[𝑡] such that the map 𝑡 ↦ (𝑋(𝑡), 𝑌 (𝑡)) is a (birational, i.e., generically one-to-one) parametrization of 𝐶.
A projective algebraic curve 𝐶 = {𝐹(𝑥, 𝑦, 𝑧) = 0} ⊂ ℙ 2 is called polynomial if 𝐶 does not contain the line at infinity 𝐿 ∞ = {𝑧 = 0}, and the portion of 𝐶 contained in the affine (𝑥, 𝑦)-plane (i.e., the curve {𝐹(𝑥, 𝑦, 1) = 0}) is an affine polynomial curve. Remark 4.3. Not every polynomial map defines a polynomial parametrization. For example, 𝑡 ↦ (𝑡 2 -𝑡, 𝑡 4 -2𝑡 3 + 𝑡) is not a polynomial (or birational) parametrization, since it is not generically one-to-one: (𝑋(𝑡), 𝑌 (𝑡)) = (𝑋(1 -𝑡), 𝑌 (1 -𝑡)).
Example 4.4. The cubic 𝑦 2 = 𝑥 2 (𝑥 -1) is a real polynomial curve, with a polynomial parametrization 𝑡 ↦ (𝑡 2 + 1, 𝑡(𝑡 2 + 1)). Note that this curve has an elliptic node (0, 0), attained for imaginary parameter values 𝑡 = ±√-1.
Example 4.5. The "witch of Agnesi" cubic 𝑥 2 𝑦 + 𝑦 -1 is rational but not polynomial. Indeed, if 𝑋(𝑡) is a positive-degree polynomial in 𝑡, then 𝑌 (𝑡) = 1 (𝑋(𝑡)) 2 +1 is not. 𝑇 𝑎 (cos 𝜑) = cos(𝑎𝜑).
The polynomial 𝑇 𝑎 (𝑥) has integer coefficients, and is an even (resp., odd) function of 𝑥 when 𝑎 is even (resp., odd). We note that 𝑇 𝑎 (𝑇 𝑏 (𝑡)) = 𝑇 𝑏 (𝑇 𝑎 (𝑡)) = 𝑇 𝑎𝑏 (𝑡).
Let 𝑎 and 𝑏 be coprime positive integers. The Chebyshev curve with parameters (𝑎, 𝑏) is given by the equation
It is not hard to see that this curve is polynomial; let us briefly sketch why. (For a detailed exposition, see [15, Section 3.9].) Without loss of generality, let us assume that 𝑎 is odd. Then the (𝑎, 𝑏)-Chebyshev curve has a polynomial parametrization 𝑡 ↦ (-𝑇 𝑏 (𝑡), 𝑇 𝑎 (𝑡)). Indeed, 𝑇 𝑎 (-𝑇 𝑏 (𝑡)) + 𝑇 𝑏 (𝑇 𝑎 (𝑡)) = -𝑇 𝑎𝑏 (𝑡) + 𝑇 𝑎𝑏 (𝑡) = 0.
To illustrate, consider the Chebyshev curves with parameters (3, 2) and (3, 4) shown in Figure 5. The (3, 2)-Chebyshev curve is a nodal Weierstrass cubic
or parametrically 𝑡 ↦ (-2𝑡 2 + 1, 4𝑡 3 -3𝑡). The (3, 4)-Chebyshev curve (see Figure 5) is a quartic given by the equation
or by the polynomial parametrization 𝑡 ↦ (-8𝑡 4 + 8𝑡 2 -1, 4𝑡 3 -3𝑡). Lemma 4.7. For a real plane algebraic curve 𝐶, the following are equivalent:
(1) 𝐶 is polynomial;
(2) 𝐶 has a parametrization 𝑡 ↦ (𝑋(𝑡), 𝑌 (𝑡)) with 𝑋(𝑡), 𝑌 (𝑡) ∈ ℝ[𝑡];
(3) 𝐶 is rational, with a unique local branch at infinity.
Proof. The equivalence (1)⇔(3) (for complex curves) is well known; see, e.g., [1]. The implication (2)⇒( 1) is obvious. It remains to show that (3)⇒( 2). The local branch of 𝐶 at infinity must by real, since otherwise complex conjugation would yield another such branch. Consequently, the set of real points of 𝐶 has a one-dimensional connected component which contains the unique point 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ . It follows that the normalization map 𝑛 ∶ ℙ 1 → 𝐶 ↪ ℙ 2 (which is nothing but a rational parametrization of 𝐶) pulls back the complex conjugation on 𝐶 to the antiholomorphic involution 𝑐 ∶ ℙ 1 → ℙ 1 which is a reflection with a fixed point set Fix(𝑐) ≃ 𝑆 1 , and 𝑝 lifts to a point in Fix(𝑐). (There is another possible antiholomorphic involution on ℙ 1 , the antipodal one, corresponding to real plane curves with a finite real point set.) Thus, we can choose coordinates (𝑡 0 , 𝑡 1 ) on ℙ 1 so that 𝑐(𝑡 0 , 𝑡 1 ) = (𝑡 0 , 𝑡 1 ) and the preimage of 𝑝 is (0, 1). Hence the map 𝑛 can be expressed as 𝑥 = 𝑋(𝑡 0 , 𝑡 1 ), 𝑦 = 𝑌 (𝑡 0 , 𝑡 1 ), 𝑧 = 𝑡 𝑑 0 , where (𝑡 0 , 𝑡 1 ) ∈ ℙ 1 , 𝑋 and 𝑌 are bivariate homogeneous polynomials of degree 𝑑 = deg 𝐶, and by construction 𝑥 = 𝑋(𝑡 0 , 𝑡 1 ), 𝑦 = 𝑌 (𝑡 0 , 𝑡 1 ), which means that 𝑋 and 𝑌 have real coefficients. □
Recall that a trigonometric polynomial is a finite linear combination of functions of the form 𝑡 ↦ sin(𝑘𝑡) and/or 𝑡 ↦ cos(𝑘𝑡), with 𝑘 ∈ ℤ ≥0 . Definition 4.8. We say that a real algebraic curve 𝐶 in the affine (𝑥, 𝑦)-plane is a trigonometric curve if there exist real trigonometric polynomials 𝑋(𝑡) and 𝑌 (𝑡) such that 𝑡 ↦ (𝑋(𝑡), 𝑌 (𝑡)) is a parametrization of 𝐶 ℝ , the set of real points of 𝐶, generically one-to-one for 𝑡 ∈ [0, 2𝜋).
A projective real algebraic curve 𝐶 = {𝐹(𝑥, 𝑦, 𝑧) = 0} ⊂ ℙ 2 is called trigonometric if 𝐶 does not contain the line at infinity 𝐿 ∞ = {𝑧 = 0}, and the portion of 𝐶 contained in the affine (𝑥, 𝑦)-plane is an affine trigonometric curve. Remark 4.9. Not every trigonometric map gives a trigonometric parametrization. For example, 𝑡 ↦ (cos(𝑡), cos(2𝑡)) is not a trigonometric parametrization of its image (a segment of the parabola 𝑦 = 2𝑥 2 -1), since it is not generically one-to-one on [0, 2𝜋). The algebraic equation for this curve is To illustrate, the (2, 3)-Lissajous curve is given by the equation
or by the trigonometric parametrization (4.7) 𝑡 ↦ (cos(3𝑡), sin(2𝑡)).
Several Lissajous curves, including this one, are shown in Figure 6. Example 4.12 (Rose curves). A rose curve with parameter 𝑞 = 𝑎 𝑏 ∈ ℚ >0 is defined in polar coordinates by 𝑟 = cos(𝑞𝜃), 𝜃 ∈ [0, 2𝜋𝑏). While a general rose curve has a complicated singularity at the origin, it becomes nodal when 𝑞 = 1 2𝑘+1 , with 𝑘 ∈ ℤ >0 . In that case, we get a "multi-limaçon," defined in polar coordinates by 𝑟 = cos( 𝜃 2𝑘+1 ), or equivalently by (4.8) 𝑟𝑇 2𝑘+1 (𝑟) -𝑥 = 0.
Note that the left-hand side of (4.8) is a polynomial in 𝑟 2 = 𝑥 2 + 𝑦 2 , so it is an algebraic equation in 𝑥 and 𝑦. This is a trigonometric curve, with a trigonometric parametrization given by 𝑡 ↦ ( cos(𝑘𝑡) + cos((𝑘 + 1)𝑡) 2 , sin(𝑘𝑡) + sin((𝑘 + 1)𝑡) 2
).
The cases 𝑘 = 1, 2, 3 are shown in Figure 7.
In the special case 𝑘 = 1, we get 𝑟𝑇 3 (𝑟) = 4𝑟 4 -3𝑟 2 , and equation (4.8) becomes (4.9)
This quartic curve is one of the incarnations of the limaçon of Étienne Pascal.
Lemma 4.13. For a real plane algebraic curve 𝐶, the following are equivalent:
(1) 𝐶 is trigonometric; (2) there exist polynomials 𝑃(𝜑), 𝑄(𝜑) ∈ ℂ[𝜑] such that the map ℂ * → 𝔸 2 given by (4.10) 𝜑 ⟼ (𝑃(𝜑) + 𝑃(𝜑 -1 ), 𝑄(𝜑) + 𝑄(𝜑 -1 )) is a birational parametrization of 𝐶. (3) 𝐶 is rational, with two complex conjugate local branches at infinity and with an infinite real point set. Proof. (1) ⇒ (2) The correspondence 𝑡 ↔ 𝜑 = exp(𝑡√-1) establishes a bianalytic isomorphism between (0, 2𝜋) and 𝑆 1 ⧵ {1}. (Here 𝑆 1 = {|𝜑| = 1} ⊂ ℂ.) Under this correspondence, we have (4.11) 𝑎 cos(𝑘𝑡) + 𝑏 sin(𝑘𝑡) = 𝑎-𝑏√-1 2 𝜑 𝑘 + 𝑎+𝑏√-1 2 𝜑 -𝑘 (𝑎, 𝑏 ∈ ℝ), so any trigonometric polynomial in 𝑡 transforms into a Laurent polynomial in 𝜑 of the form 𝑃(𝜑) + 𝑃(𝜑 -1 ). Thus a trigonometric parametrization of a curve 𝐶 yields its parametrization of the form (4.10); this parametrization is generically one-to-one along 𝑆 1 and therefore extends to a birational map ℙ 1 → 𝐶. (2) ⇒ (1) A parametrization (4.10) of a curve 𝐶 sends the circle 𝑆 1 generically one-to-one to 𝐶 ℝ , the real point set of 𝐶, see the formula (4.11). The same formula (4.11) converts the parametrization (4.10) restricted to the circle 𝑆 1 into a trigonometric parametrization 𝑡 ∈ [0, 2𝜋) ↦ (𝑋(𝑡), 𝑌 (𝑡)) of 𝐶 ℝ .
(2) ⇒ (3) A parametrization (4.10) intertwines the standard real structure in ℙ 2
and the real structure defined by the involution 𝑐(𝜑) = 𝜑 -1 on ℂ ⧵ {0}. Thus, it takes the set 𝑆 1 = Fix(𝑐) to the set 𝐶 ℝ of real points of 𝐶, while the conjugate points 𝜑 = 0 and 𝜑 = ∞ of ℙ 1 go to the points of 𝐶 at infinity determining two complex conjugate local branches at infinity.
(3) ⇒ ( 2) Assuming (3), the normalization map 𝑛 ∶ ℙ 1 → 𝐶 pulls back the standard complex conjugation in ℙ 2 to the standard complex conjugation on ℙ 1 , while the circle ℝℙ 1 maps to the one-dimensional connected component of 𝐶 ℝ , and some complex conjugate points 𝛼, 𝛼 ∈ ℙ 1 go to infinity. The automorphism of ℙ 1 defined by
takes the points 𝑠 = 𝛼 and 𝑠 = 𝛼 to 0 and ∞, respectively, the circle ℝℙ 1 to the circle 𝑆 1 , and the standard complex conjugation to the involution 𝑐, see above. Hence the parametrization 𝑛 ∶ ℙ
1 → 𝐶 goes to a parametrization (4.10). □ Lemma 4.14. Let 𝐶 be a real polynomial (resp., trigonometric) nodal plane curve, with a parametrization {(𝑋(𝑡), 𝑌 (𝑡))} as in Definition 4.2 (resp., Definition 4.8). Assume that 𝐶 has no elliptic nodes. Then 𝐶 ℝ = {(𝑋(𝑡), 𝑌 (𝑡)) | 𝑡 ∈ ℝ}.
Proof. This lemma follows from the well known fact (see, e.g., [25, Proposition 1.9]) that the real point set of a real nodal rational curve 𝐶 in ℝℙ 2 is the disjoint union of a circle ℝℙ 1 generically immersed in ℝℙ 2 and a finite set of elliptic nodes. □
If we allow elliptic nodes, the conclusion of Lemma 4.14 can fail, cf. Example 4.4.
Recall that an irreducible nodal plane curve of degree 𝑑 has at most
nodes, with the upper bound only attained for rational curves. In the case of trigonometric or polynomial curves, maximizing the number of nodes has direct geometric consequences: Proposition 4.15 (G. Ishikawa [23,Proposition 1.4]). Let 𝐶 be a trigonometric curve of degree 𝑑 with
real hyperbolic nodes. Then 𝐶 has no inflection points.
Proposition 4.16. A (complex) plane polynomial curve of degree 𝑑 with
nodes has no inflection points.
Proof. By Hironaka's genus formula (3.7), the projective closure Ĉ of 𝐶 has a single smooth point 𝑝 on 𝐿 ∞ , with ( Ĉ ⋅ 𝐿 ∞ ) 𝑝 = 𝑑. We then determine the number of inflection points of 𝐶 (in the affine plane) using Plücker's formula (see, e.g., [35, Chapter IV, Sections 6.2-6.3]):
Example 4.17. The curve (4.12) 𝑡 ↦ (cos((𝑘 -1)𝑡) + 𝑎 cos(𝑘𝑡), sin((𝑘 -1)𝑡) -𝑎 sin(𝑘𝑡))
is a trigonometric curve of degree 𝑑 = 2𝑘. (It is a special kind of hypotrochoid, cf. Definition 7.12.) For suitably chosen real values of 𝑎, this curve has
= (𝑘 -1)(2𝑘 -1) real hyperbolic nodes, as in Proposition 4.15. See Figures 8 and 9.
In the special case 𝑘 = 1 illustrated in Figure 8, we get a three-petal hypotrochoid, a quartic trigonometric curve with 3 nodes given by the parametrization (4.13) 𝑡 ↦ (cos(𝑡) + 𝑎 cos(2𝑡), sin(𝑡) -𝑎 sin(2𝑡)),
or by the algebraic equation Figure 9. Five-petal hypotrochoids {(cos(2𝑡) + 𝑎 cos(3𝑡), sin(2𝑡) -𝑎 sin(3𝑡))}, with 𝑎 = 5 6 (left) and 𝑎 = 2 (right). Each is a trigonometric curve of degree 𝑑 = 6, with (𝑑-1)(𝑑-2) 2 = 10 nodes.
• 𝐶 ℝ is connected, and contains at least two (hence infinitely many) points.
Remark 5.2. Let 𝐺(𝑥, 𝑦) be a real polynomial with real Morse critical points. Then each double point of 𝑉 ℝ (𝐺) must be a critical point of 𝐺 (a saddle). Also, each bounded connected component of ℝ 2 ⧵𝑉 ℝ (𝐺) must contain at least one critical point (an extremum). Thus, for 𝐺 to be expressive, it must have the smallest possible number of complex critical points that is allowed by the topology of 𝑉 ℝ (𝐺): a saddle at each double point, one extremum within each bounded component of ℝ 2 ⧵ 𝑉 ℝ (𝐺), and nothing else. The following quadratic polynomials are not expressive: • 𝐺 is squarefree (i.e., not divisible by a square of a nonscalar polynomial);
• all saddle points of 𝐺 lie on 𝑉 ℝ (𝐺); they are precisely the singular points of 𝑉(𝐺); • each bounded connected component of ℝ 2 ⧵ 𝑉 ℝ (𝐺) is simply connected.
Proof. As the critical points of 𝐺 are real and Morse, each of them is either a saddle or a local (strict) extremum of 𝐺, viewed as a function ℝ → ℝ. The extrema must be located outside 𝐶 ℝ , one per bounded connected component of ℝ 2 ⧵ 𝐶 ℝ . The saddles must lie on 𝐶 ℝ , so they are precisely the double points of it. We conclude that 𝐺 has finitely many critical points. Consequently 𝐺 is squarefree. Finally, since 𝐶 ℝ is connected, each bounded component of ℝ 2 ⧵ 𝐶 ℝ must be simply connected. □ Definition 5.1 naturally extends to homogeneous polynomials in three variables, and to algebraic curves in the projective plane: Definition 5.5. Let 𝐹(𝑥, 𝑦, 𝑧) ∈ ℝ[𝑥, 𝑦, 𝑧] ⊂ ℂ[𝑥, 𝑦, 𝑧] be a homogeneous polynomial with real coefficients, and 𝐶 = 𝑍(𝐹) the corresponding projective algebraic curve. Assume that 𝐹(𝑥, 𝑦, 𝑧) is not divisible by 𝑧. (In other words, 𝐶 does not contain the line at infinity 𝐿 ∞ .) We call 𝐹 and 𝐶 expressive if the bivariate polynomial 𝐹(𝑥, 𝑦, 1) is expressive in the sense of Definition 5.1, or equivalently the affine curve 𝐶⧵𝐿 ∞ ⊂ 𝔸 2 = ℙ 2 ⧵𝐿 ∞ is expressive.
In the rest of this section, we examine examples of expressive and nonexpressive curves and polynomials.
Example 5.6 (Conics). Among real conics (cf. Example 5.3), a parabola, an ellipse, and a pair of crossing real lines are expressive, whereas a hyperbola, a pair of parallel (or identical) lines, and a pair of complex conjugate lines (an elliptic node) are not.
Example 5.7. Let 𝑓 1 (𝑥), 𝑓 2 (𝑥) ∈ ℝ[𝑥] be two distinct real univariate polynomials of degrees ≤ 𝑑 such that 𝑓 1 -𝑓 2 has 𝑑 distinct real roots. (Thus, at least one of 𝑓 1 , 𝑓 2 has degree 𝑑.) We claim that the polynomial
is expressive. To prove this, we first introduce some notation. Let 𝑥 1 , . . . , 𝑥 𝑑 be the roots of 𝑓 1 -𝑓 2 , and let 𝑧 1 , . . . , 𝑧 𝑑-1 be the roots of its derivative 𝑓 ′ 1 -𝑓 ′ 2 ; they are also real and distinct, by Rolle's theorem. The critical points of 𝐺 satisfy
It is straightforward to see that these equations have 2𝑑 -1 solutions: 𝑑 hyperbolic nodes (𝑥 𝑘 , 𝑓 1 (𝑥 𝑘 )) = (𝑥 𝑘 , 𝑓 2 (𝑥 𝑘 )), for 𝑘 = 1, . . . , 𝑑, as well as 𝑑 -1 extrema at the points (𝑧 𝑘 , ) = (𝑥, 𝑦 -𝑓 1 (𝑥)) to reduce the problem to the easy case when one of the two polynomials is 0.) Example 5.8 (Lemniscates). The lemniscate of Huygens (or Gerono) is given by the equation 𝑦 2 + 4𝑥 4 -4𝑥 2 = 0, or by the parametrization 𝑡 ↦ (cos(𝑡), sin(2𝑡)). This curve, shown in Figure 6 on the far left, is a Lissajous curve with parameters (2, 1), cf. Example 4.11. The polynomial 𝐺(𝑥, 𝑦) = 𝑦 2 + 4𝑥 4 -4𝑥 2 has three real critical points: a saddle at the hyperbolic node (0, 0), plus two extrema (± 1 √2 , 0) inside the two bounded connected components of ℝ 2 ⧵ 𝑉 ℝ (𝐺). Thus, this lemniscate is expressive.
By contrast, another quartic curve with a similar name (and a similar-looking set of real points), the lemniscate of Bernoulli (5.1)
is not expressive, as the polynomial 𝐺(𝑥, 𝑦) = (𝑥 2 + 𝑦 2 ) 2 -2𝑥 2 + 2𝑦 2 has critical points (0, ±𝑖) outside ℝ 2 .
Many more examples of expressive and nonexpressive polynomials (or curves) are given in Tables 1 and 2, and later in the paper. Remark 5.9. Definition 5.1 can be generalized to allow arbitrary "hyperbolic" singular points, i.e., isolated real singular points all of whose local branches are real.
Remark 5.10. In can be verified by an exhaustive case-by-case analysis that every expressive curve of degree 𝑑 ≤ 4 is 𝐿 ∞ -regular. Starting with 𝑑 ≥ 5, this is no longer the case, cf. Example 5.11: an expressive curve 𝐶 need not be 𝐿 ∞ -regular, even when 𝐶 is rational. Still, examples like this one are rare.
Example 5.11. Consider the real rational quintic curve 𝐶 parametrized by
The set of its real points in the (𝑥, 𝑦)-plane consists of two interval components corresponding to the negative and positive values of 𝑡, respectively. These components intersect at the point (1, 1), attained for 𝑡 = 1 and 𝑡 = -1. The algebraic equation of 𝐶 is obtained as follows:
The Newton triangle of 𝐶 is conv{(0, 0), (3, 2), (2, 0)}, which means that
• at the point 𝑝 1 = (1, 0, 0), the curve 𝐶 has a type 𝐴 2 (ordinary cusp) singularity, tangent to the axis 𝑦 = 0; • at the point 𝑝 2 = (0, 1, 0), the curve 𝐶 has a type 𝐸 8 singularity, tangent to the axis 𝑥 = 0. In projective coordinates, we have 𝐶 = 𝑍(𝐹) where
It is now easy to check that the only critical point of 𝐹(𝑥, 𝑦, 1) is the hyperbolic node (1, 1) discussed above. It follows that the curve 𝐶 is expressive.
We next examine the behaviour of the polar curves at infinity. In a neighborhood of the point 𝑝 2 = (0, 1, 0), we set 𝑦 = 1 and obtain why not expressive? 𝑥 3 -𝑦 2 semicubic parabola critical point is not Morse 𝑥 3 -3𝑥 -𝑦 2 two-component elliptic curve saddle in an unbounded region 𝑥 3 -3𝑥 + 3 -𝑦 2 one-component elliptic curve saddle in an unbounded region 𝑥 3 + 3𝑥 -𝑦 2 one-component elliptic curve two nonreal critical points 𝑥 3 + 𝑥𝑦 2 + 4𝑥𝑦 + 𝑦 2 oblique strophoid two nonreal critical points 𝑥 2 𝑦 -𝑥 2 + 2𝑦 2 Newton's species #54 two nonreal critical points 𝑦𝑥 2 -𝑦 2 -𝑥𝑦 + 𝑥 2 Newton's species #51 two nonreal critical points 𝑥 3 + 𝑦 3 nodal Weierstrass cubic 3 (𝑥 -1)𝑥𝑦 three lines, two of them parallel
parabola crossed by a line 3 (𝑥 + 𝑦 -1)𝑥𝑦 three generic lines 3 (𝑥 2 + 𝑦 2 -1)𝑥 ellipse crossed by a line 4 𝑦 -𝑥 4 quartic parabola 4 (𝑦 -𝑥 2 ) 2 -𝑥 4 (𝑦 -𝑥 2 ) 2 -𝑥 2 two aligned parabolas 4 (𝑦 -𝑥 2 ) 2 + 𝑥 2 -1 4 (𝑦 -𝑥 3 )𝑥 cubic parabola and its axis 4 (𝑦 -𝑥 2 ) 2 -𝑥𝑦 4 (𝑦 -𝑥 2 )𝑥(𝑥 -1) parabola + two lines parallel to its axis 4 𝑦 2 -(𝑥 2 -1) 2 co-oriented parabolas crossing at two points 4 𝑦 2 + 4𝑥 4 -4𝑥 2 lemniscate of Huygens 4 𝑥(𝑥 -1)(𝑥 + 1)𝑦 three parallel lines crossed by a fourth 4 4(𝑥 2 + 𝑦 2 ) 2 -3(𝑥 2 + 𝑦 2 ) -𝑥 limaçon 4 𝑥(𝑥 -1)𝑦(𝑦 -1) two pairs of parallel lines 4 (𝑥 2 + 𝑦 2 -1)(𝑥 2 -2𝑥 + 𝑦 2 ) two circles crossing at two points 4 (𝑦 -𝑥 3 + 𝑥)𝑦 cubic parabola + line 4 (𝑥 3
parabola + line + line parallel to the axis 4 (𝑦 -𝑥 2 )(𝑦 -1)(𝑦 -2) parabola + two parallel lines 4 (𝑦 -𝑥 2 + 1)(𝑥 -𝑦 2 + 1)
two parabolas crossing at four points 4 see (4. 14) three-petal hypotrochoid 4 (𝑥 3 -3𝑥 + 2 -𝑦 2 )(𝑥 + 𝑦 -𝑎) nodal Weierstrass cubic + line 4 (𝑥 + 𝑦)(𝑥 -𝑦)(𝑥 -1)(𝑥 -𝑎) line + line + two parallel lines 4 (𝑥 2 + 𝑦 2 -1)𝑥(𝑥 -𝑎) ellipse + two parallel lines 4 (4𝑦 -𝑥 2 )(𝑥 + 𝑦)(𝑎𝑥 + 𝑦 + 1) parabola + two lines 4 (𝑥 2 + 𝑦 2 -1)(𝑦 -4𝑥 2 + 2) ellipse and parabola crossing at four points 4 𝑥(𝑦 + 1)(𝑥 -𝑦)(𝑥 + 𝑦 -1) four lines 4 (𝑥 2 + 𝑦 2 -1)𝑥(𝑥 + 𝑦 -𝑎) ellipse + two lines 4 (𝑥 2 + 4𝑦 2 )(4𝑥 2 + 𝑦 2 ) two ellipses crossing at four points Example 5.12 (Lissajous-Chebyshev curves). Let 𝑎 and 𝑏 be positive integers. The Lissajous-Chebyshev curve with parameters (𝑎, 𝑏) is given by the equation (5.2) 𝑇 𝑎 (𝑥) + 𝑇 𝑏 (𝑦) = 0. When 𝑎 and 𝑏 are coprime, with 𝑎 odd, we recover the polynomial Chebyshev curve with parameters (𝑎, 𝑏), see (4.2). When both 𝑎 and 𝑏 are even, with 𝑎 2 and 𝑏 2 coprime, we recover the trigonometric Lissajous curve with parameters ( 𝑎 2 , 𝑏 2
), see (4.5). This explains our use of the term "Lissajous-Chebyshev curve." The general construction appears in the work of S. Guseın-Zade [20], who observed (without using this terminology) that the Lissajous-Chebyshev curve with parameters (𝑎, 𝑏) provides a morsification of an isolated quasihomogeneous singularity of type (𝑎, 𝑏).
There is also a variant of the Lissajous-Chebyshev curve defined by
When 𝑎 (resp., 𝑏) is odd, this curve is a mirror image of the Lissajous-Chebyshev curve (5.2), under the substitution 𝑥 ≔ -𝑥 (resp., 𝑦 ≔ -𝑦). However, when both 𝑎 and 𝑏 are even, the two curves differ. For example, for (𝑎, 𝑏) = (4, 2), the curve defined by (5.2) is the lemniscate of Huygens (see Example 5.8), whereas the curve defined by (5.3) is a union of two parabolas. It is not hard to verify that every Lissajous-Chebyshev curve (and every curve 𝑉(𝑇 𝑎 (𝑥) -𝑇 𝑏 (𝑦))) is expressive. The critical points of 𝑇 𝑎 (𝑥) ± 𝑇 𝑏 (𝑦) are found from the equations 𝑇 ′ 𝑎 (𝑥) = 𝑇 ′ 𝑏 (𝑦) = 0, so they are of the form (𝑥 𝑖 , 𝑦 𝑗 ) where 𝑥 1 , . . . , 𝑥 𝑎-1 (resp., 𝑦 1 , . . . , 𝑦 𝑏-1 ) are the (distinct) roots of 𝑇 𝑎 (resp., 𝑇 𝑏 ). Since the total number of nodes and bounded components of ℝ 2 ⧵ 𝐶 ℝ is easily seen to be exactly (𝑎 -1)(𝑏 -1), the claim follows. Recall that the multi-limaçon with parameter 𝑘 is a trigonometric curve 𝐶 of degree 2𝑘 + 2 given by equation (4.8). It is straightforward to verify that the corresponding polynomial has 2𝑘 + 1 critical points, all of them located on the 𝑥 axis. Comparing this to the shape of the curve 𝐶 ℝ , we conclude that 𝐶 is expressive.
The notion of a divide was first introduced and studied by N. A'Campo [4,24]. The version of this notion that we use in this paper differs slightly from A'Campo's, and from the version used in [16]. Definition 6.1. Let 𝐃 be a disk in the real plane ℝ 2 . A divide 𝐷 in 𝐃 is the image of a generic relative immersion of a finite set of intervals and circles into 𝐃 satisfying the conditions listed below. The images of these immersed intervals and circles are called the branches of 𝐷. They must satisfy the following conditions:
• the immersed circles do not intersect the boundary 𝜕𝐃;
• the endpoints of the immersed intervals lie on 𝜕𝐃, and are pairwise distinct;
• these immersed intervals intersect 𝜕𝐷 transversally;
• all intersections and self-intersections of the branches are transversal. We view divides as topological objects, i.e., we do not distinguish between divides related by a diffeomorphism between their respective ambient disks.
A divide is called connected if the union of its branches is connected.
The connected components of the complement 𝐃 ⧵ 𝐷 which are disjoint from 𝜕𝐃 are the regions of 𝐷. If 𝐷 is connected, then each region of 𝐷 is simply connected. We refer to the singular points of 𝐷 as its nodes. See Figure 11. Remark 6.2. Although all divides of interest to us are connected, we forego any connectivity requirements in Definition 6.1 of a divide, which therefore is slightly more general than [16, Definition 2.1].
The main focus of [16] was on the class of algebraic divides coming from real morsifications of isolated plane curve singularities, see [16,Definition 2.3]. Here we study a different (albeit related) class of divides which arise from real algebraic curves: Definition 6.3. Let 𝐺(𝑥, 𝑦) ∈ ℝ[𝑥, 𝑦] be a real polynomial such that each real singular point of the curve 𝑉(𝐺) ⊂ ℂ 2 is a hyperbolic node (an intersection of two smooth real local branches). Then the portion of 𝑉 ℝ (𝐺) contained in a sufficiently large disk (6.1)
gives a divide in 𝐃 𝑅 . Moreover this divide does not depend (up to homeomorphism) on the choice of 𝑅 ≫ 0. We denote this divide by 𝐷 𝐺 .
Proposition 6.4. Let 𝐺(𝑥, 𝑦) ∈ ℝ[𝑥, 𝑦] be a real polynomial with 𝜉 < ∞ critical points. Assume that the real algebraic set 𝑉 ℝ (𝐺) = {𝐺 = 0} ⊂ ℝ 2 is nonempty, and each singular point of 𝑉 ℝ (𝐺) is a hyperbolic node. Let 𝜈 be the number of such nodes, and let 𝜄 be the number of interval branches of the divide 𝐷 𝐺 . Then (6.2) 𝜉 ≥ 2𝜈 -𝜄 + 1, with equality if and only if the polynomial 𝐺 is expressive. Proof. Let 𝐾 denote the union of the divide 𝐷 𝐺 and all its regions, viewed as a closed subset of ℝ 2 . Let 𝜎 be the number of connected components of 𝐾. Since all these components are simply connected, the Euler characteristic of 𝐾 is equal to 𝜎. On the other hand, 𝐾 can be split into 0-dimensional cells (the nodes, plus the ends of interval branches), 1-dimensional cells (curve segments of 𝐷 connecting nodes), ovals (smooth closed components of 𝐷 𝐺 ), and regions. The number 𝛽 of 1-dimensional cells satisfies 2𝛽 = 2𝜄 + 4𝜈 (by counting endpoints), implying 𝛽 = 𝜄 + 2𝜈. We thus have
where 𝜌 is the number of regions in 𝐷 𝐺 , and ℎ denotes the total number of holes (the sum of first Betti numbers) over all regions. Denote 𝑢 = 𝜉 -𝜈 -𝜌. The set of critical points of 𝐺 contains all of the nodes, plus at least one extremum per region. Thus 𝑢 = 0 if 𝐺 has no other critical points, and 𝑢 > 0 otherwise.
Putting everything together, we obtain: 𝜉 = 𝜈 + 𝜌 + 𝑢 = 𝜈 + 𝜎 -𝜄 + 𝜈 + ℎ + 𝑢 = 2𝜈 -𝜄 + 𝜎 + ℎ + 𝑢.
Since 𝜎 ≥ 1 and ℎ, 𝑢 ≥ 0, we get (6.2). Moreover 𝜉 = 2𝜈 -𝜄 + 1 if and only if 𝐾 is connected, all regions are simply connected, and 𝐺 has exactly 𝜈 + 𝜌 critical points. All these conditions are satisfied if 𝐺 is expressive; conversely, they ensure expressivity.
(𝑉 ℝ (𝐺) is connected if 𝐾 is connected and each region is simply connected.) □ Remark 6.5. Table 2 does not list any expressive quartic with 𝜉 = 4. We can now explain why. First, by Proposition 6.4, 𝜉 = 4 and 𝑑 = 4 would imply that 2𝜈 = 𝜄 + 3.
Case 1. 𝜈 = 3, 𝜄 = 3. Then our quartic 𝐶 has three real irreducible components, viz., a conic (necessarily a parabola) and two lines crossing it, with three hyperbolic nodes total, and no other nodes. Such a configuration is impossible: if both lines are parallel to the axis of the parabola, then we get two nodes; otherwise, at least four.
Case 2. 𝜈 = 2, 𝜄 = 1. Then 𝐶 is irreducible. (Otherwise, 𝐶 would split into two conicsa parabola and an ellipse-intersecting at four real points, contrary to 𝜈 = 2.) An irreducible quartic 𝐶 with 𝜄 = 1 has one real local branch 𝑄 centered at some real point 𝑝 ∈ 𝐿 ∞ , plus perhaps a pair of complex conjugate local branches centered on 𝐿 ∞ . Let us list the possibilities.
Case 2A. 𝑄 is smooth, (𝑄 ⋅ 𝐿 ∞ ) 𝑝 = 4, with no other local branches centered on 𝐿 ∞ .
Case 2B. 𝑄 is smooth, (𝑄 ⋅ 𝐿 ∞ ) 𝑝 = 2, with two smooth complex conjugate branches transversal to 𝐿 ∞ . These two local branches either cross 𝐿 ∞ at different points or at the same point 𝑝 ′ . By the genus formula (3.7) and due to 𝜈 = 2, we have 𝑝 ′ ≠ 𝑝, with a nodal singularity at 𝑝 ′ .
Case 2C. 𝑄 is singular. In this case, again using 𝜈 = 2 and the genus formula (3.7), we conclude that 𝑄 is of type 𝐴 2 . The intersection multiplicity of a cusp and a line is either 2 or 3. In our setting, (𝑄 ⋅ 𝐿 ∞ ) 𝑝 = 2, with two additional smooth complex conjugate local branches of 𝐶 centered at distinct complex conjugate points on 𝐿 ∞ .
All Cases 2A-2C are subject to Proposition 3.6, so 𝐶 is 𝐿 ∞ -regular. A direct computation then yields that ∑ 𝑞∈𝐶∩𝐿 ∞ 𝜇(𝐶, 𝑞, 𝐿 ∞ ) ≤ 3 in every case. This, however, is in contradiction with ∑ 𝑞∈𝐶∩𝐿 ∞ 𝜇(𝐶, 𝑞, 𝐿 ∞ ) = (4 -1) 2 -𝜉 = 5. 𝑔(𝐶 ′ ) + 𝜄 + ∑ 𝑝∈𝐶∩𝐿 ∞ Br(𝐶, 𝑝) ≥ 2𝑠, with equality if and only if 𝐶 is both expressive and 𝐿 ∞ -regular. On the other hand, we have the inequalities ∑ 𝐶 ′ ∈𝐂𝐨𝐦𝐩(𝐶) 𝑔(𝐶 ′ ) ≥ 0, (7.2) 𝜄 ≥ 𝑠 1 , (7.3) ∑ 𝑝∈𝐶∩𝐿 ∞ Br(𝐶, 𝑝) ≥ 𝑠 1 + 2𝑠 2 , (7.4) 2𝑠 1 + 2𝑠 2 ≥ 2𝑠, (7.5) whose sum yields (7.1). Therefore we have equality in (7.1) if and only if each of (7.2)-(7.5) is an equality. This is precisely statement (ii). □ Example 7.2. The curve 𝐶 from Example 5.11 satisfies requirements (a)-(e) of Proposition 7.1. Since 𝐶 is not 𝐿 ∞ -regular, condition (ii) must fail. Indeed, while 𝐶 is rational, it has two real local branches at infinity, centered at the points 𝑝 1 and 𝑝 2 . Proposition 7.3. Let 𝐶 be an expressive 𝐿 ∞ -regular plane curve whose irreducible components are all real. Then each component of 𝐶 is either trigonometric or polynomial. Proof. Since 𝐶 is 𝐿 ∞ -regular and expressive, with real components, the requirements (a)-(e) of Proposition 7.1 are automatically satisfied, cf. Lemma 5.4. Consequently the statement (ii) of Proposition 7.1 holds. By Lemmas 4.7 and 4.13, this would imply that each component of 𝐶 is either trigonometric or polynomial, provided the real point set of each component is infinite. It thus suffices to show that each component 𝐵 of 𝐶 has an infinite real point set in 𝔸 2 . In fact, it is enough to show that this set is nonempty, for if it were finite and nonempty, then 𝐵-hence 𝐶-would have an elliptic node, contradicting the expressivity of 𝐶 (cf. Lemma 5.4). It remains to prove that each component of 𝐶 has a nonempty real point set in 𝔸 2 .
We argue by contradiction. Let 𝐶 = 𝑍(𝐹). Suppose that 𝐵 = 𝑍(𝐺) is a component of 𝐶 without real points in 𝔸 2 . We claim that the rest of 𝐶 is given by a polynomial of the form 𝐻(𝐺, 𝑧), where 𝐻 ∈ ℝ[𝑢, 𝑣] is a bivariate polynomial. Once we establish this claim, it will follow that the polynomials 𝐹 𝑥 and 𝐹 𝑦 have a nontrivial common factor 𝜕 𝜕ᵆ (𝑢𝐻(𝑢, 𝑣))| | ᵆ=𝐺,𝑣=𝑧 , contradicting the finiteness of the intersection 𝑍(𝐹 𝑥 ) ∩ 𝑍(𝐹 𝑦 ).
Let us make a few preliminary observations. First, the degree 𝑑 = deg 𝐺 = deg 𝐵 must be even. Second, by Proposition 7.1, 𝐵 has two complex conjugate branches centered on 𝐿 ∞ . Third, in view of the expressivity of 𝐶, the affine curve 𝐵 ∩ 𝔸 2 is disjoint from any other component 𝐵 ′ of 𝐶, implying that
Consider two possibilities.
Case 1. 𝐵 ∩ 𝐿 ∞ consists of two complex conjugate points 𝑝 and 𝑝. Let 𝑄 and 𝑄 be the local branches of 𝐵 centered at 𝑝 and 𝑝, respectively. Let 𝐵 ′ = 𝑍(𝐺 ′ ) be some other component of 𝐶, of degree 𝑑 ′ = deg 𝐵 ′ = deg 𝐺 ′ . Since 𝐵 ′ is real and satisfies statement (ii) of Proposition 7.1 as well as (7.6), we get
moreover 𝐵 ′ has a unique local branch 𝑅 (resp., 𝑅) at the point 𝑝 (resp. 𝑝). We have
It follows that any curve B in the pencil Span{𝐵 ′ , (𝐿 ∞ ) 𝑑 ′ } satisfies
2 . Pick a point 𝑞 ∈ 𝐵 ∩ 𝔸 2 . Since 𝑞 ∉ 𝐵 ′ ∪ 𝐿 ∞ , there exists a curve B ∈ Span{𝐵 ′ , (𝐿 ∞ ) 𝑑 ′ } containing 𝑞. It then follows by Bézout and by (7.7) that B must contain 𝐵 as a component. In particular, 𝑑 ′ ≥ 𝑑. By symmetry, the same argument yields 𝑑 ≥ 𝑑 ′ . Hence 𝑑 ′ = 𝑑, 𝐵 ∈ Span{𝐵 ′ , (𝐿 ∞ ) 𝑑 }, and the polynomial 𝐺 ′ defining the curve 𝐵 ′ satisfies 𝐺 ′ = 𝛼𝐺 + 𝛽𝑧 𝑑 for some 𝛼, 𝛽 ∈ ℂ ⧵ {0}. The desired claim follows.
Case 2. 𝐵 ∩ 𝐿 ∞ consists of one (real) point 𝑝. Then 𝐵 has two complex conjugate branches 𝑄 and 𝑄 centered at 𝑝.
Let 𝐵 ′ = 𝑍(𝐺 ′ ) be a component of 𝐶 different from 𝐵, and let 𝐵 ′ have a unique (real) local branch 𝑅, necessarily centered at 𝑝. Denote 𝑑 ′ = deg 𝐵 ′ = deg 𝐺 ′ . Then
and then we conclude-as above-that there exists a curve B ′ ∈ Span{𝐵 ′ , (𝐿 ∞ ) 𝑑 ′ } containing 𝐵 as a component. On the other hand,
which in a similar manner implies that there exists a curve B ∈ Span{𝐵, (𝐿 ∞ ) 𝑑 } containing 𝐵 ′ as a component. We conclude that 𝑑 ′ = 𝑑, 𝐵 ∈ Span{𝐵 ′ , (𝐿 ∞ ) 𝑑 }, and finally, 𝐺 ′ = 𝛼𝐺 + 𝛽𝑧 𝑑 for 𝛼, 𝛽 ∈ ℂ ⧵ {0}, as desired. Now let 𝐵 ′ = 𝑍(𝐺 ′ ) be a component of 𝐶 different from 𝐵, and let it have a couple of complex conjugate local branches 𝑅 and 𝑅 centered at 𝑝. Since 𝐵 ′ is real, we have
and since 𝐵 is real, we have
Thus the above reasoning applies again, yielding 𝑑 ′ = 𝑑 and 𝐵 ′ ∈ Span{𝐵, (𝐿 ∞ ) 𝑑 }. Hence 𝐺 ′ = 𝛼𝐺 + 𝛽𝑧 𝑑 for 𝛼, 𝛽 ∈ ℂ ⧵ {0}, and we are done. □ Example 7.4 shows that in Proposition 7.3, the requirement that all components are real cannot be dropped. We note that such a curve also needs to be immersed, meeting itself transversally at real hyperbolic nodes (thus, no cusps, tacnodes, or triple points).
We next provide a partial converse to Corollary 7.5.
Proposition 7.6. Let 𝐶 be a real polynomial or trigonometric curve whose singular set in the affine plane 𝔸 2 = ℙ 2 ⧵ 𝐿 ∞ consists solely of hyperbolic nodes. Then 𝐶 is expressive and 𝐿 ∞ -regular.
Proof. By Lemmas 4.7 and 4.13, a real polynomial or trigonometric curve 𝐶 = 𝑍(𝐹) is a real rational curve with one real or two complex conjugate local branches at infinity, and with a nonempty set of real points in 𝔸 2 . Thus, conditions (ii), (a), (b), (c), and (e) of Proposition 7.1 are satisfied, so in order to obtain (i), we only need to establish (d).
That is, we need to show that the polynomial 𝐹 has finitely many critical points in the affine plane 𝔸 2 . We will prove this by contradiction. Denote 𝑑 = deg 𝐶 = deg 𝐹. Suppose that 𝑍(𝐹 𝑥 ) ∩ 𝑍(𝐹 𝑦 ) contains a (real, possibly reducible) curve 𝐵 of a positive degree 𝑑 ′ < 𝑑.
We first observe that 𝐵∩𝐶∩𝔸 2 = ∅. Assume not. If 𝑞 ∈ 𝐵∩𝐶∩𝔸 2 , then 𝑞 ∈ 𝐒𝐢𝐧𝐠(𝐶)∩ 𝔸 2 , so 𝑞 must be a hyperbolic node of 𝐶, implying (𝑍(𝐹 𝑥 ) ⋅ 𝑍(𝐹 𝑦 )) 𝑞 = 𝜇(𝐶, 𝑞) = 1; but since 𝑞 lies on a common component of 𝑍(𝐹 𝑥 ) and 𝑍(𝐹 𝑦 ), we must have (𝑍(𝐹 𝑥 )⋅𝑍(𝐹 𝑦 )) 𝑞 = ∞. Since 𝐵 is real, and 𝐶 has either one real or two complex conjugate local branches at infinity, it follows that 𝐵 ∩ 𝐶 = 𝐿 ∞ ∩ 𝐶.
Case 1. 𝐶 has a unique (real) branch 𝑄 at a point 𝑝 ∈ 𝐿 ∞ . Then (𝐵 ⋅ 𝑄) 𝑝 = 𝑑 ′ 𝑑. On the other hand, ((𝐿 ∞ ) 𝑑 ′ ⋅ 𝑄) 𝑝 = 𝑑 ′ 𝑑. Hence any curve B ∈ Span{𝐵, (𝐿 ∞ ) 𝑑 ′ } satisfies ( B ⋅ 𝑄) 𝑝 ≥ 𝑑 ′ 𝑑. For any point 𝑞 ∈ 𝐶 ⧵ {𝑝} there is a curve B ∈ Span{𝐵, (𝐿 ∞ ) 𝑑 ′ } passing through 𝑞. Hence 𝐶 is a component of B , in contradiction with 𝑑 ′ < 𝑑.
Case 2. 𝐶 has two complex conjugate branches 𝑄 and 𝑄 centered at (possibly coinciding) points 𝑝 and 𝑝 on 𝐿 ∞ , respectively. Since 𝐵 is real, we have
This implies that any curve B ∈ Span{𝐵, (𝐿 ∞ ) 𝑑 ′ } satisfies both ( B ⋅ 𝑄) 𝑝 ≥ 𝑑 ′ 𝑑/2 and ( Q ⋅ 𝑄) 𝑝 ≥ 𝑑 ′ 𝑑/2, resulting in a contradiction as in Case 1. □ Remark 7.7. In Proposition 7.6, the requirement concerning the singular set cannot be dropped: a real polynomial curve 𝐶 may have elliptic nodes (see Example 4.4) or cusps (consider the cubic (𝑡 2 , 𝑡 3 )), preventing 𝐶 from being expressive.
Remark 7.8. Proposition 7.6 implies that if a real curve 𝐶 is polynomial or trigonometric, and also expressive, then it is necessarily 𝐿 ∞ -regular. Indeed, expressivity in particular means that all singular points of 𝐶 are hyperbolic nodes. Example 7.14. An epitrochoid with (coprime) parameters 𝑏 and 𝑐 (here 𝑏 > -𝑐 > 0) has 𝑏 + 𝑐 inward-pointing "petals," see Figure 14.
When 𝑏 = -𝑐 + 1, we recover the "multi-limaçons" of Example 4.12 and Figure 7 (with 𝑏 ∈ {2, 3, 4}, 𝑐 = 1 -𝑏). Proposition 7.15. Let 𝐶 be an epitrochoid or hypotrochoid given by (7.9)-(7.10). (As in Definition 7.12, 𝑏 and 𝑐 are coprime integers, with 𝑏 > |𝑐|; and 𝑢, 𝑣 ∈ ℝ * .) Then 𝐶 is expressive and 𝐿 ∞ -regular if it has (𝑏 + 𝑐)(𝑏 -1) hyperbolic nodes in 𝔸 2 . In particular, this holds if |𝑣| |ᵆ| is sufficiently small. Proof. Setting 𝜏 = 𝑒 𝑖𝑡 , we convert the trigonometric parametrization (7.9)-(7.10) of 𝐶 into a rational one: 𝑥 = 1 2 (𝑢𝜏 2𝑏 + 𝑣𝜏 𝑏+𝑐 + 𝑣𝜏 𝑏-𝑐 + 𝑢), 𝑦 = -𝑖 2 (𝑢𝜏 2𝑏 -𝑣𝜏 𝑏+𝑐 + 𝑣𝜏 𝑏-𝑐 -𝑢), 𝑧 = 𝜏 𝑏 .
This curve has two points at infinity, namely (1, 𝑖, 0) and (1, -𝑖, 0), corresponding to 𝜏 = 0 and 𝜏 = ∞, respectively. Noting that the line 𝑥 + 𝑖𝑦 = 0 passes through (1, 𝑖, 0), we change the coordinates by replacing 𝑦 by 𝑥 + 𝑖𝑦 = 𝑢𝜏 2𝑏 + 𝑣𝜏 𝑏-𝑐 .
In a neighborhood of (1, 𝑖, 0), this yields the following parametrization:
Thus, at the point (1, 𝑖, 0) we have a semi-quasihomogeneous singularity of weight (𝑏 -𝑐, 𝑏), and similarly at (1, -𝑖, 0). The 𝛿-invariant at each of the two points is equal to 1 2 (𝑏 -𝑐 -1)(𝑏 -1). Hence in the affine plane 𝔸 2 , there remain nodes. So if all these nodes are real hyperbolic (and distinct), then 𝐶 is expressive and 𝐿 ∞ -regular by Proposition 7.6 (or Theorem 7.10).
It remains to show that 𝐶 has (𝑏 + 𝑐)(𝑏 -1) hyperbolic nodes as long as 𝑢, 𝑣 ∈ ℝ * are chosen appropriately (in particular, if |𝑣| |ᵆ| is sufficiently small). Consider the rational parametrization of 𝐶 given by 𝑥 + 𝑖𝑦 = 𝑢𝜏 𝑏 + 𝑣𝜏 -𝑐 , 𝑥 -𝑖𝑦 = 𝑢𝜏 -𝑏 + 𝑣𝜏 𝑐 , 𝑧 = 1.
We need to show that all solutions (𝜏, 𝜎) ∈ (ℂ * ) 2 of 𝑢𝜏 𝑏 + 𝑣𝜏 -𝑐 = 𝑢𝜎 𝑏 + 𝑣𝜎 -𝑐 , (7.11) 𝑢𝜏 -𝑏 + 𝑣𝜏 𝑐 = 𝑢𝜎 -𝑏 + 𝑣𝜎 𝑐 , (7.12) 𝜏 ≠ 𝜎 (7.13) satisfy |𝜏| = |𝜎| = 1. Let us rewrite (7.11)-(7.12) as 𝑢(𝜏 𝑏 -𝜎 𝑏 ) -𝑣(𝜏𝜎) -𝑐 (𝜏 𝑐 -𝜎 𝑐 ) = 0, -𝑢(𝜏𝜎) -𝑏 (𝜏 𝑏 -𝜎 𝑏 ) + 𝑣(𝜏 𝑐 -𝜎 𝑐 ) = 0.
The condition gcd(𝑏, 𝑐) = 1 implies that at least one of 𝜏 𝑏 -𝜎 𝑏 and 𝜏 𝑐 -𝜎 𝑐 is nonzero (or else 𝜏 = 𝜎). Consequently
meaning that 𝜔 = 𝜏𝜎 must be a root of unity: 𝜔 𝑏+𝑐 = 1. With respect to 𝜏 and 𝜔, the conditions (7.11)-(7.13) become:
𝑢(𝜏 𝑏 -𝜔 𝑏 𝜏 -𝑏 ) + 𝑣(𝜏 -𝑐 -𝜔 -𝑐 𝜏 𝑐 ) = 0, (7.15) 𝜏 2 ≠ 𝜔. (7.16) Let 𝜆 be a square root of 𝜔, i.e., 𝜆 2 = 𝜔. Then (7.16) states that 𝜏 ∉ {𝜆, -𝜆}.
For each of the 𝑏 + 𝑐 possible roots of unity 𝜔, (7.15) is an algebraic equation of degree 2𝑏 in 𝜏. We claim that if 𝑢, 𝑣 ∈ ℝ * are suitably chosen, then all 2𝑏 solutions of this equation lie on 𝑆 1 = {|𝜏| = 1}. It is easy to see that this set of solutions contains the two values 𝜏 = ±𝜆 which we need to exclude, leaving us with 2(𝑏 -1) solutions for each 𝜔 satisfying (7.14). We also claim that all these 2(𝑏 + 𝑐)(𝑏 -1) solutions are distinct, thereby yielding (𝑏 + 𝑐)(𝑏 -1) hyperbolic nodes of the curve, as desired.
Let us establish these claims. Denote 𝜀 = 𝜆 𝑏+𝑐 ∈ {-1, 1}. Replacing 𝜏 by 𝜌 = 𝜏𝜆 -1 (thus 𝜏 = 𝜆𝜌), we transform (7.15) into 𝑢𝜀(𝜌 𝑏 -𝜌 -𝑏 ) = 𝑣(𝜌 𝑐 -𝜌 -𝑐 ).
Making the substitution 𝜌 = 𝑒 𝑖𝛼 , we translate this into (7.17)
𝑢𝜀 sin(𝑏𝛼) = 𝑣 sin(𝑐𝛼).
If | 𝑣 ᵆ | is sufficiently small, then equation (7.17) clearly has 2𝑏 distinct real solutions in the interval [0, 2𝜋), as claimed. Finally, it is not hard to see that all resulting 2(𝑏+𝑐)(𝑏-1) values of 𝜏 are distinct. □
We return to the general treatment of (potentially reducible) expressive 𝐿 ∞ -regular curves. First, a generalization of Proposition 7.6: Proposition 7.16. Let 𝐶 be a reduced real plane curve such that
• each component of 𝐶 is real, and either polynomial or trigonometric;
• the singular set of 𝐶 in the affine plane 𝔸 2 consists solely of hyperbolic nodes;
• the set of real points of 𝐶 in the affine plane is connected.
Then 𝐶 is expressive and 𝐿 ∞ -regular.
Proof. The proof utilizes the approach used in the proof of Proposition 7.6. Arguing exactly as at the beginning of the latter proof, we conclude that all we need to show is that the polynomial 𝐹 defining 𝐶 has finitely many critical points in the affine plane 𝔸 2 . Once again, we argue by contradiction. Assuming that 𝑍(𝐹 𝑥 ) ∩ 𝑍(𝐹 𝑦 ) contains a real curve 𝐵 of a positive degree 𝑑 ′ < 𝑑 = deg(𝐶), and reasoning as in the earlier proof, we conclude that for any component 𝐶 ′ of 𝐶, there exists a curve B 𝐶 ′ ∈ Span{𝐵, (𝐿 ∞ ) 𝑑 ′ } containing 𝐶 ′ as a component. In view of the connectedness of 𝐶 ℝ and the fact that different members of the pencil Span{𝐵, (𝐿 ∞ ) 𝑑 ′ } are disjoint from each other in the affine plane 𝔸 2 , we establish that there is just one curve B ∈ Span{𝐵, (𝐿 ∞ ) 𝑑 ′ } that contains all the components of 𝐶-but this contradicts the inequality 𝑑 ′ < 𝑑. □ Theorem 7.17 is the main result of this paper.
Theorem 7.17. Let 𝐶 be a reduced real plane algebraic curve, with all irreducible components real. The following are equivalent:
• 𝐶 is expressive and 𝐿 ∞ -regular;
• each irreducible component of 𝐶 is either trigonometric or polynomial, all singular points of 𝐶 in the affine plane 𝔸 2 are hyperbolic nodes, and the set of real points of 𝐶 in the affine plane is connected.
Proof. Follows from Propositions 7.3 and 7.16. □ Corollary 7.18. Let 𝐶 be an 𝐿 ∞ -regular expressive plane curve whose irreducible components are all real. Let 𝐶 ′ be a subcurve of 𝐶, i.e., a union of a subset of irreducible components. If the set of real points of 𝐶 ′ in the affine plane is connected, then 𝐶 ′ is also 𝐿 ∞ -regular and expressive.
Proof. Follows from Theorem 7.17. □
We conclude this section by a corollary whose statement is entirely elementary, and in particular does not involve the notion of expressivity.
Proof. Immediate from Proposition 7.6 (or Theorem 7.10). □
We first discuss the irreducible case. By Theorem 7.10, an irreducible plane curve is expressive and 𝐿 ∞ -regular if and only if it is either trigonometric or polynomial, and all its singular points outside 𝐿 ∞ are real hyperbolic nodes. The last condition is usually the trickiest to verify.
One simple case is when the number of hyperbolic nodes attains its maximum:
Corollary 8.1. Let 𝐶 be a real polynomial or trigonometric curve of degree 𝑑 with (𝑑-1)(𝑑-2) 2 hyperbolic nodes. Then 𝐶 is expressive and 𝐿 ∞ -regular.
Proof. In view of Hironaka's formula (3.7), the curve 𝐶 has no other singular points besides the given hyperbolic nodes. The claim follows by Proposition 7.6. □
Examples illustrating Corollary 8.1 include Lissajous-Chebyshev curves (5.2) with parameters (𝑑, 𝑑 -1) as well as hypotrochoids with parameters (𝑘, 𝑘 -1) (cf. (4.12), with suitably chosen value of 𝑎). Remark 8.2. In Corollary 8.1, the requirement that the curve 𝐶 is polynomial or trigonometric cannot be dropped. For example, there exists an irreducible real quadric with three hyperbolic nodes which is not expressive. Corollary 8.1 can be generalized as follows.
Corollary 8.3. Let 𝐶 = 𝑉(𝐺(𝑥, 𝑦)) be a real polynomial or trigonometric curve with 𝜈 hyperbolic nodes. Suppose that the Newton polygon of 𝐺(𝑥, 𝑦) has 𝜈 interior integer points. Then 𝐶 is expressive and 𝐿 ∞ -regular.
Proof. It is well known (see [9] or [17,Section 4.4]) that the maximal possible number of nodes of an irreducible plane curve with a given Newton polygon (equivalently, the arithmetic genus of a curve in the linear system spanned by the monomials in the Newton polygon, on the associated toric surface) equals the number of interior integer points in the Newton polygon. The claim then follows by Proposition 7.6. □
It is natural to seek an algorithm for verifying whether a given immersed real polynomial or trigonometric curve 𝐶, say one given by an explicit parametrization, is expressive. By Theorem 7.10, this amounts to checking that each point of self-intersection of 𝐶 in the affine plane 𝔸 2 corresponds to two real values of the parameter. In the case of a polynomial curve 𝑡 ↦ (𝑃(𝑡), 𝑄(𝑡)), this translates into requiring that
• the resultant (with respect to either variable 𝑠 or 𝑡) of the polynomials In this case,
The resultant of P and Q (which we computed using Sage) is equal to -256(2𝑠 2 -3)(2𝑠 2 -2𝑠 -1)(2𝑠 2 + 2𝑠 -1)(8𝑠 6 -24𝑠 4 -4𝑠 3 + 18𝑠 2 + 6𝑠 -1).
All its 12 roots are real, so the curve is expressive, with 6 nodes. See Figure 15. In this case,
The resultant of P and Q (which we computed using Sage) is equal to -256(2s 2 -3)(2s 2 -2s -1)(2s 2 + 2s -1)(8s 6 -24s 4 -4s 3 + 18s 2 + 6s -1).
All its 12 roots are real, so the curve is expressive, with 6 nodes. See Figure 15.
Figure 15: The curve
The case of a trigonometric curve can be treated in a similar way. Let C be a trigonometric curve with a Laurent parametrization t → (x(t), y(t)) = (P (t) + P (t -1 ),
We write down the differences
clear the denominators by multiplying by an appropriate power of ts, and require that all values of t and s for which the resulting polynomials vanish (equivalently, all roots of their resultants) lie on the unit circle.
Proof. By Bézout's theorem, the right-hand side of (8.1) is the number of critical points of G in the affine plane, counted with multiplicities. Since each region of D G contains at least one (real) critical point, and each node of C R is a critical point, the only way for (8.1) to hold is for C to be expressive. 2 ) 2 )
The case of a trigonometric curve can be treated in a similar way. Let 𝐶 be a trigonometric curve with a Laurent parametrization clear the denominators by multiplying by an appropriate power of 𝑡𝑠, and require that all values of 𝑡 and 𝑠 for which the resulting polynomials vanish (equivalently, all roots of their resultants) lie on the unit circle. Proposition 8.5. Let 𝐶 = 𝑉(𝐺) be an 𝐿 ∞ -regular curve, with 𝐺(𝑥, 𝑦) ∈ ℝ[𝑥, 𝑦]. Assume that 𝐶 ℝ is connected, and each singular point of it is a hyperbolic node, as in Definition 6.3. Let 𝜈 denote the number of such nodes, and let 𝜌 be the number of regions of the divide 𝐷 𝐺 . Then 𝐶 is expressive if and only if
Proof. By Bézout's theorem, the right-hand side of (8.1) is the number of critical points of 𝐺 in the affine plane, counted with multiplicities. Since each region of 𝐷 𝐺 contains at least one (real) critical point, and each node of 𝐶 ℝ is a critical point, the only way for (8.1) to hold is for 𝐶 to be expressive. □
In this section, we describe several transformations which can be used to construct new examples of expressive curves from existing ones. The simplest transformation of this kind is the "bending" procedure based on the following observation: If, in addition, 𝐶 is 𝐿 ∞ -regular, with real components, then so is C.
Proposition 9.1 is illustrated in Figure 16. Proof. The automorphism (9.1) is an invertible change of variables that restricts to a diffeomorphism of the real plane ℝ 2 . As such, it sends (real) critical points to (real) critical points, does not change the divide of the curve, and preserves expressivity.
Let us now assume that 𝐶 is expressive and 𝐿 ∞ -regular, with real components. In view of Theorem 7.17, all we need to show is that all components of C are polynomial or trigonometric. Geometrically, a polynomial (resp., trigonometric) component is a Riemann sphere punctured at one real point (resp., two complex conjugate points) and equivariantly immersed into the plane. This property is preserved under real biregular automorphisms of 𝔸 2 . □ Remark 9.2. It is well known [34] that the group of automorphisms of the affine plane is generated by affine transformations together with the transformations of the form (𝑥, 𝑦) ↦ (𝑥, 𝑦 + 𝑃(𝑥)), for 𝑃 a polynomial. This holds over any field of characteristic zero, in particular over the reals.
Example 9.3. Several examples of bending can be extracted from Table 3. Applying the automorphism (𝑥, 𝑦) ↦ (𝑥, 𝑦 + 𝑥 -𝑥 𝑚 ) to the line 𝑉(𝑥 -𝑦), we get 𝑉(𝑥 𝑚 -𝑦).
The automorphism (𝑥, 𝑦) ↦ (𝑥, 𝑦 -𝑥 2 ) transforms the parabola 𝑉(𝑦 2 -𝑥) into 𝑉((𝑦 -𝑥 2 ) 2 -𝑥), the ellipse 𝑉(𝑥 2 + 𝑦 2 -1) into 𝑉(𝑥 2 + (𝑦 -𝑥 2 ) 2 -1), and the nodal cubic 𝑉(𝑦 2 -𝑥𝑦-𝑥 3 ) into 𝑉((𝑦-𝑥 2 ) 2 -𝑥𝑦). (In turn, 𝑉(𝑦 2 -𝑥𝑦-𝑥 3 ) and 𝑉(4𝑥 3 -3𝑥+2𝑦 2 -1) are related to each other by an affine transformation.)
Example 9.4. The polynomial expressive curves shown in Figure 17 (in red) are obtained by bending a parabola and a nodal cubic.
We next discuss the "doubling" construction which transforms a plane curve 𝐶 = 𝑉(𝐺(𝑥, 𝑦)) into a new curve C = 𝑉(𝐺(𝑥, 𝑦 2 )). Proposition 9.5 shows that, under certain conditions, this procedure preserves expressivity. See Figure 17. Proposition 9.5. Let 𝐶 be an expressive 𝐿 ∞ -regular curve whose components are all real (hence polynomial or trigonometric, see Proposition 7.3). Suppose that • each component 𝐵 = 𝑉(𝐺(𝑥, 𝑦)) of 𝐶, say with deg 𝑥 (𝐺) = 𝑑, intersects 𝑍(𝑦) in 𝑑 real points (counting multiplicities), all of which are smooth points of 𝐶; moreover, ∘ if 𝐵 is trigonometric, then it intersects 𝑍(𝑦) at 𝑑 2 points of quadratic tangency; ∘ if 𝐵 is polynomial, then it intersects 𝑍(𝑦) at 𝑑 2 , 𝑑-1 2 or 𝑑-2 2 points of quadratic tangency, with 0, 1, or 2 points of transverse intersection, respectively; • all nodes of 𝐶 lie in the real half-plane {𝑦 > 0}; • at each point of quadratic tangency between 𝐶 and 𝑍(𝑦), the local real branch of 𝐶 lies in the upper half-plane {𝑦 > 0}; • each connected component of the set {(𝑥, 𝑦) ∈ 𝐶 ∩ ℝ 2 | 𝑦 < 0} is unbounded. Then the curve C = 𝑉(𝐺(𝑥, 𝑦 2 )) is expressive and 𝐿 ∞ -regular.
Proof. The curve C is nodal by construction. It is not hard to see that the set Cℝ is connected, and all the nodes of C are hyperbolic. (See the proof of Proposition 9.8 for a more involved version of the argument.) In view of Theorem 7.17, it remains to show that all components of C are real polynomial or trigonometric.
We only treat the trigonometric case, since the argument in the polynomial case is similar. (Cf. also the proof of Lemma 9.7, utilizing such an argument in a more complicated context.) The natural map C → 𝐶 lifts to a two-sheeted ramified covering 𝜌 ∶ C∨ → 𝐶 ∨ between respective normalizations. The restriction of 𝜌 to C∨ ⧵ 𝜌 -1 (𝑍(𝑦)) is an unramified two-sheeted covering, and each component of C contains a one-dimensional fragment of the real point set, hence is real. In fact, 𝜌 is not ramified at all, since each point in 𝐶 ∩ 𝑍(𝑦) lifts to a node, and hence to two preimages in C∨ . Since 𝐶 ∨ = ℂ * , it follows that C∨ is a union of at most two disjoint copies of ℂ * . We conclude that C consists of one or two trigonometric components. □ If a curve 𝐶 = 𝑉(𝐺(𝑥, 𝑦)) satisfies the conditions in Proposition 9.5 with respect to each of the coordinate axes 𝑍(𝑥) and 𝑍(𝑦), with all points in 𝐶 ∩ 𝑍(𝑥) (resp., 𝐶 ∩ 𝑍(𝑦)) located on the positive ray {𝑥 = 0, 𝑦 > 0} (resp., {𝑦 = 0, 𝑥 > 0}), the one can apply the doubling transformation twice, obtaining an expressive curve C = 𝑉(𝐺(𝑥 2 , 𝑦 2 )). A couple of examples are shown in Figure 18.
ρ : C ∨ → C ∨ between respective normalizations. The restriction of ρ to C ∨ \ρ -1 (Z(y)) is an unramified two-sheeted covering, and each component of C contains a onedimensional fragment of the real point set, hence is real. In fact, ρ is not ramified at all, since each point in C ∩ Z(y) lifts to a node, and hence to two preimages in C ∨ . Since C ∨ = C * , it follows that C ∨ is a union of at most two disjoint copies of C * . We conclude that C consists of one or two trigonometric components.
If a curve C = V (G(x, y)) satisfies the conditions in Proposition 9.5 with respect to each of the coordinate axes Z(x) and Z(y), with all points in C ∩Z(x) (resp., C ∩Z(y)) located on the positive ray {x = 0, y > 0} (resp., {y = 0, x > 0}), the one can apply the doubling transformation twice, obtaining an expressive curve C = V (G(x 2 , y 2 )). A couple of examples are shown in Figure 18. The remainder of this section is devoted to the discussion of "unfolding." This is a transformation of algebraic curves that utilizes the coordinate change (9.2) (x, y) = (x, T m (u)). (As before, T m denotes the mth Chebyshev polynomial of the first kind, see (4.1).) A precise description of unfolding is given in Proposition 9.8 below. To get a general idea of how this construction works, take a look at the examples in Figures 19-20. As these examples illustrate, the bulk of the unfolded curve (viewed up to an isotopy of the real plane) is obtained by stitching together m copies of the input curve C, or more precisely the portion of C contained in the strip {-1 ≤ y ≤ 1}. Lemma 9.6. Let C = V (F (x, y)) be a trigonometric curve. Suppose that • the strip {-1 < y < 1} contains a one-dimensional fragment of C R ; 1 Jun 2022 08:44:25 PDT 201027-Fomin Version 2 -Submitted to Commun. Amer. Math. Soc.
ρ : C ∨ → C ∨ between respective normalizations. The restriction of ρ to C ∨ \ρ -1 (Z(y)) is an unramified two-sheeted covering, and each component of C contains a onedimensional fragment of the real point set, hence is real. In fact, ρ is not ramified at all, since each point in C ∩ Z(y) lifts to a node, and hence to two preimages in C ∨ . Since C ∨ = C * , it follows that C ∨ is a union of at most two disjoint copies of C * . We conclude that C consists of one or two trigonometric components.
If a curve C = V (G(x, y)) satisfies the conditions in Proposition 9.5 with respect to each of the coordinate axes Z(x) and Z(y), with all points in C ∩Z(x) (resp., C ∩Z(y)) located on the positive ray {x = 0, y > 0} (resp., {y = 0, x > 0}), the one can apply the doubling transformation twice, obtaining an expressive curve C = V (G(x 2 , y 2 )). A couple of examples are shown in Figure 18. The remainder of this section is devoted to the discussion of "unfolding." This is a transformation of algebraic curves that utilizes the coordinate change (9.2) (x, y) = (x, T m (u)). (As before, T m denotes the mth Chebyshev polynomial of the first kind, see (4.1).) A precise description of unfolding is given in Proposition 9.8 below. To get a general idea of how this construction works, take a look at the examples in Figures 19-20. As these examples illustrate, the bulk of the unfolded curve (viewed up to an isotopy of the real plane) is obtained by stitching together m copies of the input curve C, or more precisely the portion of C contained in the strip {-1 ≤ y ≤ 1}. Lemma 9.6. Let C = V (F (x, y)) be a trigonometric curve. Suppose that • the strip {-1 < y < 1} contains a one-dimensional fragment of C R ; 1 Jun 2022 08:44:25 PDT 201027-Fomin Version 2 -Submitted to Commun. Amer. Math. Soc. • all of these points are smooth points of quadratic tangency between 𝐶 and 𝑍(𝑦 2 -1).
Then for any 𝑚 ∈ ℤ >0 , the curve 𝐶 (𝑚) defined by
is a union of trigonometric curves.
Proof. The natural map 𝐶 (𝑚) → 𝐶 given by (9.2) lifts to the 𝑚-sheeted ramified covering map 𝜌 ∶ 𝐶 ∨ (𝑚) → 𝐶 ∨ between the normalizations. The restriction 𝜌 ∶ 𝐶 ∨ (𝑚) ⧵ 𝜌 -1 (𝑍(𝑦 2 -1)) → 𝐶 ∨ ⧵ 𝑍(𝑦 2 -1) • C intersects each of the lines y = ±1 in d/2 points; • all of these points are smooth points of quadratic tangency between C and Z(y 2 -1).
Then for any m ∈ Z >0 , the curve C (m) defined by
is a union of trigonometric curves.
Proof. The natural map C (m) → C given by (9.2) lifts to the m-sheeted ramified covering map ρ : C ∨ (m) → C ∨ between the normalizations. The restriction is an unramified 𝑚-sheeted covering, and each of the components of 𝐶 (𝑚) contains a one-dimensional fragment of the real point set, hence is real. Let us show that 𝜌 is not ramified at all. If 𝑝 ∈ 𝐶 ∩ 𝑍(𝑦 -1) and 𝑇 -1 𝑚 (1) consists of 𝑎 simple roots and 𝑏 double roots, then 𝑝 lifts to 𝑎 smooth points (where 𝐶 (𝑚) is quadratically tangent to the lines 𝑢 = 𝜆 with 𝜆 running over all these simple roots) and 𝑏 nodes, totaling 𝑎 + 2𝑏 = 𝑚 preimages in 𝐶 ∨ (𝑚) . As 𝐶 is trigonometric, 𝐶 ∨ = ℂ * . Since a cylinder can only be covered by a cylinder, 𝐶 ∨ (𝑚) is a union of disjoint copies of ℂ * (not necessarily 𝑚 of them), so 𝐶 (𝑚) is a union of trigonometric components. □ Lemma 9.7. Let 𝐶 = 𝑉(𝐹(𝑥, 𝑦)) be a real polynomial curve. Let 𝑑 = deg 𝑥 𝐹(𝑥, 𝑦). Suppose that • the strip {-1 < 𝑦 < 1} contains a one-dimensional fragment of 𝐶 ℝ ; • 𝐶 intersects 𝑍(𝑦 2 -1) in 2𝑑 points (counting multiplicities), all of which are smooth points of 𝐶; • these points include 𝑑 -1 quadratic tangencies and two transverse intersections. Then for any 𝑚 ∈ ℝ >0 , the curve 𝐶 (𝑚) defined by (9.3) is a union of polynomial or trigonometric components.
Proof. It is not hard to see that 𝐶 has a polynomial parametrization 𝑡 ↦ (𝑃(𝑡), 𝑄(𝑡)) with 𝑄(𝑡) = ±𝑇 𝑑 (𝑡). This follows from "Chebyshev's equioscillation theorem" of approximation theory (due to E. Borel and A. Markov, see, e.g., [31,Section 1.1] or [28,Theorem 3.4]). In the rest of the proof, we assume that 𝑄(𝑡) = 𝑇 𝑑 (𝑡), as the case 𝑄(𝑡) = -𝑇 𝑑 (𝑡) is completely similar.
We note that one can slightly vary the coefficients of 𝑃 while keeping the intersection properties of 𝐶 with 𝑍(𝑦 2 -1) (and maintaining expressivity if 𝐶 has this property). Thus, we can assume that 𝑃(𝑡) is a generic polynomial (in particular, with respect to 𝑄(𝑡) = 𝑇 𝑑 (𝑡)). One can sort out which of these components are distinct by taking into account that 𝑥 is invariant with respect to the substitutions 𝜃 ↦ -𝜃 and 𝜃 ↦ 𝜃 + 2𝜋 𝑗 𝑟 . Finally, we note that the curve 𝐶 (𝑚) has no multiple components. To see that, take a line 𝑥 = 𝑥 0 such that the polynomial 𝐹(𝑥 0 , 𝑦) has only simple roots 𝑦 1 , . . . , 𝑦 𝑟 which differ from ±1. (Such a line 𝑥 = 𝑥 0 does exist, since otherwise 𝐶 would have multiple components or contain one of the lines 𝑦 = ±1.) Then the polynomial 𝐹(𝑥 0 , 𝑇 𝑚 (𝑢)) has only simple roots which all come from the equations 𝑇 𝑚 (𝑢) = 𝑦 𝑖 , 𝑖 = 1, . . . , 𝑟. □ Proposition 9.8. Let 𝐶 = 𝑉(𝐺(𝑥, 𝑦)) be an expressive 𝐿 ∞ -regular curve all of whose components are real (hence polynomial or trigonometric, cf. Proposition 7.3). Suppose that
• each component 𝑉(𝐹(𝑥, 𝑦)) of 𝐶, say with deg 𝑥 (𝐹) = 𝑑, intersects 𝑍(𝑦 2 -1) in 2𝑑 real points (counting multiplicities), all of which are smooth points of 𝐶; • moreover, these intersections occur in one of the following two ways:
• all nodes of 𝐶 lie in the strip {-1 < 𝑦 < 1}; • at each point of quadratic tangency between 𝐶 and 𝑍(𝑦 2 -1), the local real branch of 𝐶 lies in the strip {-1 ≤ 𝑦 ≤ 1}. Then the curve 𝐶 (𝑚) = 𝑉(𝐺(𝑥, 𝑇 𝑚 (𝑦))) is expressive and 𝐿 ∞ -regular.
Proof. By construction, the curve 𝐶 (𝑚) is nodal. By Lemmas 9.6 and 9.7, the components of 𝐶 (𝑚) are real polynomial or trigonometric. In view of Theorem 7.17, it remains to show that the set 𝐶 (𝑚),ℝ is connected, and that all the nodes of 𝐶 (𝑚) are hyperbolic.
The set of real points 𝐶 ℝ is connected by Definition 5.1. Since all the nodes of 𝐶 lie inside the strip {-1 < 𝑦 < 1}, it follows that the set 𝐶 ℝ ∩ {-1 ≤ 𝑦 ≤ 1} is connected.
Let {𝑎 1 < ⋯ < 𝑎 𝑚+1 } be the set of roots of the equation 𝑇 2 𝑚 (𝑎) = 1. Then each set (9.5)
onto the strip {𝑎 𝑗 ≤ 𝑦 ≤ 𝑎 𝑗+1 }. Furthermore, for each 𝑗 = 2, . . . , 𝑚, the sets
are attached to each other at their common points along the line 𝑍(𝑦 -𝑎 𝑗 ). (This set of common points is nonempty since it includes the images of the intersection of 𝐶 ℝ with one of the two lines 𝑍(𝑦 ± 1).) To obtain the entire set 𝐶 (𝑚),ℝ , we attach to the (connected) union of the 𝑚 sets (9.5) the diffeomorphic images of the intervals forming the set 𝐶 ℝ ⧵ {-1 ≤ 𝑦 ≤ 1} (if any). We conclude that 𝐶 (𝑚),ℝ is connected.
Regarding the nodes of 𝐶 (𝑚) , we observe that they come in two flavours. First, as one of the 𝑚 preimages of a node of 𝐶 contained in the strip {-1 < 𝑦 < 1}; all these preimages are real, hence hyperbolic. Second, as a preimage of a tangency point between 𝐶 and 𝑍(𝑦 2 -1); this again yields a hyperbolic node. □
We next discuss ways of putting together several expressive curves to create a new (reducible) expressive curve. Our key tool is Corollary 10.1.
Corollary 10.1. Let 𝐶 1 , . . . , 𝐶 𝑘 be expressive and 𝐿 ∞ -regular plane curves such that
• each pair 𝐶 𝑖 and 𝐶 𝑗 intersect each other in 𝔸 2 at (distinct) hyperbolic nodes, and
Then 𝐶 is expressive and 𝐿 ∞ -regular.
Proof. Follows from Theorem 7.17. □
One easy consequence of Corollary 10.1 is the following construction. 1 k
polynomial f i (x) -f j (x) has real roots, and all such roots (over all pairs {i, j}) are pairwise distinct. Then the curve V ( i (y -f i (x))) is expressive and L ∞ -regular.
Example 10.3 (Line arrangements). A connected arrangement of distinct real lines in the plane forms an expressive and L ∞ -regular curve, as long as no three lines intersect at a point. Parallel lines are allowed. See Figure 21. Example 10.4 (Arrangements of parabolas). Let C be a union of distinct real parabolas in the affine plane. Then C is an expressive and L ∞ -regular curve provided • the set of real points of C is connected;
• no three parabolas intersect at a point;
• all intersections between parabolas are transverse;
• for each pair of parabolas P 1 and P 2 , one of the following options holds:
-P 1 and P 2 differ by a shift of the plane, or -P 1 and P 2 have parallel (or identical) axes, and intersect at 2 points, or -P 1 and P 2 intersect at 4 points. See Figures 22 and 23 Example 10.4 (Arrangements of parabolas). Let 𝐶 be a union of distinct real parabolas in the affine plane. Then 𝐶 is an expressive and 𝐿 ∞ -regular curve provided
• the set of real points of 𝐶 is connected;
• no three parabolas intersect at a point; • all intersections between parabolas are transverse;
• for each pair of parabolas 𝑃 1 and 𝑃 2 , one of the following options holds:
-𝑃 1 and 𝑃 2 differ by a shift of the plane, or -𝑃 1 and 𝑃 2 have parallel (or identical) axes, and intersect at 2 points, or -𝑃 1 and 𝑃 2 intersect at 4 points.
See Figures 22 and 23. Examples 10.3 and 10.4 have a common generalization: Example 10.5 (Arrangements of lines and parabolas).
Let 𝐶 be a union of distinct real lines and parabolas in the affine plane. Then 𝐶 is an expressive and 𝐿 ∞ -regular curve provided
• the set of real points of 𝐶 is connected;
• no three of these curves intersect at a point; • all intersections between these lines and parabolas are transverse;
• each line intersects every parabola at one or two points;
• each pair of parabolas intersects in one of the ways listed in Example 10.4.
Another elegant application of Corollary 10.1 involves arrangements of circles:
Example 10.6 (Circle arrangements). Let {𝐶 𝑖 } be a collection of circles on the real affine plane such that each pair of circles intersects transversally at two real points, with no triple intersections. Then the curve ⋃ 𝐶 𝑖 is expressive and 𝐿 ∞ -regular. See Figure 24. Here is a common generalization of Examples 10.3 and 10.6:
Example 10.7 (Arrangements of lines and circles). Let {C i } be a collection of lines and circles on the real affine plane such that each circle intersects every line (resp., every other circle) transversally at two points, with no triple intersections. Then the curve C i is expressive and L ∞ -regular. See Figure 25. 11. Shifts and dilations
In this section, we obtain lower bounds for an intersection multiplicity (at a point 𝑝 ∈ 𝐿 ∞ ) between a plane curve 𝐶 and another curve obtained from 𝐶 by a shift or dilation. In Sections 12-13, we will use these estimates to derive expressivity criteria for unions of polynomial or trigonometric curves.
Without loss of generality, we assume that 𝑝 = (1, 0, 0) throughout this section.
To state our bounds, we will need some notation involving Newton diagrams: Assume that 𝑝 = (1, 0, 0) ∈ 𝐶 ∩ 𝐿 ∞ . Then
Proof. In the coordinates (𝑦, 𝑧), the dilation 𝐶 ; 𝐶 𝑐 can be regarded as a transition from the polynomial 𝐹 1 (see (11.1)) to the polynomial 𝐹 𝑐 (𝑦, 𝑧) = 𝐹(𝑐, 𝑐𝑦, 𝑧). The polynomials 𝐹 1 and 𝐹 𝑐 have the same Newton diagram at 𝑝 (resp., Newton polygon). Furthermore, let 𝐺(𝑦, 𝑧) be a polynomial with the same Newton polygon and with generic coefficients. By the lower semicontinuity of the intersection number, we have We denote by 𝐹 𝐸 (𝑦, 𝑧) = 𝐹 𝐸(𝑄) (𝑦, 𝑧) the truncation of 𝐹 1 (see 11.1) along the edge 𝐸 = 𝐸(𝑄). The polynomial 𝐹 𝐸 is quasihomogeneous with respect to the weighting of 𝑦 and 𝑧 by 𝑛 𝑦 and 𝑛 𝑧 , respectively. We denote
𝜂(𝑄) = multiplicity of (𝑧 𝑛 𝑦 -𝜌𝑦 𝑛 𝑧 ) as a factor of 𝐹 𝐸 (𝑦, 𝑧). (11.9) It is not hard to see that 𝜌(𝑄) is well defined, and that 𝜂(𝑄) ≥ 1.
Proposition 11.4. Let 𝐶 = 𝑍(𝐹(𝑥, 𝑦, 𝑧)) be a projective curve not containing the line 𝑍(𝑧) = 𝐿 ∞ as a component. Assume that 𝑝 = (1, 0, 0) ∈ 𝐶 ∩ 𝐿 ∞ , and that 𝐶 is not tangent to the line 𝑍(𝑦) at 𝑝. Let 𝑎, 𝑏 ∈ ℂ, and let 𝐶 𝑎,𝑏 denote the shifted curve (11.10) 𝐶 𝑎,𝑏 = 𝑍(𝐹(𝑥 + 𝑎𝑧, 𝑦 + 𝑏𝑧, 𝑧)).
Then
where the sum is over all local branches 𝑄 of 𝐶 at 𝑝 which are tangent to 𝐿 ∞ .
(For the definitions of 𝑆 -(Γ(𝐶, 𝑝)), 𝑟(𝑄) and 𝜂(𝑄), see Definition 11.1, (11.5)-(11.7) and (11.8)-(11.9), respectively.)
Proof. Since the intersection multiplicity is lower semicontinuous, we may assume that 𝑎 and 𝑏 are generic complex numbers. Let us consider the family of curves {𝐶 𝜆𝑎,𝜆𝑏 } 0≤𝜆≤1 interpolating between 𝐶 = 𝐶 0,0 and 𝐶 𝑎,𝑏 . Being equisingular at the point 𝑝, this deformation preserves the number of local branches at 𝑝 as well as their topological characteristics. It therefore descends to families of individual local branches at 𝑝, yielding an equisingular bijection 𝒬(𝐶, 𝑝) ⟶ 𝒬(𝐶 𝑎,𝑏 , 𝑝), 𝑄 ⟼ 𝑄 𝑎,𝑏 .
Since the fixed point set of the shift (𝑥, 𝑦, 𝑧) ↦ (𝑥 + 𝑎𝑧, 𝑦 + 𝑏𝑧, 𝑧) is the line 𝐿 ∞ , this bijection restricts to bijections 𝒬 0 (𝐶, 𝑝) → 𝒬 0 (𝐶 𝑎,𝑏 , 𝑝) and 𝒬 1 (𝐶, 𝑝) → 𝒬 1 (𝐶 𝑎,𝑏 , 𝑝).
To obtain the desired lower bound on (𝐶 ⋅ 𝐶 𝑎,𝑏 ) 𝑝 , we will exploit the decomposition (11.12)
where we use the notation
(𝑄 ⋅ 𝑄 ′ 𝑎,𝑏 ) 𝑝 , for 𝜀, 𝛿 ∈ {0, 1}.
Lemma 11.5. Suppose 𝒬 1 (𝐶, 𝑝) ≠ ∅. Then (11.13)
where 𝑆 -(𝐸 (1,1) ) denotes the area of the trapezoid bounded by the edge 𝐸 (1,1) , the coordinate axes, and the vertical line through the rightmost endpoint of 𝐸 (1,1) .
Proof. For 𝑄 ∈ 𝒬 1 (𝐶, 𝑝), the tangent lines to 𝑄 and 𝑄 𝑎,𝑏 differ from each other. The foregoing discussion implies that, for any 𝑄 ∈ 𝒬 1 (𝐶, 𝑝) and 𝑄 ′ ∈ 𝒬(𝐶, 𝑝), we have (𝑄 ⋅ 𝑄 ′ 𝑎,𝑏 ) 𝑝 = (𝑄 𝑎,𝑏 ⋅ 𝑄 ′ ) 𝑝 = mult 𝑄 ⋅ mult 𝑄 ′ . We then observe that ∑ 𝑄∈𝒬 0 (𝐶,𝑝) mult 𝑄 = ℓ 0 ≝ length of the projection of Γ 0 (𝐶, 𝑝) to the vertical axis, ∑ 𝑄∈𝒬 1 (𝐶,𝑝) mult 𝑄 = ℓ 1 ≝ length of the projection of 𝐸 (1,1) to either of the axes, implying
In light of (11.12) and (11.13), it remains to obtain the desired lower bound for the summand Σ 00 . To simplify notation, we will pretend, for the time being, that all local branches are tangent to 𝐿 ∞ , so that 𝒬(𝐶, 𝑝) = 𝒬 0 (𝐶, 𝑝) and Γ(𝐶, 𝑝) = Γ 0 (𝐶, 𝑝).
Let 𝒬(𝐸) denote the set of local branches of 𝐶 at 𝑝 associated with an edge 𝐸 of Γ(𝐶, 𝑝). Equivalently, 𝒬(𝐸) = {𝑄 ∈ 𝒬(𝐶, 𝑝) | 𝐸(𝑄) = 𝐸}. Lemma 11.6. Let 𝐸 1 and 𝐸 2 be two distinct edges of the Newton diagram Γ(𝐶, 𝑝). Assume that 𝐸 2 is located above and to the left of 𝐸 1 , so that
1 , 𝑗 1 ), (𝑖 ′ 1 , 𝑗 ′ 1 )], 𝐸 2 = [(𝑖 2 , 𝑗 2 ), (𝑖 ′ 2 , 𝑗 ′ 2 )], 𝑖 ′ 2 < 𝑖 2 ≤ 𝑖 ′ 1 < 𝑖 1 , 𝑗 1 < 𝑗 ′ 1 ≤ 𝑗 2 < 𝑗 ′ 2 . Then (11.14) ∑ 𝑄∈𝒬(𝐸 1 ) ∑ 𝑄 ′ ∈𝒬(𝐸 2 ) (𝑄 ⋅ 𝑄 ′ 𝑎,𝑏 ) 𝑝 = ∑ 𝑄∈𝒬(𝐸 1 ) ∑ 𝑄 ′ ∈𝒬(𝐸 2 )
(𝑄 𝑎,𝑏 ⋅ 𝑄 ′ ) 𝑝 = (𝑗 ′ 1 -𝑗 1 )(𝑖 2 -𝑖 ′ 2 ).
Note that (𝑗 ′ 1 -𝑗 1 )(𝑖 2 -𝑖 ′ 2 ) is the area of the rectangle formed by the intersections of the horizontal lines passing through 𝐸 1 with the vertical lines passing through 𝐸 2 . See Figure 27. We provide two proofs of Lemma 11.6. The first proof, based on the Bernstein-Kouchnirenko mixed volume formula, relies on a genericity assumption whose justification we do not provide. (Note however that this assumption is not needed for the proof of the "≥" inequality in (11.14). This inequality will be sufficient for the upcoming proof of Proposition 11.4.) The second proof is rigorous but more technical.
Proof 1 of Lemma 11.6 (sketch). Assume that the truncations of 𝐹(1, 𝑦, 𝑧) along the edges 𝐸 1 and 𝐸 2 are square-free. Construct the right triangles 𝜏 1 and 𝜏 2 as shown in Figure 28(a). By Bernstein's theorem [11], the left and middle terms in (11.14) can be bounded from below by the difference between the mixed area of 𝐸 1 , 𝐸 2 and the mixed area of 𝜏 1 , 𝜏 2 (see Figure 28 2 . (c) Mixed area of 𝜏 1 , 𝜏 2 . Proof 2 of Lemma 11.6. In the coordinates (1, 𝑦, 𝑧), the shift transformation (1, 𝑦, 𝑧) ↦ (1 + 𝑎𝑧, 𝑦 + 𝑏𝑧, 𝑧) converts the polynomial 𝐹 1 (𝑦, 𝑧) = 𝐹(1, 𝑦, 𝑧) defining the affine curve 𝐶 ⧵ 𝑍(𝑥) into the polynomial 𝐹 𝑎,𝑏 (𝑦, 𝑧) = 𝐹(1 + 𝑎𝑧, 𝑦 + 𝑏𝑧, 𝑧) defining 𝐶 𝑎,𝑏 ⧵ 𝑍(𝑥). Under this transformation, each monomial 𝑦 𝑖 𝑧 𝑗 of 𝐹 1 becomes (11.15) 𝑦 𝑖 𝑧 𝑗 + ∑ 𝑖 ′ ,𝑗 ′ ≥0 𝑖 ′ +𝑗 ′ >0
𝑐 𝑖 ′ 𝑗 ′ 𝑦 𝑖-𝑖 ′ 𝑧 𝑗+𝑖 ′ +𝑗 ′ .
In view of the assumption 𝒬(𝐶, 𝑝) = 𝒬 0 (𝐶, 𝑝), the monomials appearing in the sum above correspond to integer points lying strictly above the Newton diagram Γ(𝐶, 𝑝) of 𝐹 1 . In particular, 𝐹 𝑎,𝑏 has the same Newton diagram Γ(𝐶, 𝑝), and the same truncations to its edges. Furthermore, each local branch 𝑄 ∈ 𝒬(𝐶, 𝑝) and its counterpart 𝑄 𝑎,𝑏 ∈ 𝒬(𝐶 𝑎,𝑏 , 𝑝) are associated with the same edge of Γ(𝐶, 𝑝). The union of the local branches of 𝐶 𝑎,𝑏 associated with the edge 𝐸 1 can be defined by an analytic equation 𝑓(𝑦, 𝑧) = 0 whose Newton diagram at the origin is the line segment [(𝑖 1 -𝑖 ′ 1 , 0), (0, 𝑗 ′ 1 -𝑗 1 )] (cf. the Newton-Puiseux algorithm [18,Algorithm I.3.6]). where
, we obtain 𝑓(𝜑(𝑡), 𝜓(𝑡)) = 𝑡 𝑚(𝑖 1 -𝑖 ′ 1 ) (𝛽 + 𝑂(𝑡)), 𝛽 ≠ 0.
The statement of the lemma now follows from the fact that the total multiplicity of the local branches of 𝐶 at 𝑝 associated with the edge 𝐸 2 equals 𝑗 ′ 2 -𝑗 2 . □
We are now left with the task of computing (11.16)
where the first sum runs over the edges 𝐸 of the Newton diagram. In this part of the proof, we continue to assume, for the sake of simplifying the exposition, that all local branches are tangent to 𝐿 ∞ . Since the shift (𝑎, 𝑦, 𝑧) ↦ (1 + 𝑎𝑧, 𝑦 + 𝑏𝑧, 𝑧) acts independently on the analytic factors of 𝐹 1 (𝑦, 𝑧), we furthermore assume, in our computation of ∑ 𝑄,𝑄 ′ ∈𝒬(𝐸) (𝑄 ⋅ 𝑄 ′ 𝑎,𝑏 ) 𝑝 (see (11.16)), that the Newton diagram Γ(𝐶, 𝑝) consists of a single edge 𝐸 = [(0, 𝑀), (𝐷, 0)], with 𝑀 < 𝐷 and 𝐧(𝐸) = (𝑛 𝑦 , 𝑛 𝑧 ).
Pick a local branch 𝑄 ∈ 𝒬(𝐸). It admits an analytic parametrization of the form (11.17 Before providing a proof of Lemma 11.7, we note that the weaker inequality (11.19) ∑ 𝑄 ′ ∈𝒬(𝐸) (𝑄 ⋅ 𝑄 ′ 𝑎,𝑏 ) 𝑝 ≥ 𝑑𝑀 -𝑚 + 𝑑 can be deduced directly from the Bernstein-Kouchnirenko formula, similarly to Proof 1 of Lemma 11.6 provided above. As in the case of Lemma 11.6, this lower bound would be sufficient to complete the upcoming proof of Proposition 11.4, restricted to the case of a Newton nondegenerate singularity (𝐶, 𝑝).
Proof. The left-hand side of (11.18) is the minimal exponent of 𝑡 appearing in the expansion of 𝐹 𝑎,𝑏 (𝜑(𝑡), 𝜓(𝑡)) into a power series in 𝑡. Since 𝐹 1 (𝜑(𝑡), 𝜓(𝑡)) = 0, we may instead substitute (11.17) into the difference 𝐹 𝑎,𝑏 (𝑦, 𝑧) -𝐹 1 (𝑦, 𝑧), or equivalently into the monomials of the second summand in (11.15) (corresponding to individual monomials 𝑦 𝑖 𝑧 𝑗 of 𝐹 1 (𝑦, 𝑧)). Evaluating 𝑦 𝑖-𝑖 ′ 𝑧 𝑗+𝑖 ′ +𝑗 ′ at 𝑦 = 𝜑(𝑡), 𝑧 = 𝜓(𝑡), we obtain (𝑡 𝑚 ) 𝑖-𝑖 ′ (𝑡 𝑑 (𝛼 + 𝑂(𝑡))) 𝑗+𝑖 ′ +𝑗 ′ = 𝑡 𝑚𝑖+𝑑𝑗+(𝑑-𝑚)𝑖 ′ +𝑑𝑗 ′ (𝛼 𝑗+𝑖 ′ +𝑗 ′ + 𝑂(𝑡)).
To get the minimal value of the exponent 𝑚𝑖 + 𝑑𝑗 + (𝑑 -𝑚)𝑖 ′ + 𝑑𝑗 ′ , we need to minimize 𝑚𝑖 + 𝑑𝑗 (which is achieved for (𝑖, 𝑗) ∈ 𝐸) and take 𝑖 ′ = 1 and 𝑗 ′ = 0. Developing 𝐹 𝑎,𝑏 (𝑦, 𝑧) = 𝐹(1 + 𝑎𝑧, 𝑦 + 𝑏𝑧, 𝑧) into a power series in 𝑎 and 𝑏, we see that the corresponding monomials 𝑦 𝑖-𝑖 ′ 𝑧 𝑗+𝑖 ′ +𝑗 ′ = 𝑦 𝑖-1 𝑧 𝑗+1 in 𝐹 𝑎,𝑏 (𝑦, 𝑧) -𝐹(1, 𝑦, 𝑧) add up to 𝑏𝑧𝐹 𝑦 (1, 𝑦, 𝑧). We conclude that the desired minimal exponent of 𝑡 occurs when we substitute (𝑦, 𝑧) = (𝜑(𝑡), 𝜓(𝑡)) either into 𝑏𝑧𝐹 𝐸 𝑦 (𝑦, 𝑧) or into a monomial 𝑦 𝑖-1 𝑧 𝑗+1 such that (𝑖, 𝑗) is one of the integral points closest to the edge 𝐸 and lying above 𝐸. The latter condition reads 𝑛 𝑦 𝑖 + 𝑛 𝑧 𝑗 = 𝑛 𝑧 𝑀 + 1.
The truncation 𝐹 𝐸 (𝑦, 𝑧) of 𝐹
1 (𝑦, 𝑧) has the form (11.20) 𝐹 𝐸 (𝑦, 𝑧) = 𝑛 ∏ 𝑘=1 (𝑧 𝑛 𝑦 -𝛽 𝑘 𝑦 𝑛 𝑧 ) 𝑟 𝑘 , where 𝛽 1 , . . . , 𝛽 𝑛 ∈ ℂ are distinct, and 𝑟 1 , . . . , 𝑟 𝑛 ∈ ℤ >0 . Developing 𝐹 𝐸 (𝑦, 𝑧) into a power series in 𝑡, we see that the monomials of 𝐹 𝐸 yield the minimal exponent of 𝑡. Since 𝐹 1 (𝜑(𝑡), 𝜓(𝑡)) = 0, these minimal powers of 𝑡 must cancel out, implying that, for some 𝑘 0 ∈ {1, . . . , 𝑛}, we have (cf. (11.8), (11.9)): 𝜌 = 𝜌(𝑄) = lim 𝑡→0 (𝛼𝑡 𝑑 ) 𝑛 𝑦 (𝑡 𝑚 ) 𝑛 𝑧 = 𝛼 𝑛 𝑦 = 𝛽 𝑘 0 , 𝜂 = 𝜂(𝑄) = 𝑟 𝑘 0 . The factorization formula (11.20) implies that 𝑏𝑧𝐹 𝐸 𝑦 (𝑦, 𝑧) = 𝑏𝑛 𝑧 𝑦 𝑛 𝑧 -1 𝑧 𝑛 ∑ 𝑘=1 ((-𝛽 𝑘 ) 𝑟 𝑘 (𝑧 𝑛 𝑦 -𝛽 𝑘 𝑦 𝑛 𝑧 ) 𝑟 𝑘 -1 ∏ 𝑙≠𝑘 (𝑧 𝑛 𝑦 -𝛽 𝑙 𝑦 𝑛 𝑧 ) 𝑟 𝑙 ). Putting everything together, we obtain (11.11).
In this section, we generalize Example 10.5 to arrangements of polynomial curves obtained from a given curve by shifts, dilations, and/or rotations. We start by obtaining upper bounds on the number of intersection points of two polynomial curves related by one of these transformations. These bounds lead to expressivity criteria for arrangements consisting of such curves.
Recall that for 𝑎, 𝑏 ∈ ℂ and 𝑐 ∈ ℂ * , we denote by 𝐶 𝑎,𝑏 = 𝑍(𝐹(𝑥 + 𝑎𝑧, 𝑦 + 𝑏𝑧, 𝑧)), (12.1) 𝐶 𝑐 = 𝑍(𝐹(𝑐𝑥, 𝑐𝑦, 𝑧)) (12.2) the curves obtained from a plane curve 𝐶 = 𝑍(𝐹(𝑥, 𝑦, 𝑧)) by a shift and a dilation, respectively. We will also use the notation (12.3) 𝐶 𝑎,𝑏,𝑐 = 𝑍(𝐹(𝑐𝑥 + 𝑎𝑧, 𝑐𝑦 + 𝑏𝑧, 𝑧))
for a curve obtained from 𝐶 by a combination of a shift and a dilation.
As before, we identify a projective curve 𝐶 = 𝑍(𝐹(𝑥, 𝑦, 𝑧)) with its restriction to the affine plane 𝔸 2 = ℙ 2 ⧵ 𝐿 ∞ given by 𝐶 = 𝑉(𝐺(𝑥, 𝑦)), where 𝐺(𝑥, 𝑦) = 𝐹(𝑥, 𝑦, 1). Under this identification, we have 𝐶 𝑎,𝑏 = 𝑉(𝐺(𝑥 + 𝑎, 𝑦 + 𝑏)), (12.4) 𝐶 𝑐 = 𝑉(𝐺(𝑐𝑥, 𝑐𝑦)), (12.5) 𝐶 𝑎,𝑏,𝑐 = 𝑉(𝐺(𝑐𝑥 + 𝑎, 𝑐𝑦 + 𝑏)). (12.6) Remark 12.1. Unless 𝑐 = 1 (the case of a pure shift), the transformation 𝐶 ; 𝐶 𝑎,𝑏,𝑐 can be viewed as a pure dilation centered at some point 𝑜 ∈ ℂ (where 𝑜 may be different from 0). Let 𝑎, 𝑏, 𝑐 ∈ ℂ, 𝑐 ≠ 0, and let 𝐶 𝑎,𝑏,𝑐 be the shifted and dilated curve given by (12.3).
Assume that 𝐶 and 𝐶 𝑎,𝑏,𝑐 have 𝑁 intersection points in 𝔸 2 = ℙ 2 ⧵ 𝐿 ∞ , counting with multiplicity. Then 𝑁 ≤ 𝑑𝑑 ′ .
Proof. Without loss of generality, assume that 𝑑 > 𝑑 ′ . By Bézout's theorem, the intersection multiplicity (𝐶 ⋅ 𝐶 𝑎,𝑏,𝑐 ) 𝑝 at the point 𝑝 = (1, 0, 0) ∈ 𝐿 ∞ is at most 𝑑 2 -𝑁. Applying (12.7), with 𝑚 = 𝑑 -𝑑 ′ , we obtain 𝑑 2 -𝑁 ≥ 𝑑(𝑑 -𝑑 ′ ). The claim follows. □ Proposition 12.3 and Theorem 7.17 imply that if 𝐶 and 𝐶 𝑎,𝑏,𝑐 intersect at 𝑑𝑑 ′ hyperbolic nodes in the real affine plane, then the union 𝐶 ∪ 𝐶 𝑎,𝑏,𝑐 is expressive.
Example 12.4. Let 𝐶 be the (2, 3)-Chebyshev curve, the singular cubic given by (12.9) 2𝑥 2 -1 + 4𝑦 3 -3𝑦 = 0 or parametrically by 𝑡 ↦ (4𝑡 3 -3𝑡, -2𝑡 2 + 1), cf. (4.3). Applying Proposition 12.3 (with 𝑑 = 3 and 𝑑 ′ = 2), we see that the curve 𝐶 and its dilation 𝐶 𝑎,𝑏,𝑐 (𝑐 ≠ 1) can intersect in the real affine plane in at most 6 points. When this bound is attained, the union 𝐶 ∪ 𝐶 𝑎,𝑏,𝑐 is expressive. Figure 30 shows one such example, with 𝑎 = 𝑏 = 0 and 𝑐 = -1 (so 𝐶 𝑎,𝑏,𝑐 is a reflection of 𝐶). Cf. also Figure 38. Let 𝑎, 𝑏 ∈ ℂ, and let 𝐶 𝑎,𝑏 be the shifted curve given by (12.1). Assume that 𝐶 and 𝐶 𝑎,𝑏 have 𝑁 intersection points in 𝔸 2 = ℙ 2 ⧵ 𝐿 ∞ , counting with multiplicity. Then (12.10) 𝑁 ≤ 𝑑𝑑 ′ -𝑑 ′ -gcd(𝑑, 𝑑 ′ ) + 1. Proof. We use the same arguments as in the proof of Proposition 12.3 above, with the lower bound (12.7) replaced by (12.8): 𝑁 ≤ 𝑑 2 -(𝐶 ⋅ 𝐶 𝑎,𝑏 ) 𝑝 ≤ 𝑑 2 -(𝑑 -1)(𝑚 + 1) -gcd(𝑑, 𝑚) = 𝑑 2 -(𝑑 -1)(𝑑 -𝑑 ′ + 1) -gcd(𝑑, 𝑑 ′ ) = 𝑑𝑑 ′ -𝑑 ′ + 1 -gcd(𝑑, 𝑑 ′ ). □
For special choices of shifts, the bound (12.10) can be strengthened. Here is one example, involving the Lissajous-Chebyshev curves, see Example 5.12. (Note that such a curve does not have to be polynomial: it could be trigonometric or reducible.) Proposition 12.6. Let 𝐶 be a Lissajous-Chebyshev curve given by (12.11) 𝑇 𝑘 (𝑥) + 𝑇 ℓ (𝑦) = 0.
Let 𝑎 ∈ ℂ, and let 𝐶 𝑎,0 (resp., 𝐶 0,𝑎 ) be the shift of 𝐶 in the 𝑥 (resp., 𝑦) direction. Assume that 𝐶 and 𝐶 𝑎,0 (resp., 𝐶 0,𝑎 ) intersect at 𝑁 𝑥 (resp., 𝑁 𝑦 ) points in 𝔸 2 . Then 𝑁 𝑥 ≤ (𝑘 -1)ℓ, (12.12) 𝑁 𝑦 ≤ 𝑘(ℓ -1). (12.13) We note that in the case when gcd(𝑘, ℓ) = 1 and 𝑘 < ℓ, the bound (12.13) matches (12.10), with 𝑑 = 𝑘 and 𝑑 ′ = ℓ, whereas (12.12) gives a stronger bound.
Proof. Due to symmetry, it suffices to prove (12.13). The intersection of the curves 𝐶 and 𝐶 0,𝑎 is given by
The equation 𝑇 ℓ (𝑦 + 𝑎) -𝑇 ℓ (𝑦) = 0 has at most ℓ -1 roots; each of these values of 𝑦 then gives at most 𝑘 possible values of 𝑥. □ Example 12.7. As in Example 12.4, let 𝐶 be the (2, 3)-Chebyshev curve (12.9). Applying Proposition 12.6 (with 𝑘 = 2 and ℓ = 3), we see that the curve 𝐶 and its vertical shift 𝐶 0,𝑏 (𝑏 ≠ 0) can intersect in the affine plane in at most 4 points. More generally, Proposition 12.5 (with 𝑑 = 3 and 𝑑 ′ = 2) gives the upper bound of 4 for the number of intersection points between 𝐶 and its nontrivial shift 𝐶 𝑎,𝑏 . On the other hand, in the case of a horizontal shift, we get at most 3 points of intersection. When these bounds are attained, with all intersection points real, the union 𝐶 ∪ 𝐶 𝑎,𝑏 is expressive. See Figure 31. In addition to shifts and dilations, we can consider other linear changes of variables that can be used to construct new expressive curves. Let us illustrate one such construction using the example of Lissajous-Chebyshev curves: Proof. Since 𝑇 𝑘 (𝑥) = 2 𝑘-1 𝑥 𝑘 + 𝑂(𝑥 𝑘-1 ) and 𝑇 ℓ (𝑦) = 2 ℓ-1 𝑦 ℓ + 𝑂(𝑦 ℓ-1 ), the equations defining 𝐶 and 𝐶 [𝑞] can be written as 2 𝑘-1 𝑥 𝑘 + 𝑂(𝑥 𝑘-2 ) + 2 ℓ-1 𝑦 ℓ + 𝑂(𝑦 ℓ-2 ) = 0, (12.16) 2 𝑘-1 𝑞 -𝑘ℓ 𝑥 𝑘 + 𝑂(𝑥 𝑘-2 ) + 2 ℓ-1 𝑞 -𝑘ℓ 𝑦 ℓ + 𝑂(𝑦 ℓ-2 ) = 0. (12.17) Multiplying (12.16) by 𝑞 𝑘ℓ and subtracting (12.16), we get an equation of the form (12.18) 𝑂(𝑥 𝑘-2 ) + 𝑂(𝑦 ℓ-2 ) = 0.
We thus obtain a system of two algebraic equations of the form (12.16) and (12.18). Their Newton polygons are contained in the triangles with vertices (0, 0), (𝑘, 0), (0, ℓ) and (0, 0), (𝑘 -2, 0), (0, ℓ -2), respectively. The mixed area of these two triangles is equal to 𝑘(ℓ -2) (here we use that ℓ > 𝑘 ≥ 2 and therefore ℓ-2 𝑘-2 > ℓ 𝑘 ). By Bernstein's theorem [11], this system of equations has at most 𝑘(ℓ -2) solutions. □ Example 12.9. Let 𝐶 be the (2, 3)-Chebyshev curve (12.9), as in Examples 12.4 and 12.7. Applying Proposition 12.8 (with 𝑘 = 2 and ℓ = 3), we see that the curve 𝐶 and the rescaled curve 𝐶 [𝑞] defined by (12.14) can intersect in the affine plane in at most 2 points. When they do intersect at 2 real points, the union 𝐶 ∪ 𝐶 𝑎,𝑏 is expressive. See Figure 32. Remark 12.10. Let 𝐶 ′ be a curve obtained from a plane curve 𝐶 = 𝑉(𝐺(𝑥, 𝑦)) of degree 𝑑 by an arbitrary affine change of variables: 𝐶 ′ = 𝑉(𝐺(𝑐 11 𝑥 + 𝑐 12 𝑦 + 𝑎, 𝑐 21 𝑥 + 𝑐 22 𝑦 + 𝑏)). Then 𝐶 and 𝐶 ′ intersect in 𝔸 2 in at most 𝑑 2 points. Thus, if they intersect at 𝑑 2 hyperbolic nodes, then 𝐶 ∪ 𝐶 ′ is expressive. See Figure 33. Example 12.11. Figures 34-35 show five different ways to arrange two Chebyshev curves (the (2, 3)-Chebyshev cubic and the (3, 4)-Chebyshev quartic, respectively) related to each other by an affine transformation of the plane 𝔸 2 so that the resulting two-component curve is expressive. These pictures illustrate: (a) Proposition 12.8, cf. Example 12.9; (b, c) Proposition 12.6, cf. Example 12.7; (d) Proposition 12.3, cf. Example 12.4; and (e) Remark 12.10. 64 SERGEY FOMIN AND EUGENII SHUSTIN Figure 33: Two expressive cubics related by a 90 • rotation, and intersecting at 9 real points. The resulting two-component curve is expressive. Example 12.11. Figures 34-35 show five different ways to arrange two Chebyshev curves (the (2, 3)-Chebyshev cubic and the (3, 4)-Chebyshev quartic, respectively) related to each other by an affine transformation of the plane A 2 so that the resulting two-component curve is expressive. These pictures illustrate: (a) Proposition 12.8, cf. Example 12.9; (b,c) Proposition 12.6, cf. Example 12.7; (d) Proposition 12.3, cf. Example 12.4; and (e) Remark 12.10. (a) (b) (c) (d) (e) Figure 34: Two expressive cubics forming a two-component expressive curve. The two components intersect at 2, 3, 4, 6, and 9 real points, respectively. (a) (b) (c) (d) (e) Figure 34. Two expressive cubics forming a two-component expressive curve. The two components intersect at 2, 3, 4, 6, and 9 real points, respectively. (a) (b) (c) (d) (e) Figure 35: Two expressive quartics forming a two-component expressive curve. The two components intersect at 6, 8, 10, 12, and 16 real points, respectively. Remark 12.12. More generally, consider a collection of expressive polynomial curves related to each other by affine changes of variables (equivalently, affine transformations of the plane A 2 ). Suppose that for every pair of curves in this collection, the number of hyperbolic nodes in their intersection attains the upper bound for an appropriate version of Proposition 12.3, 12.5, 12.6, 12.8, or Remark 12.10. Then the union of the curves in the given collection is an expressive curve. See Figures 36-37. 1 Jun 2022 08:44:25 PDT 201027-Fomin Version 2 -Submitted to Commun. Amer. Math. Soc. Remark 12.12. More generally, consider a collection of expressive polynomial curves related to each other by affine changes of variables (equivalently, affine transformations of the plane 𝔸 2 ). Suppose that for every pair of curves in this collection, the number of hyperbolic nodes in their intersection attains the upper bound for an appropriate version of Proposition 12.3, 12.5, 12.6, 12.8, or Remark 12.10. Then the union of the curves in the given collection is an expressive curve. See Figures 36-37. EXPRESSIVE CURVES 65 Figure 36: An expressive curve whose three components are translations of the same nodal cubic. Each pair of components intersect at 4 hyperbolic nodes.
In this section, we generalize Example 10.6 to arrangements of curves obtained from a given trigonometric curve by shifts, dilations, and/or rotations.
We continue to use the notation (12.1)-(12.6) for the shifted and dilated curves. Example 13.2. Let 𝐶 be an epitrochoid with parameters (2, -1), i.e., a limaçon. It has two conjugate points at infinity, each an ordinary cusp transversal to 𝐿 ∞ . This is a quartic trigonometric curve, so by Corollary 13.1 (with 𝑑 = 2), any two shifts/dilations of 𝐶 intersect in at most 8 points in the affine plane. Thus, if they intersect at 8 hyperbolic nodes, then 𝐶 ∪ 𝐶 𝑎,𝑏,𝑐 is expressive by Theorem 7.17. More generally, an arrangement of limaçons related to each other by shifts and dilations gives an expressive curve if any two of these limaçons intersect at 8 hyperbolic nodes. See Figure 39. Corollary 13.3. Let 𝐶 be a trigonometric curve of degree 2𝑑, with two local branches at infinity centered at distinct points 𝑝, 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ . Suppose that 𝑚 = mult(𝐶, 𝑝) < 𝑑, i.e., 𝐶 is tangent to 𝐿 ∞ . Then we have, for 𝑎, 𝑏 ∈ ℂ and 𝑐 ∈ ℂ * : (𝐶 ⋅ 𝐶 𝑎,𝑏,𝑐 ) 𝑝 ≥ 𝑑𝑚, (13.2) 3-petal hypotrochoids related to each other by a shift (resp., dilation) is expressive if they intersect at 10 (resp., 12) hyperbolic nodes Corollary 13.5. Let 𝐶 = 𝑍(𝐹) be a trigonometric projective curve of degree 2𝑑, with two local complex conjugate branches 𝑄, 𝑄 centered at the same point 𝑝 ∈ 𝐶 ∩ 𝐿 ∞ . Denote mult(𝐶, 𝑝) = 2 mult 𝑄 = 2𝑚. Then we have, for 𝑎, 𝑏 ∈ ℂ and 𝑐 ∈ ℂ * : (𝐶 ⋅ 𝐶 𝑎,𝑏,𝑐 ) 𝑝 ≥ 4𝑑𝑚, (13.6) (𝐶 ⋅ 𝐶 𝑎,𝑏 ) 𝑝 ≥ { 4𝑑𝑚 -2𝑚 + 2𝑑 + 2 gcd(𝑑, 𝑚) -2 if (𝑄 ⋅ 𝑄) 𝑝 = 𝑑𝑚, 4𝑑𝑚 -2𝑚 + 2𝑑 + 2 gcd(𝑑, 𝑚) if (𝑄 ⋅ 𝑄) 𝑝 > 𝑑𝑚. (13.7) Proof. Once again, we apply Propositions 11. For special choices of shifts and dilations, the bounds in the corollaries above can be strengthened, leading to examples of expressive curves whose components intersect in fewer real points than one would ordinarily expect. Here are two examples: Example 13.7. Let C be the lemniscate (13.8). Since C is a Lissajous-Chebyshev curve with parameters (4, 2), by Proposition 12.6, its vertical shift C 0,b intersects C in A 2 in at most 4 points. Hence C ∪ C 0,b is expressive for b ∈ (-2, 2). See Figure 41. Example 13.8. Let C = V (G(x, y)) be the trigonometric curve defined by the polynomial (13.9) G(x, y) = x 2 + y 4 -11y 2 + 18y -8 = x 2 + (y + 4)(y -1) 2 (y -2). For special choices of shifts and dilations, the bounds in the corollaries above can be strengthened, leading to examples of expressive curves whose components intersect in fewer real points than one would ordinarily expect. Here are two examples: Example 13.7. Let 𝐶 be the lemniscate (13.8). Since 𝐶 is a Lissajous-Chebyshev curve with parameters (4, 2), by Proposition 12.6, its vertical shift 𝐶 0,𝑏 intersects 𝐶 in 𝔸 2 in at most 4 points. Hence 𝐶 ∪ 𝐶 0,𝑏 is expressive for 𝑏 ∈ (-2, 2). See Figure 41. cf. (7.8). As we saw in Example 7.4, 𝐺 is expressive while G is not. Consequently, the affine algebraic curve 𝑉(𝐺) is expressive while its real point set, the real topological curve 𝒞 is not-because G, rather than 𝐺, is the minimal polynomial of 𝒞.
Example 14.3. The real polynomials 𝐺(𝑥, 𝑦) and G (𝑥, 𝑦) given by 𝐺(𝑥, 𝑦) = 𝑥𝑦(𝑥 2 + 𝑦 2 + 1), G(𝑥, 𝑦) = 𝑥𝑦 define the same real algebraic set 𝒞 = 𝑉 ℝ (𝐺) = 𝑉 ℝ ( G). Clearly, G is expressive, and so is the affine curve 𝑉( G), or the projective curve 𝑍(𝑥𝑦). Since G is the minimal polynomial of 𝒞, this real algebraic set is expressive as well. On the other hand, 𝐺 𝑥 = 3𝑥 2 𝑦 + 𝑦 3 + 𝑦 = 𝑦(3𝑥 2 + 𝑦 2 + 1), 𝐺 𝑦 = 𝑥 3 + 3𝑥𝑦 2 + 𝑥 = 𝑥(𝑥 2 + 3𝑦 2 + 1), and we see that 𝐺 has 9 critical points:
(0, 0), (0, ±𝑖), (±𝑖, 0), ( 12 𝑖, ± 1 2 𝑖), (-1 2 𝑖, ± 1 2 𝑖). Since 8 of these points are not real, the polynomial 𝐺 is not expressive; nor are the curves 𝑉(𝐺) and 𝑍(𝑥𝑦(𝑥 2 + 𝑦 2 + 𝑧 2 )).
The following criterion is a direct consequence of Theorem 7.17. Corollary 14.4. Let 𝒞 ⊂ ℝ 2 be the set of real points of a real affine algebraic curve. Assume that 𝒞 is connected, and contains at least two (hence infinitely many) points. Then the following are equivalent:
• the minimal polynomial of 𝒞 (or the Zariski closure 𝐶 = 𝒞, cf. Definition 14.1) is expressive and 𝐿 ∞ -regular; • each component of 𝐶 = 𝒞 is trigonometric or polynomial, and all singular points of 𝐶 in the complex affine plane 𝔸 2 are real hyperbolic nodes.
We conclude this section by discussing the challenges involved in extending the notion of expressivity to arbitrary (i.e., not necessarily polynomial) smooth real functions of two real arguments. Remark 14.5. Let 𝐺 ∶ ℝ 2 → ℝ be a smooth function. Suppose that 𝐺 satisfies the following conditions:
• the set 𝑉 ℝ (𝐺) = {𝐺(𝑥, 𝑦) = 0} ⊂ ℝ 2 is connected; • 𝑉 ℝ (𝐺) is a union of finitely many immersed circles and open intervals which intersect each other and themselves transversally, in a finite number of points; • the complement ℝ 2 ⧵ 𝑉 ℝ (𝐺) is a union of a finite number of disjoint open sets; • all critical points of 𝐺 in ℝ have a nondegenerate Hessian; • these critical points are located as follows:
∘ one critical point inside each bounded connected component of ℝ 2 ⧵ 𝑉 ℝ (𝐺); ∘ no critical points inside each unbounded connected component of ℝ 2 ⧵ 𝑉 ℝ (𝐺); ∘ a saddle at each double point of 𝑉 ℝ (𝐺).
One may be tempted to call such a (topological, not necessarily algebraic) curve 𝑉 ℝ (𝐺) expressive. Unfortunately, this definition turns out to be problematic, as one and the same curve 𝒞 = ℝ 2 can be defined by two different smooth functions one of which satisfies the above-listed conditions whereas the other does not. An example is shown in Figure 2, with the polynomial 𝐺 given by 𝐺(𝑥, 𝑦) = ( 𝑥 2 16 + 𝑦 2 -1)((𝑥 -1) 2 + (𝑦 -1) 2 -1). As Figure 2 demonstrates, the function 𝐺 is not expressive in any reasonable sense. At the same time, 𝒞 can be transformed into an expressive curve (a union of two circles) by a diffeomorphism of ℝ 2 , and consequently can be represented as the vanishing set of a smooth function satisfying the conditions listed above.
It is natural to wonder which divides arise from expressive curves (perhaps satisfying additional technical conditions).
If 𝐺(𝑥, 𝑦) is an expressive polynomial, then all singular points of 𝑉(𝐺) are (real) hyperbolic nodes, so the divide 𝐷 𝐺 is well defined, see Definition 6.3. The class of divides arising in this way (with or without the additional requirement of 𝐿 ∞ -regularity) is however too broad to be a natural object of study: as Example 14.2 demonstrates, a nonexpressive polynomial may become expressive upon multiplication by a polynomial with an empty set of real zeroes.
With this in mind, we propose Definition 15.1.
Definition 15.1. A divide 𝐷 is called regular-expressive if there exists an 𝐿 ∞ -regular expressive curve 𝐶 = 𝑉(𝐺) with real irreducible components such that 𝐷 = 𝐷 𝐺 .
Remark 15.2. Some readers might prefer to just call such divides "expressive" rather than regular-expressive. We decided against the shorter term, as it would misleadingly omit a reference to the 𝐿 ∞ -regularity requirement.
By Theorem 7.17, a connected divide is regular-expressive if and only if it arises from an algebraic curve whose components are real and either polynomial or trigonometric, and all of whose singular points in the affine plane are hyperbolic nodes.
We note that a regular-expressive divide is isotopic to an expressive algebraic subset of ℝ 2 , in the sense of Definition 14.1, or more precisely to an intersection of such a subset with a sufficiently large disk 𝐃 𝑅 , see (6.1).
Numerous examples of regular-expressive divides are scattered throughout this paper.
Problem 15.3. Find a criterion for deciding whether a given divide is regularexpressive.
Remark 15.4. Problem 15.3 appears to be very difficult. It seems to be even harder to determine whether a given divide can be realized by an expressive curve of a specified degree. For example, the divide can be realized by an expressive sextic (the (2, 6)-Lissajous curve, see Figure 10) but not by an expressive quadric-even though there exists a (nonexpressive) quadric realizing this divide.
Here is a nonobvious nonexample: Proof. Suppose on the contrary that the divide 𝐷 in Figure 43 is regular-expressive. By Theorem 7.17, 𝐷 must come from a plane curve 𝐶 consisting of two polynomial components: 𝐶 = 𝐾 ∪ 𝐿. One of them, say 𝐿, is smooth. By the Abhyankar-Moh theorem [2, Theorem 1.6], there exists a real automorphism of the affine plane that transforms 𝐿 into a real straight line. So without loss of generality, we can simply assume that 𝐿 is a line. The other component 𝐾 has degree 𝑑 ≥ 4. Since 𝐶 is expressive, the projective line L must intersect the projective closure K of 𝐾 at the unique point 𝑝 ∈ 𝐾 ∩ 𝐿 ∞ with multiplicity 𝑑 -2. The curve K has a unique tangent line 𝐿 ∞ at 𝑝. Therefore any projective line L′ ≠ 𝐿 ∞ passing through 𝑝 intersects K at 𝑝 with multiplicity 𝑑 -2. Equivalently, every affine line 𝐿 ′ parallel to 𝐿 intersects 𝐾 in at most two points (counting multiplicities). However, shifting 𝐿 in a parallel way until it intersects 𝐾 at a node for the first time, we obtain a real line 𝐿 ′ parallel to 𝐿 and crossing 𝐾 in at least four points in the affine plane, a contradiction. □
We recall the following terminology, adapting it to our current needs.
Definition 15.6. A (simple) pseudoline arrangement is a connected divide whose branches are embedded intervals any two of which intersect at most once. A pseudoline arrangement is called stretchable if it is isotopic to a configuration of straight lines, viewed within a sufficiently large disk. Proposition 15.7. Let D be a pseudoline arrangement in which any two pseudolines intersect. Then the divide D is regular-expressive if and only if it is stretchable. Proof. The "if" direction has already been established, see Example 10.3. Let us prove the converse. Suppose that a pseudoline arrangement 𝐷 with 𝑛 pseudolines 𝐷 1 , . . . , 𝐷 𝑛 is the divide of an 𝐿 ∞ -regular expressive curve 𝐶 with real irreducible components. We need to show that 𝐷 is stretchable. By Theorem 7.17, 𝐶 must consist of 𝑛 polynomial components 𝐶 1 , . . . , 𝐶 𝑛 . Since 𝐶 1 is smooth, the Abhyankar-Moh theorem [2, Theorem 1.6] implies that a suitable real automorphism of the affine plane takes 𝐶 1 to a straight line (and leaves the other components polynomial). So without loss of generality, we can assume that 𝐶 1 = {𝑥 = 0}. Let 𝑖 ∈ {2, . . . , 𝑛} be such that deg 𝐶 𝑖 = 𝑑 ≥ 2. (If there is no such 𝑖, we are done.) The projective line Ĉ 1 intersects the projective curve Ĉ 𝑖 at the point 𝑝 = (0, 1, 0) = Ĉ 𝑖 ∩ 𝐿 ∞ either with multiplicity 𝑑 -1 (if 𝐶 1 and 𝐶 𝑖 intersect in the affine plane) or with multiplicity 𝑑 (if 𝐶 1 and 𝐶 𝑖 are disjoint there). Note that Ĉ 1 cannot be tangent to Ĉ 𝑖 at 𝑝, since Ĉ 𝑖 is unibranch at 𝑝, and the infinite line 𝐿 ∞ is the tangent to Ĉ 𝑖 at 𝐿 ∞ . It follows that ( Ĉ 1 ⋅ Ĉ 𝑖 ) 𝑝 < ( Ĉ 𝑖 ⋅ 𝐿 ∞ ) 𝑝 = 𝑑 and therefore ( Ĉ 1 ⋅ Ĉ 𝑖 ) 𝑝 = 𝑑 -1. Since Ĉ 1 is transversal to Ĉ 𝑖 at 𝑝, we have mult( Ĉ 𝑖 , 𝑝) = 𝑑 -1. It follows that the (affine) equation of 𝐶 𝑖 does not contain monomials with exponents of 𝑦 higher than 1. That is, 𝐶 𝑖 = {𝑦𝑃 𝑖 (𝑥) = 𝑄 𝑖 (𝑥)} with 𝑃 𝑖 , 𝑄 𝑖 coprime real polynomials. Moreover, the polynomiality of 𝐶 𝑖 implies that 𝑃 𝑖 (𝑥) is a nonzero constant, say 1. Finally, recall that any two components 𝐶 𝑖 and 𝐶 𝑗 , 2 ≤ 𝑖 < 𝑗 ≤ 𝑛, intersect in at most one point in the affine plane, and if they do, the intersection is transversal. So if 𝐶 𝑖 = {𝑦 = 𝑄 𝑖 (𝑥)} and 𝐶 𝑗 = {𝑦 = 𝑄 𝑗 (𝑥)}, then deg(𝑄 𝑖 -𝑄 𝑗 ) ≤ 1. It follows that a suitable automorphism of the affine plane (𝑥, 𝑦) ↦ (𝑥, 𝑦 -𝑅(𝑥)), with 𝑅(𝑥) ∈ ℝ[𝑥], leaves 𝐶 1 invariant while simultaneously taking all other components 𝐶 2 , . . . , 𝐶 𝑛 to straight lines. □
In this section, we compare two classes of divides: • the algebraic divides, which arise from real morsifications of isolated singularities of real plane curves, see, e.g., [16, Definition 2.2]; • the regular-expressive divides, which arise from 𝐿 ∞ -regular expressive curves with real components, see Section 15.
There are plenty of divides (such as, e.g., generic real line arrangements) which are both algebraic and regular-expressive. The same argument works for the other divides except for the left divides in Figures 8 and 9, which require a slightly more complicated treatment. We leave it to the reader as an exercise.
(b) Suppose on the contrary that our divide 𝐷 is algebraic. The corresponding singularity must be unibranch, with Milnor number 12. Let 𝑇 be the sole region of 𝐷 whose closure has zero-dimensional intersection with the union of closures of the unbounded connected components. Every real straight line crossing 𝑇 intersects 𝐷 in at least 4 points (counting multiplicities). It follows that the underlying singularity has multiplicity ≥ 4. The simplest unibranch singularity of multiplicity 4 is 𝑦 4 -𝑥 5 = 0 (up to topological equivalence); its Milnor number is 12. Hence we cannot encounter any other (more complicated) singularity. The link of the singularity 𝑦 4 -𝑥 5 = 0 is a (4, 5) torus knot. Its Alexander polynomial is (see, e.g., [32, p. 131, formula (1)]) (16.1) (1 -𝑡 4⋅5 )(1 -𝑡) (1 -𝑡 4 )(1 -𝑡 5 ) = 1 -𝑡 + 𝑡 4 -𝑡 6 + 𝑡 8 -𝑡 11 + 𝑡 12 . On the other hand, as shown by N. A'Campo [5, Theorem 2] (reproduced in [16, Theorem 7.6]), the link of a singularity is isotopic to the link of the divide of its morsification, as defined by A'Campo; see, e.g., [16, Definition 7.1]. In the terminology of [16, Section 11], the divide 𝐷 is scannable of multiplicity 4. Its link can be computed as the closure of a 4-strand braid 𝛽 constructed by the Couture-Perron algorithm [12, Proposition 2.3] (reproduced in [16, Definition 11.3]): 𝛽 = 𝜎 1 𝜎 3 𝜎 2 𝜎 3 𝜎 2 𝜎 1 𝜎 3 𝜎 2 𝜎 3 𝜎 2 𝜎 1 𝜎 3 𝜎 2 𝜎 3 𝜎 2 . is versal if 𝑑 ≥ 𝜇(𝑓, 0)-1; here 𝐁 𝑁 𝜀 ⊂ ℂ 𝑁 denotes a sufficiently small open disc centered at the origin. If 𝐶 = 𝑉(𝐹(𝑥, 𝑦)) is an affine curve and 𝑝 1 , . . . , 𝑝 𝑟 are some of its isolated singularities, then the deformation 𝐹(𝑥, 𝑦) + ∑ 𝑖+𝑗≤𝑑 𝑡 𝑖𝑗 𝑥 𝑖 𝑦 𝑗 = 0, (𝑡 𝑖𝑗 ) ∈ 𝐁 𝑁 𝜀 , 𝑁 = ( 𝑑+2 2 ), where 𝑑 ≥ ∑ 𝑖 𝜇(𝐶, 𝑝 𝑖 )-1, is a joint versal deformation of the singular points 𝑝 1 , . . . , 𝑝 𝑟 . That is, it simultaneously induces arbitrary individual deformations at all these singular points; see the versality criterion [18, Theorem II.1.16] and [19, Proposition 3.4.6]. Let Since the total Milnor number of the Pappus configuration (i.e., the sum of Milnor numbers at all singular points) is less than 𝜇(𝐹 0 , 0) = 64, the deformation (16.2) is joint versal for all the singularities of any given curve 𝐹 𝑡 (𝑥, 𝑦) = 0, 0 < 𝑡 ≪ 1.
We now construct a morsification of the singularity 𝐹 0 = 0 which is isotopic to our divide 𝐷. We begin by deforming it into a Pappus configuration 𝐹 𝑡 = 0 with 0 < 𝑡 ≪ 1. Then, by variation of the monomials up to degree 63, we deform each triple point of the latter curve into appropriate three double intersections (while keeping the nodes of the Pappus configuration). Since 𝑡 can be chosen arbitrarily small, the curve isotopic to the constructed one appears arbitrarily close to the original germ {𝐹 0 = 0}. In view of the fact that all the strata in the real part of the discriminant in the versal deformation base are semialgebraic sets, by the arc selection lemma [29, Lemma 3.1], there exists a real analytic deformation F𝑡 (𝑥, 𝑦) = 0, 0 ≤ 𝑡 ≪ 1, of F0 = 𝐹 0 = 0, whose members F𝑡 = 0, 𝑡 ≠ 0, realize the desired morsification. □
Jun
08:44:25 PDT 201027-Fomin Version 2 -Submitted to Commun. Amer. Math. Soc.