More Like this

1. Smooth points on semi-algebraic sets

   https://doi.org/10.1145/3457341.3457347  

   Harris, Katherine; Hauenstein, Jonathan D.; Szanto, Agnes (September 2020, ACM Communications in Computer Algebra)

   Many algorithms for determining properties of semi-algebraic sets rely upon the ability to compute smooth points [1]. We present a simple procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected bounded component of a real atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for then= 4 case. We also apply our method to design an efficient algorithm to compute the real dimension of algebraic sets, the original motivation for this research. 

   more » « less
2. Semi-Algebraic Off-Line Range Searching and Biclique Partitions in the Plane

   https://doi.org/10.4230/LIPICS.SOCG.2024.4  

   Agarwal, Pankaj K; Ezra, Esther; Sharir, Micha (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Mulzer, Wolfgang; Phillips, Jeff M (Ed.)

   Let P be a set of m points in ℝ², let Σ be a set of n semi-algebraic sets of constant complexity in ℝ², let (S,+) be a semigroup, and let w: P → S be a weight function on the points of P. We describe a randomized algorithm for computing w(P∩σ) for every σ ∈ Σ in overall expected time O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n), where s > 0 is a constant that bounds the maximum complexity of the regions of Σ, and where the O^*(⋅) notation hides subpolynomial factors. For s ≥ 3, surprisingly, this bound is smaller than the best-known bound for answering m such queries in an on-line manner. The latter takes O^*(m^{s/(2s-1)}n^{(2s-2)/(2s-1)} + m + n) time. Let Φ: Σ × P → {0,1} be the Boolean predicate (of constant complexity) such that Φ(σ,p) = 1 if p ∈ σ and 0 otherwise, and let Σ_Φ P = {(σ,p) ∈ Σ× P ∣ Φ(σ,p) = 1}. Our algorithm actually computes a partition ℬ_Φ of Σ_Φ P into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) O^*(m^{2s/(5s-4)}n^{(5s-6)/(5s-4)} + m^{2/3}n^{2/3} + m + n). It is straightforward to compute w(P∩σ) for all σ ∈ Σ from ℬ_Φ. Similarly, if η: Σ → S is a weight function on the regions of Σ, ∑_{σ ∈ Σ: p ∈ σ} η(σ), for every point p ∈ P, can be computed from ℬ_Φ in a straightforward manner. We also mention a few other applications of computing ℬ_Φ. 

   more » « less
3. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

   more » « less
4. Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems

   Agarwal, Pankaj K; Aronov, Boris; Ezra, Esther; Katz, Matthew J; Sharir, Micha (January 2022, Proceedings of the 38th Annual Symposium on Computational Geometry)

   Let T be a set of n planar semi-algebraic regions in R^3 of constant complexity (e.g., triangles, disks), which we call _plates_. We wish to preprocess T into a data structure so that for a query object gamma, which is also a plate, we can quickly answer various intersection queries, such as detecting whether gamma intersects any plate of T, reporting all the plates intersected by gamma, or counting them. We also consider two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree algebraic arcs in R^3 (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in R^3. Besides being interesting in their own right, the data structures for these two special cases form the building blocks for handling the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we obtain a variety of results with different storage and query-time bounds, depending on the complexity of the input and query objects. For example, if T is a set of plates and the query objects are arcs, we obtain a data structure that uses O^*(n^(4/3)) storage (where the O^*(...) notation hides subpolynomial factors) and answers an intersection query in O^*(n^(2/3)) time. Alternatively, by increasing the storage to O^*(n^(3/2)), the query time can be decreased to O^*(n^(rho)), where rho = (2t-3)/(3(t-1)) < 2/3 and t≤3 is the number of parameters needed to represent the query arcs. 

   more » « less
5. Efficient simplicial replacement of semialgebraic sets

   https://doi.org/10.1017/fms.2023.36  

   Basu, Saugata; Karisani, Negin (January 2023, Forum of Mathematics, Sigma)

   Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set$$S \subset \mathbb {R}^k$$by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$, is semialgebraically homeomorphic toS. In this paper, we consider a weaker version of this question. We prove that for any$$\ell \geq 0$$, there exists an algorithm which takes as input a description of a semialgebraic subset$$S \subset \mathbb {R}^k$$given by a quantifier-free first-order formula$$\phi $$in the language of the reals and produces as output a simplicial complex$$\Delta $$, whose geometric realization,$$|\Delta |$$is$$\ell $$-equivalent toS. The complexity of our algorithm is bounded by$$(sd)^{k^{O(\ell )}}$$, wheresis the number of polynomials appearing in the formula$$\phi $$, andda bound on their degrees. For fixed$$\ell $$, this bound issingly exponentialink. In particular, since$$\ell $$-equivalence implies that thehomotopy groupsup to dimension$$\ell $$of$$|\Delta |$$are isomorphic to those ofS, we obtain a reduction (having singly exponential complexity) of the problem of computing the first$$\ell $$homotopy groups ofSto the combinatorial problem of computing the first$$\ell $$homotopy groups of a finite simplicial complex of size bounded by$$(sd)^{k^{O(\ell )}}$$. 

   more » « less