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1. Polynomial Identity Testing via Evaluation of Rational Functions

   https://doi.org/10.4086/toc.2024.v020a001  

   Hu, Ivan; Melkebeek, Dieter van; Morgan, Andrew (July 2024, Theory of Computing)

   We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. We establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials. We initiate a systematic analytic study of the power of hitting set generators by characterizing their vanishing ideals, i.e., the sets of polynomials that they fail to hit. We provide two such characterizations for our generator. First, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition class size of set-multilinearity in the vanishing ideal. Second, inspired by a connection to alternating algebra, we develop a structured deterministic membership test for the multilinear part of the vanishing ideal. We present a derivation based on alternating algebra along with the required background, as well as one in terms of zero substitutions and partial derivatives, avoiding the need for alternating algebra. As evidence of the utility of our analytic approach, we rederive known derandomization results based on the generator by Shpilka and Volkovich and present a new application in derandomization / lower bounds for read-once oblivious algebraic branching programs. 

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2. Algorithms for computing greatest common divisors of parametric multivariate polynomials

   Kapur, Deepak; Lu, Dong; Monagan, Micheal; Sun, Yao; Wang, Dingkang (January 2021, Journal of symbolic computation)

   Two new efficient algorithms for computing greatest common divisors (gcds) of parametric multivariate polynomials over k[U][X]are presented. The key idea of the first algorithm is that the gcd of two non-parametric multivariate polynomials can be obtained by dividing their product by the generator of the intersection of two principal ideals generated by the polynomials. The second algorithm is based on another simple insight that the gcd can be extracted using the generator of the ideal quotient of a polynomial with respect to the second polynomial. Since the ideal intersection and ideal quotient in these cases are also principal ideals, their generators can be obtained by computing minimal Gröbner bases of the ideal intersection and ideal quotient, respectively. To avoid introducing new variables which can adversely affect the efficiency, minimal Gröbner bases computations are performed on modules. Both of these constructions generalize to the parametric case as shown in the paper. Comprehensive Gröbner system constructions are used for the parametric ideal intersection and ideal quotient using the Kapur-Sun-Wang’s algorithm. It is proved that whether in a minimal comprehensive Gröbner system of a parametric ideal 20intersection or in that of a parametric ideal quotient, each branch of the specializations corresponds to a principal parametric ideal with a single generator. Using this generator, the parametric gcd of that branch is obtained by division. For the case of more than two parametric polynomials, we can use the above two algorithms to compute gcds recursively, and get an extended algorithm by generalizing the idea of the second algorithm. Algorithms do not suffer from having to apply expensive steps such as ensuring whether parametric polynomials are primitive w.r.t. the main variable as used in both the algorithms proposed by Nagasaka (ISSAC, 2017). The resulting algorithms are not only conceptually simple to understand but are more efficient in practice. The proposed algorithms and both of Nagasaka’s algorithms have been implemented in Singular, and their performance is compared on a number of examples. 

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3. Some bounds arising from a polynomial ideal associated to any $$t$$-design

   https://doi.org/10.13069/jacodesmath.729446  

   MARTIN, William J; STINSON, Douglas R (May 2020, Journal of Algebra Combinatorics Discrete Structures and Applications)

   null (Ed.)

   We consider ordered pairs (X,B) where X is a finite set of size v and B is some collection of k-element subsets of X such that every t-element subset of X is contained in exactly λ "blocks'' b ∈B for some fixed λ. We represent each block b by a zero-one vector c_b of length v and explore the ideal I(B) of polynomials in v variables with complex coefficients which vanish on the set { c_b ∣ b ∈ B}. After setting up the basic theory, we investigate two parameters related to this ideal: γ1(B) is the smallest degree of a non-trivial polynomial in the ideal I(B) and γ2(B) is the smallest integer s such that I(B) is generated by a set of polynomials of degree at most s. We first prove the general bounds t/2 < γ1(B) ≤ γ2(B) ≤ k. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have γ2(B) ≤ t. But we expect γ2(B) to be closer to k for less structured designs and we indicate this by constructing infinitely many triple systems satisfying γ2(B) = k. 

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4. Efficient Algorithms for Least Square Piecewise Polynomial Regression

   Lokshtanov, Daniel; Suri, Subhash; Xue, Jie (September 2021, ESA21: Proceedings of European Symposium on Algorithms)

   null (Ed.)

   We present approximation and exact algorithms for piecewise regression of univariate and bivariate data using fixed-degree polynomials. Specifically, given a set S of n data points (x1, y1), . . . , (xn, yn) ∈ Rd × R where d ∈ {1, 2}, the goal is to segment xi’s into some (arbitrary) number of disjoint pieces P1, . . . , Pk, where each piece Pj is associated with a fixed-degree polynomial fj : Rd → R, to minimize the total loss function λk+􏰄ni=1(yi −f(xi))2, where λ ≥ 0 is a regularization term that penalizes model complexity (number of pieces) and f : 􏰇kj=1 Pj → R is the piecewise polynomial function defined as f|Pj = fj. The pieces P1,...,Pk are disjoint intervals of R in the case of univariate data and disjoint axis-aligned rectangles in the case of bivariate data. Our error approximation allows use of any fixed-degree polynomial, not just linear functions. Our main results are the following. For univariate data, we present a (1 + ε)-approximation algorithm with time complexity O(nε log1ε), assuming that data is presented in sorted order of xi’s. For bivariate data, we √ present three results: a sub-exponential exact algorithm with running time nO( n); a polynomial-time constant- approximation algorithm; and a quasi-polynomial time approximation scheme (QPTAS). The bivariate case is believed to be NP-hard in the folklore but we could not find a published record in the literature, so in this paper we also present a hardness proof for completeness. 

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5. Certified Hermite Matrices from Approximate Roots - Univariate Case

   https://doi.org/10.1007/978-3-030-43120-4_1  

   Ayyildiz Akoglu, T; Szanto, A. (March 2020, Part of the Lecture Notes in Computer Science book series (LNCS, volume 11989))

   Let 𝑓1,…,𝑓𝑚 be univariate polynomials with rational coefficients and I:=⟨𝑓1,…,𝑓𝑚⟩⊂ℚ[𝑥] be the ideal they generate. Assume that we are given approximations {𝑧1,…,𝑧𝑘}⊂ℚ[𝑖] for the common roots {𝜉1,…,𝜉𝑘}=𝑉(I)⊆ℂ . In this study, we describe a symbolic-numeric algorithm to construct a rational matrix, called Hermite matrix, from the approximate roots {𝑧1,…,𝑧𝑘} and certify that this matrix is the true Hermite matrix corresponding to the roots V(I) . Applications of Hermite matrices include counting and locating real roots of the polynomials and certifying their existence. 

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