On Zeros of a Polynomial in a Finite Grid

Bishnoi, Anurag; Clark, Pete L.; Potukuchi, Aditya; Schmitt, John R.

doi:10.1017/s0963548317000566

Cited by 30 publications

(22 citation statements)

References 30 publications

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“…, k} d . The latter case, with sets R * defined as in the proof of Lemma 4.12, recovers several, but not all, versions of the Schwartz-Zippel lemma in [BCPS18]. (1) For every 1 ≤ l ≤ d and every…”

Section: Transversalitymentioning

confidence: 78%

Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems

Guo

Zorin-Kranich

2020

Advances in Mathematics

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We prove p L p decoupling inequalities for a class of moment manifolds. These inequalities imply optimal mean value estimates for multidimensional Weyl sums of the kind considered by Arkhipov, Chubarikov, and Karatsuba and by Parsell.In our proofs we take a new point of view on the Bourgain-Demeter-Guth induction on scales argument. This point of view substantially simplifies even the proof of 2 L p decoupling for the moment curve.

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Section: Transversalitymentioning

confidence: 78%

Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems

Guo

Zorin-Kranich

2020

Advances in Mathematics

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“…In this subsection, we obtain a weaker but more concise version of the bound (3.1) for a single polynomial, which has as particular cases the bounds by DeMillo and Lipton [8], Zippel [26,Theorem 1], [27,Proposition 3] and Alon and Füredi [2,Theorem 5]. We observe that Counterexample 7.4 in [4] shows that a straightforward extension of these bounds to standard multiplicities as in (1.1) is not possible, in contrast with the bound given by Schwartz in [25,Lemma 1], which has been already extended in [9,Lemma 8]. Proof.…”

Section: Bounds By Demillo Lipton Zippel Alon and Füredimentioning

confidence: 85%

“…The bound in [25,Lemma 1] can also be derived by those given by DeMillo and Lipton [8], and Zippel [26,Theorem 1], [27,Proposition 3] (see Proposition 5.5 below), and is referred to as the Schwartz-Zippel bound in many works in the literature [9,12,19,20]. Interestingly, an extension of such a bound for standard multiplicities in the sense of (1.1) has recently been given in [9,Lemma 8], but as Counterexample 7.4 in [4] shows, no straightforward extension of the footprint bound in the sense of (1.1) seems possible (recall that we will give a footprint bound in the sense of (1.2)). To conclude this work, we give an extension of the Schwartz-Zippel bound in the sense of (1.1) to derivatives with weighted order less than a given positive integer, which we will call weighted multiplicities.…”

Section: Introductionmentioning

confidence: 98%

Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

Geil¹,

Martínez-Peñas²

2018

Combinator. Probab. Comp.

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We upper bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations on the parameters of evaluation codes with consecutive derivatives [19], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [7], Schwartz [24], Zippel [25,26], and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection with our extension of the footprint bound.If V ≥r (F (x)) denotes the set of zeros of F (x) of multiplicity at least r, then a weaker, but still sharp, bound is the following:( 2) In the multivariate case, the standard approach is to consider the first r consecutive Hasse derivatives as those whose multiindices have order less than r, where the order of a multiindex (i 1 , i 2 , . . . , i m ) is defined as m j=1 i j . We will use the terms standard multiplicities to refer to this type of multiplicities. In this work, we consider arbitrary finite families J of multiindices that are consecutive in a coordinate-wise sense: if (i 1 , i 2 , . . . , i m ) belongs to J and k j ≤ i j , for j = 1, 2, . . . , m, then (k 1 , k 2 , . . . , k m ) also belongs to J . Obviously, the (finite) family J of multiindices of order less than a given positive integer r satisfies this property, hence is a particular case.Our main contribution is an upper bound on the number of common zeros over a grid of a family of polynomials and their (Hasse) derivatives corresponding to a finite set J of consecutive multiindices. This upper bound makes use of the technique from Gröbner basis theory known as footprint [10,15], and can be seen as an extension of the classical footprint bound [6, Section 5.3] in the sense of (2). A first extension for standard multiplicities has been given as Lemma 2.4 in the expanded version of [23].We will then show that this bound is sharp for ideals of polynomials, characterize those which satisfy equality, and give as applications extensions of known results in algebra and combinatorics: Alon's combinatorial Nullstellensatz [1,3,5,20,22], existence and uniqueness of Hermite interpolating polynomials [9,18,21], estimations on the parameters of evaluation codes with consecutive derivatives [11,18,19], and the bounds by DeMillo and Lipton [7], Zippel [25,26], and Alon and Füredi [2], and a particular case of the bound given by Schwartz in [24, Lemma 1].The bound in [24, Lemma 1] can also be derived by those given by DeMillo and Lipton [7], and Zippel [25, Theorem 1], [26, Proposition 3] (see Proposition 3 below), and is referred to as the Schwartz-Zippel bound in many works i...

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“…The article by Bishnoi et al [7], which also mentions Ore's paper, contains a nice section on various versions of this lemma and its connections to the Alon-Füredi theorem [2]. Lipton's blog too [15] has an interesting discussion on the Polynomial Identity Lemma.…”

Section: The Polynomial Identity Lemmamentioning

confidence: 99%

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Arvind¹,

Joglekar²,

Mukhopadhyay³

et al. 2019

Theory of Comput.

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In this paper we show that black-box polynomial identity testing (PIT) for nvariate noncommutative polynomials f of degree D with at most t nonzero monomials can be done in randomized poly(n, logt, log D) time, and consequently in randomized poly(n, logt, s) time if f is computable by a circuit of size s. This result makes progress on a question that has been open for over a decade. Our algorithm is based on efficiently isolating a monomial using nondeterministic automata. The above result does not yield an efficient randomized PIT for noncommutative circuits in general, as noncommutative circuits of size s can compute polynomials with a doubleexponential (in s) number of monomials. As a first step, we consider a natural class of homogeneous noncommutative circuits, that we call +-regular circuits, and give a white-box polynomial-time deterministic PIT for them. These circuits can compute noncommutative polynomials with number of monomials double-exponential in the circuit size. Our algorithm combines some new structural results for +-regular circuits with known PIT results for noncommutative algebraic branching programs, a rank bound for commutative depth-3 A preliminary version of this paper appeared in the Proceedings of the 49th ACM Symp. on Theory of Computing (STOC'17) [4]. * Part of the work was done while the author was a postdoctoral fellow at Chennai Mathematical Institute, India.

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On Zeros of a Polynomial in a Finite Grid

Cited by 30 publications

References 30 publications

Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems

Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems

Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

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