The Combinatorial Nullstellensätze Revisited

Clark, Pete L.

doi:10.37236/4359

Cited by 13 publications

(10 citation statements)

References 16 publications

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“…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 [1,3,6,21,23], existence and uniqueness of Hermite interpolating polynomials [10,19,22], estimations of the parameters of evaluation codes with consecutive derivatives [12,19,20], and the bounds by DeMillo and Lipton [8], Zippel [26,27] and Alon and Füredi [2], and a particular case of the bound given by Schwartz in [25,Lemma 1].…”

Section: Introductionmentioning

confidence: 88%

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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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Section: Introductionmentioning

confidence: 88%

“…where ρ is the projection to the quotient ring. We may then extend the notion of reduction of a polynomial as follows (see [6,Section 3.1] and [10, Section 6.3], for instance). Given F(x) ∈ F[x], we define its reduction over the set S with derivatives in J as…”

Section: Hermite Interpolation Over Grids With Consecutive Derivativesmentioning

confidence: 99%

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

Geil¹,

Martínez-Peñas²

2018

Combinator. Probab. Comp.

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“…Moreover, (3) certainly implies that #Z X = 1, so we recover Theorem 1.3b). Thus Theorem 1.4 is a simultaneous generalization of the Chevalley-Warning Theorem and the Restricted Variable Chevalley Theorem, so in [Cl14] this result is called the "Restricted Variable Chevalley-Warning Theorem. "…”

Section: Supposementioning

confidence: 99%

“…The second author has taken the perspective (e.g. in [Cl14]) that Theorem 1.1a) is a precursor of the following celebrated result.…”

Section: Introductionmentioning

confidence: 99%

Restricted Variable Chevalley-Warning Theorems

Bishnoi¹,

Clark²

2022

Preprint

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We pursue various restricted variable generalizations of the Chevalley-Warning theorem for low degree polynomial systems over a finite field. Our first such result involves variables restricted to Cartesian products of the Vandermonde subsets of Fq defined by Gács-Weiner and Sziklai-Takáts. We then define an invariant ω(X) of a nonempty subset of F n q . Our second result involves X-restricted variables when the degrees of the polynomials are small compared to ω(X). We end by exploring various classes of subsets for which ω(X) can be bounded from below.1 Some further information about the history of these results can be found in [CGS21, §1].

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“…Following [12] and [32], a non-empty subset S ⊂ R is said to satisfy Condition (D) if, for all x = y ∈ S, the element x − y ∈ R is not a zero divisor. A finite grid is a subset A = ∏ n i=1 A i of R n (for some n ∈ Z + ) with each A i a finite, non-empty subset of R. We say that A satisfies Condition (D) if each A i does.…”

Section: Notationmentioning

confidence: 99%

On Zeros of a Polynomial in a Finite Grid

Bishnoi¹,

Clark²,

Potukuchi³

et al. 2018

Combinator. Probab. Comp.

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A 1993 result of Alon and Füredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to polynomials over an arbitrary commutative ring, assuming a certain "Condition (D)" on the grid which holds vacuously when the ring is a domain. In the first half of this paper we give a further Generalized Alon-Füredi Theorem which provides a sharp upper bound when the degrees of the polynomial in each variable are also taken into account. This yields in particular a new proof of Alon-Füredi. We then discuss the relationship between Alon-Füredi and results of DeMillo-Lipton, Schwartz and Zippel. A direct coding theoretic interpretation of Alon-Füredi Theorem and its generalization in terms of Reed-Muller type affine variety codes is shown which gives us the minimum Hamming distance of these codes. Then we apply the Alon-Füredi Theorem to quickly recover -and sometimes strengthen -old and new results in finite geometry, including the Jamison/Brouwer-Schrijver bound on affine blocking sets. We end with a discussion of multiplicity enhancements.

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The Combinatorial Nullstellensätze Revisited

Cited by 13 publications

References 16 publications

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

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

Restricted Variable Chevalley-Warning Theorems

On Zeros of a Polynomial in a Finite Grid

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