An Application of the combinatorial Nullstellensatz to a graph labelling problem

THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972720001197 ◽

2020 ◽

pp. 1-10

Author(s):

NGUYEN CONG MINH ◽

LUU BA THANG ◽

TRAN NAM TRUNG

Keyword(s):

Polynomial Ring ◽

Polynomial System ◽

Combinatorial Nullstellensatz ◽

Roots Of A Polynomial

Abstract Let I be a zero-dimensional ideal in the polynomial ring $K[x_1,\ldots ,x_n]$ over a field K. We give a bound for the number of roots of I in $K^n$ counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.

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GRAPH LABELLING TECHNIQUES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000667 ◽

2018 ◽

Vol 98 (3) ◽

pp. 512-513

Author(s):

DUSHYANT KIRITBHAI TANNA

Keyword(s):

Graph Labelling

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Combinatorial Nullstellensatz Modulo Prime Powers and the Parity Argument

The Electronic Journal of Combinatorics ◽

10.37236/4124 ◽

2014 ◽

Vol 21 (4) ◽

Cited By ~ 1

Author(s):

László Varga

Keyword(s):

Complexity Class ◽

Combinatorial Nullstellensatz ◽

Search Problem ◽

Search Problems ◽

Computational Search

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA).

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Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

The Electronic Journal of Combinatorics ◽

10.37236/734 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 12

Author(s):

Uwe Schauz

Keyword(s):

Combinatorial Nullstellensatz ◽

Algebraic Solution ◽

Explicit Solutions ◽

Polynomial Map ◽

Combinatorial Number Theory ◽

Special Cases ◽

The Matrix ◽

Grid Points ◽

Algebraic Solutions ◽

Solvable Problems

The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map $P|_{\mathfrak{X}_1\times\cdots\times\mathfrak{X}_n}$ when only incomplete information about the polynomial $P(X_1,\dots,X_n)$ is given.In a very general working frame, the grid points $x\in \mathfrak{X}_1\times\cdots\times\mathfrak{X}_n$ which do not vanish under an algebraic solution – a certain describing polynomial $P(X_1,\dots,X_n)$ – correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that "owns" both, a set ${\cal S}$, which may be called the set of solutions; and a subset ${\cal S}_{\rm triv}\subseteq{\cal S}$, the set of trivial solutions.We give several examples of how to find algebraic solutions, and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries.We obtain a permanent formula by applying our coefficient formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of:1. Ryser's permanent formula.2. Alon's Permanent Lemma.3. Alon and Tarsi's Theorem about orientations and colorings of graphs.Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of planar $n$-regular graphs, the formula contains as very special cases:4. Scheim's formula for the number of edge $n$-colorings of such graphs.5. Ellingham and Goddyn's partial answer to the list coloring conjecture.

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

Combinatorics Probability Computing ◽

10.1017/s0963548318000342 ◽

2018 ◽

Vol 28 (2) ◽

pp. 253-279

Author(s):

O. GEIL ◽

U. MARTÍNEZ-PEÑAS

Keyword(s):

Upper Bound ◽

Existence And Uniqueness ◽

Combinatorial Nullstellensatz ◽

Finite Collection ◽

Multivariate Polynomials ◽

Grobner Basis ◽

Evaluation Codes ◽

Number Of Zeros ◽

Interpolating Polynomials ◽

Common Zeros

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 of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

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