Combinatorial Nullstellensatz

2021 ◽  
pp. 7-16
Author(s):  
Xuding Zhu ◽  
R. Balakrishnan
Keyword(s):  
Combinatorial Nullstellensatz

THE NUMBER OF ROOTS OF A POLYNOMIAL SYSTEM

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.

scholarly journals Combinatorial Nullstellensatz Modulo Prime Powers and the Parity Argument

10.37236/4124 ◽  
2014 ◽  
Vol 21 (4) ◽  
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).

scholarly journals Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions

10.37236/734 ◽  
2008 ◽  
Vol 15 (1) ◽  
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.

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

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.

Neighbor sum distinguishing chromatic index of sparse graphs via the combinatorial Nullstellensatz

2018 ◽  
Vol 34 (1) ◽  
pp. 135-144
Author(s):  
Xiao-wei Yu ◽  
Yu-ping Gao ◽  
Lai-hao Ding
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Chromatic Index ◽  
Sparse Graphs

scholarly journals Proof of the List Edge Coloring Conjecture for Complete Graphs of Prime Degree

10.37236/4084 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Group Action ◽  
Edge Coloring ◽  
Latin Squares ◽  
Combinatorial Nullstellensatz ◽  
Complete Graphs ◽  
Chromatic Index ◽  
Prime Degree ◽  
List Edge Coloring ◽  
Class 1

We prove that the list-chromatic index and paintability index of $K_{p+1}$ is $p$, for all odd primes $p$. This implies that the List Edge Coloring Conjecture holds for complete graphs with less then 10 vertices. It also shows that there are arbitrarily big complete graphs for which the conjecture holds, even among the complete graphs of class 1. Our proof combines the Quantitative Combinatorial Nullstellensatz with the Paintability Nullstellensatz and a group action on symmetric Latin squares. It displays various ways of using different Nullstellensätze. We also obtain a partial proof of a version of Alon and Tarsi's Conjecture about even and odd Latin squares.

Combinatorial Nullstellensatz

2021 ◽  
Author(s):  
Xuding Zhu ◽  
R. Balakrishnan
Keyword(s):  
Combinatorial Nullstellensatz

Combinatorial Nullstellensatz approach to polynomial expansion

2014 ◽  
Vol 165 (3) ◽  
pp. 279-282 ◽  
Author(s):  
Fedor Petrov
Keyword(s):  
Polynomial Expansion ◽  
Combinatorial Nullstellensatz

Neighbor distinguishing total choice number of sparse graphs via the Combinatorial Nullstellensatz

2016 ◽  
Vol 32 (2) ◽  
pp. 537-548 ◽  
Author(s):  
Cun-quan Qu ◽  
Lai-hao Ding ◽  
Guang-hui Wang ◽  
Gui-ying Yan
Keyword(s):  
Combinatorial Nullstellensatz ◽  
Sparse Graphs ◽  
Choice Number

scholarly journals Proof of the Combinatorial Nullstellensatz over Integral Domains, in the Spirit of Kouba

10.37236/463 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Heinig
Keyword(s):  
General Result ◽  
Duality Theory ◽  
Integral Domain ◽  
Present Note ◽  
Direct Proof ◽  
Vector Spaces ◽  
Combinatorial Nullstellensatz ◽  
Integral Domains ◽  
Sole Purpose ◽  
Cramer’S Rule

It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba [A duality based proof of the Combinatorial Nullstellensatz, Electron. J. Combin. 16 (2009), #N9] one obtains a direct proof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz for polynomials over an arbitrary integral domain. The proof relies on Cramer's rule and Vandermonde's determinant to explicitly describe a map used by Kouba in terms of cofactors of a certain matrix. That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon's work and emphasized in recent articles of Michałek and Schauz; the sole purpose of the present note is to point out that not only is it not necessary to invoke duality of vector spaces, but by not doing so one easily obtains a more general result.

Sign in / Sign up

Export Citation Format

Share Document