Roman Karasev
We interpret the Combinatorial Nullstellensatz of Noga Alon as a multidimensional residue formula, describe some consequences of this interpretation and related open problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate tha…
We revisit and further explore the celebrated Combinatorial Nullstellensätze of N. Alon in several different directions.
In this note we give a short, direct proof of the well known Combinatorial Nullstellensatz.
We discuss here some computational aspects of the Combinatorial Nullstellensatz argument. Our main result shows that the order of magnitude of the symmetry group associated with permutations of the variables in algebraic constraints, determines the performance of algorithms naturally deduced from Alon's Combinatorial Nullstellensatz arguments. Finally we present a primal-dual polynomial constructions for certifying the existence or the non-existence of solutions to combinatorial problems.
In this note we give an extended version of Combinatorial Nullstellensatz, with weaker assumption on nonvanishing monomial. We also present an application of our result in a situation where the original theorem does not seem to work.
The Generalized Combinatorial Lason-Alon-Zippel-Schwartz Nullstellensatz Lemma
本文综述了Alon组合零点定理与Schwartz-Zippel引理的若干强化与推广，证明了当变量在足够大范围内独立取值时多元多项式非零值的存在性。
Combinatorial Nullstellensatz Techniques
本文介绍多种将组合零点定理用于有限域多项式的技术，推广了Noga Alon的相关定理并提出若干新定理。
Applying techniques similar to Combinatorial Nullstellensatz we prove a lower estimate of $|f(A,B)|$ for finite subsets $A$, $B$ of a field, and polynomial $f(x,y)$ of the form $f(x,y)=g(x)+yh(x)$, where degree of $g$ is greater then degree of $h$.
We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon-Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable graphs. We also extend a result of Akbari, Mirrokni, and Sadjad (2006) on unique list colorability to the context of DP-coloring. We establish a sufficient algebraic condition for a graph $G…
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…
Motivated exposition of combinatorial Nullstellensatz
本文通过启发式方法展示组合判别式定理在解决基础问题中的应用，并引导读者进行推广。
By a well-known theorem of Thomassen and a planar graph depicted by Voigt, we know that every planar graph is $5$-choosable, and the bound is tight. In 1999, Lam, Xu and Liu reduced $5$ to $4$ on $C_4$-free planar graphs. In the paper, by applying the famous Combinatorial Nullstellensatz, we design an effective algorithm to deal with list coloring problems. At the same time, we prove that a planar graph $G$ is $4$-choosable if any two $4$-cycles having distance at least $5$ in $G$, which extends the result of Lam et al.
The long-standing Erdős-Faber-Lovász conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős-Faber-Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations,…
A Gröbner Basis Approach to Combinatorial Nullstellensatz
本文通过刻画首一多项式构成的Gröbner基，从Gröbner基视角建立一般零点定理，并由此得到多重集零点定理、带重数零点定理及穿孔零点定理等推论，进而推广超平面覆盖结果并给出广义Alon-Füredi定理的新证明。
Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert's Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments ba…
Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by systems of polynomial equations satisfying special properties. If the equations are infeasible, Hilbert's Nullstellensatz gives a certificate of this fact. These certificates have been studied and exhibit combinatorial meaning. In this paper, we generalize some known results and show that the Nullstellensatz certificate can be viewed as enumerating combinatorial …
Combinatorial Nullstellensatz and Turán numbers of complete $r$-partite $r$-uniform hypergraphs
本文利用 Lasoń 推广的组合零点定理框架构造完全 r-部 r-一致超图 Turán 数的下界，并给出 Erdős 盒子问题的简短显式构造，证明当 r≤4 时渐近匹配已知最佳界。
The Combinatorial Nullstellensatz, Chevalley-Warning Theorem and weak Finitesatz in skew polynomial rings
该研究针对多元斜多项式环，推广了组合零点定理，并在除环为有限域时，证明了Chevalley-Warning定理、Ax引理及Terjanian有限定理弱情形的斜类比结果。
Repeatedly applying the Combinatorial Nullstellensatz for Zero-sum Grids to Martin Gardner's minimum no-3-in-a-line problem
本文通过重复应用零和网格的组合零点定理，证明了 Martin Gardner 提出的最小无三连线问题在 $n \equiv 1 \pmod 4$ 时下界为 $n+1$ 的猜想。
In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field $\mathbb{k}$. In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3-colorable in the first iteration of this algorithm when $\mathbb{k}=GF(2)$. This greatly simplifies the work of De Loera et. al, as we express t…
Application of the Combinatorial Nullstellensatz to magic-type graph labelings
本文利用组合零点定理，对简单图\(G\)，在给定整数\(k\geq 2\)下，研究边标记\(h\)诱导顶点标记的情况，分析\(G\)成为\(t -\)和\(\mathbb Z_k -\)魔术图以及\(\mathbb{Z}_{k}-\)反魔术图的条件。
Alon's combinatorial Nullstellensatz, and in particular the resulting nonvanishing criterion is one of the most powerful algebraic tools in combinatorics, with many important applications. In this paper we extend the nonvanishing theorem in two directions. We prove a version allowing multiple points. Also, we establish a variant which is valid over arbitrary commutative rings, not merely over subrings of fields. As an application, we prove extensions of the theorem of Alon and Füredi on hyperplane coverings of discrete cubes.
The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi's Combinatorial Nullstellensatz, which provides some information about the polynomial map $P|_{\X_1\times...\times\X_n}$ when only incomplete information about the polynomial $P(X_1,...c,X_n)$ is given. In a very general working frame, the grid points $x\in\X_1\times\...b\times\X_n$ which do not vanish under an algebraic solution -- a certain describing polynomial $P(X_1,...c,X_n)$ -- correspond to the explicit solutions of a problem. As a consequence of the coefficient formula, we prove that th…
The toric residue is a map depending on n+1 semi-ample divisors on a complete toric variety of dimension n. It appears in a variety of contexts such as sparse polynomial systems, mirror symmetry, and GKZ hypergeometric functions. In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to s…
Alon's combinatorial Nullstellensatz (Theorem 1.1 from \cite{Alon1}) is one of the most powerful algebraic tools in combinatorics, with a diverse array of applications. Let $\F$ be a field, $S_1,S_2,..., S_n$ be finite nonempty subsets of $\F$. Alon's theorem is a specialized, precise version of the Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing on the set $S=S_1\times S_2\times ... \times S_n\subseteq \F^n$. From this Alon deduces a simple and amazingly widely applicable nonvanishing criterion (Theorem 1.2 in \cite{Alon1}). It provides a sufficient condition f…
In this article, we use the Combinatorial Nullstellensatz to give new proofs of the Cauchy-Davenport, the Dias da Silva-Hamidoune and to generalize a previous addition theorem of the author. Precisely, this last result proves that for a set A $\subset$ Fp such that A $\cap$ (--A) = $\emptyset$ the cardinality of the set of subsums of at least $α$ pairwise distinct elements of A is: |$Σ$$α$(A)| $\ge$ min (p, |A|(|A| + 1)/2 -- $α$($α$ + 1)/2 + 1) , the only cases previously known were $α$ $\in$ {0, 1}. The Combinatorial Nullstellensatz is used, for the first time, in a direct and in a reverse wa…
In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in $\mathbb{R}^n$, such that all the vertices of $\{0, 1\}^n \setminus \{\vec{0}\}$ are covered at least $k$ times, and $\vec{0}$ is uncovered? The $k=1$ case is the well-known Alon-Füredi theorem which says a minimum of $n$ affine hyperplanes is required, proved by the Combinatorial Nullstellensatz. We develop an analogue of the Lubell-Yamamoto-Meshalkin inequality for subset sums, and completely solve the fractional version of this problem, which also provides an asymptotic answer to the integr…
The coefficient of x^{-1} of a formal Laurent series f(x) is called the formal residue of f(x). Many combinatorial numbers can be represented by the formal residues of hypergeometric terms. With these representations and the extended Zeilberger's algorithm, we generate recurrence relations for summations involving combinatorial sequences such as Stirling numbers. As examples, we give computer proofs of several known identities and derive some new identities. The applicability of this method is also studied.
We generalize a parity result of Fleishner and Stiebitz that being combined with Alon--Tarsi polynomial method allowed them to prove that a 4-regular graph formed by a Hamiltonian cycle and several disjoint triangles is always 3-choosable. Also we present a modification of polynomial method and show how it gives slightly more combinatorial information about colourings than direct application of Alon's Combinatorial Nullstellensatz.
As an application of the Combinatorial Nullstellensatz, we give a short polynomial proof of the q-analogue of Dyson's conjecture formulated by Andrews and first proved by Zeilberger and Bressoud.
Structured and Punctured Nullstellensätze
本文推广零点定理，统一绍茨与尼卡的结果，并利用多项式除法中单项式不变性，将非零计数定理推广至穿孔网格。
As starting point, we formulate a corollary to the Quantitative Combinatorial Nullstellensatz. This corollary does not require the consideration of any coefficients of polynomials, only evaluations of polynomial functions. In certain situations, our corollary is more directly applicable and more ready-to-go than the Combinatorial Nullstellensatz itself. It is also of interest from a numerical point of view. We use it to explain a well-known connection between the sign of 1-factorizations (edge colorings) and the List Edge Coloring Conjecture. For efficient calculations and a better understandi…
Grothendieck polynomials are important objects in the study of the $K$-theory of flag varieties. Their many remarkable properties have been studied in the context of algebraic geometry and tableaux combinatorics. We explore a new tool, similar to generating sequences, which we call the iterated residue technique. We prove new formulas on the calculus of iterated residues and use them to prove straightening laws and multiplication formulas for stable Grothendieck polynomials. As a further application of our method, we give new proofs that the $K$-Pieri rule and the expansions of Grothendieck po…
On the distribution of quadratic residues
该研究运用加性组合方法探究素数域平方数集合的分布，得出当前该集合间隔数的最佳上界，且将结果拓展至小倍增集合，推进了二次剩余分布相关研究。
Let $R$ be a commutative ring with unity and $Z(R)$ and ${\rm Reg}(R)$ be the set of zero-divisors and non-zero zero-divisors of $R$, respectively. We denote by $T(Γ(R))$, the total graph of $R$, a simple graph with the vertex set $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. The induced subgraphs on $Z(R)$ and ${\rm Reg}(R)$ are denoted by $Z(Γ(R))$ and $Reg(Γ(R))$, respectively. These graphs were first introduced by D.F. Anderson and A. Badawi in 2008. In this paper, we prove the following result: let $R$ be a finite ring and one of the following condit…
We present in this article a family of new combinatorial identities via purely differential/complex geometry methods, which include as a speical case a unified and explicit formula for Chern numbers of all complex flag manifolds. Our strategy is to construct concrete circle actions with isolated fixed points on these manifolds and explicitly determine their weights. Then applying Bott's residue formula to these models yields the desired results.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。