Combinatorial Nullstellensatz

Combinatorial Nullstellensatz

We present a general algebraic technique and discuss some of its numerous applications in Combinatorial Number Theory, in Graph Theory and in Combinatorics. These applications include results in additive number theory and in the study of graph coloring problems. Many of these are known results, to which we present unified proofs, and some results are new.

Combinatorial Nullstellensatz

The Combinatorial Nullstellensatz is a theorem about the roots of a polynomial. It is related to Hilbert’s Nullstellensatz. Established in 1996 by Alon et al. [4] and generalized in 1999 by Alon [2], the Combinatorial Nullstellensatz is a powerful tool that allows the use of polynomials to solve problems in number theory and graph theory. This article introduces the Combinatorial Nullstellensat...

Circular choosability via combinatorial Nullstellensatz

A p-list assignment L of a graph G assigns to each vertex v of G a set L(v) ⊆ {0, 1, . . . , p − 1} of permissible colors. We say G is L-(p, q)-colorable if G has a (p, q)-coloring h such that h(v) ∈ L(v) for each vertex v. The circular list chromatic number χc,l(G) of a graph G is the infimum of those real numbers t for which the following holds: For any p, q, for any p-list assignment L with ...

A Generalization of Combinatorial Nullstellensatz

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.

Applications of the Combinatorial Nullstellensatz