 
Summary: Combinatorial Nullstellensatz
Noga Alon
Abstract
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 in
clude 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.
1 Introduction
Hilbert's Nullstellensatz (see, e.g., [58]) is the fundamental theorem that asserts that if F is an
algebraically closed field, and f, g1, . . . , gm are polynomials in the ring of polynomials F[x1, . . . , xn],
where f vanishes over all common zeros of g1, . . . , gm, then there is an integer k and polynomials
h1, . . . , hm in F[x1, . . . , xn] so that
fk
=
n
i=1
higi.
In the special case m = n, where each gi is a univariate polynomial of the form sSi
(xi  s), a
stronger conclusion holds, as follows.
