Summary: 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 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.
Hilbert's Nullstellensatz (see, e.g., ) 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
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.