The combinatorial Nullstellensatz is a nonvanishing theorem for polynomials on finite Cartesian grids. Let ![f in K[x_1,...,x_n]](/images/equations/CombinatorialNullstellensatz/Inline1.svg)
be a polynomial over a field 
,
and suppose the coefficient of
 |
is nonzero, where 
. If 
is a finite subset of 
with 
for each 
, then there are 
such that
 |
The theorem is a combinatorial analogue of Hilbert's Nullstellensatz and has many applications to additive combinatorics, graph coloring, and restricted-variable polynomial problems. Aichinger et al. (2026) developed structured versions that exclude additional monomials and punctured versions for grids with holes, together with corresponding lower bounds on the number of nonzeros.
