1 research outputs found
Alon's Nullstellensatz for multisets
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, 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 for a
polynomial which guarantees that is not identically zero
on the set . In this paper we extend these two results from sets of points
to multisets. We give two different proofs of the generalized nonvanishing
theorem. We extend some of the known applications of the original nonvanishing
theorem to a setting allowing multiplicities, including the theorem of Alon and
F\"uredi on the hyperplane coverings of discrete cubes.Comment: Submitted to the journal Combinatorica of the J\'anos Bolyai
Mathematical Society on August 5, 201