Proof of the Combinatorial Nullstellensatz over Integral Domains, in the Spirit of Kouba
It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba [A duality based proof of the Combinatorial Nullstellensatz, Electron. J. Combin. 16 (2009), #N9] one obtains a direct proof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz for polynomials over an arbitrary integral domain. The proof relies on Cramer's rule and Vandermonde's determinant to explicitly describe a map used by Kouba in terms of cofactors of a certain matrix.
That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon's work and emphasized in recent articles of Michałek and Schauz; the sole purpose of the present note is to point out that not only is it not necessary to invoke duality of vector spaces, but by not doing so one easily obtains a more general result.