Sharp Effective Finite-Field Nullstellensatz

Abstract

The (weak) Nullstellensatz over finite fields says that if are n-variate degree-d polynomials with no common zero over a finite field then there are polynomials such that . Green and Tao [14, Proposition 9.1] used a regularity lemma to obtain an effective proof, showing that the degrees of the polynomials Ri can be bounded independently of n, though with an Ackermann-type dependence on the other parameters m, d, and . In this paper we use the polynomial method to give a proof with a degree bound of . We also show that the dependence on each of the parameters is the best possible up to an absolute constant. We further include a generalization, offered by Pete L. Clark, from finite fields to arbitrary subsets in arbitrary fields, provided the polynomials Pi take finitely many values on said subset.