Sharp Effective Finite-Field Nullstellensatz

The (weak) Nullstellensatz over finite fields says that if $P1,\ldots,Pm$ are $n$-variate degree-$d$ polynomials with no common zero over a finite field $\mathbb{F}$ then there are polynomials $R1,\ldots,Rm$ such that $R1P1+\cdots+RmPm \equiv 1$. Green and Tao [Contrib. Discrete Math. 2009, 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 $|\mathbb{F}|$. In this paper we use the polynomial method to give a proof with a degree bound of $md(|\mathbb{F}|-1)$. 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.