Sharp Effective Finite-Field Nullstellensatz

TL;DR

This paper provides a sharp, explicit degree bound for the Nullstellensatz over finite fields using the polynomial method, improving previous bounds and establishing their optimality, with a generalization to arbitrary subsets.

Contribution

It introduces a new proof with a precise degree bound for the Nullstellensatz over finite fields and extends the result to arbitrary subsets in fields.

Findings

Degree bound of $md(| ext{field}|-1)$ for the Nullstellensatz

Bound is optimal up to a constant factor

Generalization to arbitrary subsets in fields

Abstract

The (weak) Nullstellensatz over finite fields says that if $P_1,\ldots,P_m$ are $n$-variate degree-$d$ polynomials with no common zero over a finite field $\mathbb{F}$ then there are polynomials $R_1,\ldots,R_m$ such that $R_1P_1+\cdots+R_mP_m \equiv 1$. 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 $R_i$ 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 $P_i$ take finitely many values on said subset.