Polynomial equations modulo prime numbers

TL;DR

This paper presents an elementary counting approach to polynomial equations modulo primes, providing a weak Lang-Weil bound and extending results to systems and varieties using reduction techniques.

Contribution

It introduces a simple counting method to derive a weak Lang-Weil bound and develops a reduction lemma to handle systems of equations and varieties.

Findings

Derived a weak form of the Lang-Weil bound for solutions modulo p

Extended results from single equations to systems of equations

Proved the full Lang-Weil estimate for varieties assuming hypersurfaces

Abstract

We consider polynomial equations, or systems of polynomial equations, with integer coefficients, modulo prime numbers $p$. We offer an elementary approach based on a counting method. The outcome is a weak form of the Lang-Weil lower bound for the number of solutions modulo $p$, only differing from Lang-Weil by an asymptotic $p^\epsilon$ multiplicative factor. Our second contribution is a reduction lemma to the case of a single equation which we use to extend our results to systems of equations. We show further how to use this reduction to prove the full Lang-Weil estimate for varieties, assuming it for hypersurfaces, in a version using a variant of the classical degree in the error term.