Polynomial equations modulo prime numbers

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.

The spirit of the paper was to give simple proofs around the Lang-Weil formula. However, the formula at the second line of paragraph 2.4 was pointed out as incorrect by Mirko Torresani. As a result, we no longer have an elementary proof for formula (3)