Polynomial equations modulo prime numbers

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.