The Schwartz-Zippel Lemma is a fundamental tool, that provides a bound on the size of the zero set of multivariate polynomials.
In this talk we present a classical proof of the Schwartz-Zippel Lemma based on Gröbner bases and elementary algebraic geometry, interpreting the lemma in terms of varieties, ideals and standard monomials.
One can use the Schwartz-Zippel Lemma to lower bounds the minimum distance of Reed-Muller codes.