On Bias and Rank

Abstract

Given a hypersurface $X\subset {\mathbb{P}}_{\mathbb{C}}^{N+1}$ Dimca gave a proof showing that the cohomologies of X are the same as the projective space in a range determined by the dimension of the singular locus of X. We prove the analog of Dimca's result case when $\mathbb{C}$ is replaced with an algebraically closed field of finite characteristic and singular cohomology is replaced with $\ell$-adic étale cohomology. The Weil conjectures allow relating results about éatle cohomology to counting problems over a finite field. Thus by applying this result, we are able to get a relationship between the algebraic properties of certain polynomials and the size of their zero set.