On Bias and Rank

Abstract

Given a hypersurface X ⊂ P N +1 C 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 C is replaced with an algebraically closed field of finite characteristic and singular cohomology is replaced with ℓ-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.

Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials

“…The first three examples in this list follow from [2, Theorem A]. The one about cohomology follows from the one about smoothness and a result of Dimca; see [37].…”