On Bias and Rank

TL;DR

This paper extends Dimca's cohomology results for hypersurfaces from complex numbers to finite characteristic fields using étale cohomology, linking algebraic properties of polynomials to their zero set sizes.

Contribution

It proves an analog of Dimca's theorem over finite fields, connecting algebraic geometry, étale cohomology, and counting solutions over finite fields.

Findings

Cohomology of hypersurfaces over finite fields matches projective space in certain ranges.

Relates algebraic properties of polynomials to the size of their zero sets.

Uses Weil conjectures to connect étale cohomology results to counting solutions.

Abstract

Given a hypersurface $X\subset \mathbb{P}^{N+1}_{\mathbb{C}}$ 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 $\mathbb{C}$ is replaced with an algebraically closed field of finite characteristic and singular cohomology is replaced with $\ell$-adic \'etale cohomology. The Weil conjectures allow relating results about \'eatle 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.