Betti number bounds for varieties and exponential sums

TL;DR

This paper establishes new upper bounds for Betti numbers of affine varieties using perverse sheaves, leading to improved degree bounds for zeta functions and exponential sums over finite fields, with optimal bounds in some cases.

Contribution

It introduces novel Betti number bounds for affine varieties and applies these to derive sharper degree bounds for zeta functions and exponential sums, improving classical results.

Findings

New Betti number bounds for affine varieties.

Improved degree bounds for zeta functions and L-functions.

Asymptotically optimal bounds in the complete intersection case.

Abstract

Using basic properties of perverse sheaves, we give new upper bounds for compactly supported Betti numbers for arbitrary affine varieties in $\mathbb{A}^n$ defined by $r$ polynomial equations of degrees at most $d$. As arithmetic applications, new total degree bounds are obtained for zeta functions of varieties and L-functions of exponential sums over finite fields, improving the classical results of Bombieri, Katz, and Adolphson--Sperber. In the complete intersection case, our total Betti number bound is asymptotically optimal as a function in $d$. In general, it remains an open problem to find an asymptotically optimal bound as a function in $d$.