Betti number bounds for varieties and exponential sums
Daqing Wan, Dingxin Zhang
TL;DR
The paper develops sharp, computable bounds for compactly supported Betti numbers of affine varieties defined by few polynomials or by complete intersections, using perverse sheaves and specialization principles. It connects these topological bounds to arithmetic consequences, notably improved total degree bounds for zeta functions and L-functions of exponential sums, including toric and non-toric cases, by lifting cohomological estimates to the level of L-functions. A central innovation is a specialization lemma for Betti numbers and Newton polygons, enabling upper semicontinuity results and allowing inductive estimates via Lefschetz-type reductions. The results yield asymptotically optimal bounds in the complete-intersection setting and deliver Lang–Weil type error terms with explicit constants, thereby strengthening classical bounds of Bombieri, Katz, and Adolphson–Sperber. The framework also unifies and enhances several prior approaches, providing both general theory and concrete corollaries for exponential sums over finite fields and for zeta/L-function degree bounds.
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 $d$.