Betti number bounds for varieties and exponential sums

2025/01/22 by Daqing Wan, Dingxin Zhang, Wan, Daqing +1
Mathematics · #11M38 #11T23 #14F20 #55N10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.12623

openalex publication_date 2025/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using basic properties of perverse sheaves, we give new upper bounds for compactly supported Betti numbers for arbitrary affine varieties in \mathbbAⁿ 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.

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