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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.12623 |
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| _version_ | 1866912199267581952 |
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| author | Wan, Daqing Zhang, Dingxin |
| author_facet | Wan, Daqing Zhang, Dingxin |
| contents | 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 $d$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12623 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Betti number bounds for varieties and exponential sums Wan, Daqing Zhang, Dingxin Algebraic Geometry Number Theory 14F20, 11M38, 11T23, 55N10 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 $d$. |
| title | Betti number bounds for varieties and exponential sums |
| topic | Algebraic Geometry Number Theory 14F20, 11M38, 11T23, 55N10 |
| url | https://arxiv.org/abs/2501.12623 |