The $p$-powers dividing certain exponential sums
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908999997194240 |
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| author | Phigareau, Antonine |
| author_facet | Phigareau, Antonine |
| contents | We define the notion of couple density $(D, \mathbf b)$ where $D$ is a non-empty subset of $\mathbb Z^{m}$ and $ \mathbf b$ a fixed element in $\{0, \cdots, q-2\}^{m};$ We determine a minimum in terms of the density of the couple $(D,\mathbf b)$ for the $q$-adic valuation of the sum $ S_{\ell}(F,\mathbf b)$ with $F$ a Laurent polynomial. And we show that this minimum is a bound for the $q$-adic valuation of the zeros and poles of the associated $L$-function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26234 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $p$-powers dividing certain exponential sums Phigareau, Antonine Number Theory 11T23, 11T24, 11M38 We define the notion of couple density $(D, \mathbf b)$ where $D$ is a non-empty subset of $\mathbb Z^{m}$ and $ \mathbf b$ a fixed element in $\{0, \cdots, q-2\}^{m};$ We determine a minimum in terms of the density of the couple $(D,\mathbf b)$ for the $q$-adic valuation of the sum $ S_{\ell}(F,\mathbf b)$ with $F$ a Laurent polynomial. And we show that this minimum is a bound for the $q$-adic valuation of the zeros and poles of the associated $L$-function. |
| title | The $p$-powers dividing certain exponential sums |
| topic | Number Theory 11T23, 11T24, 11M38 |
| url | https://arxiv.org/abs/2604.26234 |