A lower bound on the analytic log-canonical threshold over local fields of positive characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915592495169536 |
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| author | Glazer, Itay Hendel, Yotam I. |
| author_facet | Glazer, Itay Hendel, Yotam I. |
| contents | Given a local field $F$ of positive characteristic, an $F$-analytic manifold $X$ and an analytic function $f:X\rightarrow F$, the $F$-analytic log-canonical threshold $\mathrm{lct}_{F}(f;x_{0})$ is the supremum over the values $s\geq0$ such that $\left|f\right|_{F}^{-s}$ is integrable near $x_{0}\in X$. We show that $\mathrm{lct}_{F}(f;x_{0})>0$. Moreover, if $f$ is a regular function on a smooth algebraic $F$-variety, we obtain an effective lower bound $\mathrm{lct}_{F}(f;x_{0})>C$, where $C>0$ is explicit and depends only on the complexity class of $X$ and $f$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A lower bound on the analytic log-canonical threshold over local fields of positive characteristic Glazer, Itay Hendel, Yotam I. Algebraic Geometry Logic 14B05 (Primary) 14G20, 11S40, 11S80 (Secondary) Given a local field $F$ of positive characteristic, an $F$-analytic manifold $X$ and an analytic function $f:X\rightarrow F$, the $F$-analytic log-canonical threshold $\mathrm{lct}_{F}(f;x_{0})$ is the supremum over the values $s\geq0$ such that $\left|f\right|_{F}^{-s}$ is integrable near $x_{0}\in X$. We show that $\mathrm{lct}_{F}(f;x_{0})>0$. Moreover, if $f$ is a regular function on a smooth algebraic $F$-variety, we obtain an effective lower bound $\mathrm{lct}_{F}(f;x_{0})>C$, where $C>0$ is explicit and depends only on the complexity class of $X$ and $f$. |
| title | A lower bound on the analytic log-canonical threshold over local fields of positive characteristic |
| topic | Algebraic Geometry Logic 14B05 (Primary) 14G20, 11S40, 11S80 (Secondary) |
| url | https://arxiv.org/abs/2511.01270 |