Counting rational points on transcendental curves in valued fields
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918082594734080 |
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| author | Vermeulen, Floris |
| author_facet | Vermeulen, Floris |
| contents | We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting rational points on transcendental curves in valued fields Vermeulen, Floris Number Theory Logic We prove upper bounds on the number of rational points on transcendental curves in arbitrary $1$-h-minimal fields, similar to the Pila--Wilkie counting theorem in the o-minimal setting. These results extend results due to Cluckers--Comte--Loeser from $p$-adic fields to arbitrary valued fields of mixed characteristic. Our methods rely on parametrizations, where we avoid the usage of $r$-th power maps, combined with the determinant method. |
| title | Counting rational points on transcendental curves in valued fields |
| topic | Number Theory Logic |
| url | https://arxiv.org/abs/2506.19411 |