On sets of rational functions which locally represent all of $\mathbb{Q}$
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929460528283648 |
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| author | Klahn, Benjamin König, Joachim |
| author_facet | Klahn, Benjamin König, Joachim |
| contents | We investigate finite sets of rational functions $\{ f_{1},f_{2}, \dots, f_{r} \}$ defined over some number field $K$ satisfying that any $t_{0} \in K$ is a $K_{p}$-value of one of the functions $f_{i}$ for almost all primes $p$ of $K$. We give strong necessary conditions on the shape of functions appearing in a minimal set with this property, as well as numerous concrete examples showing that these necessary conditions are in a way also close to sufficient. We connect the problem to well-studied concepts such as intersective polynomials and arithmetically exceptional functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_12630 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On sets of rational functions which locally represent all of $\mathbb{Q}$ Klahn, Benjamin König, Joachim Number Theory We investigate finite sets of rational functions $\{ f_{1},f_{2}, \dots, f_{r} \}$ defined over some number field $K$ satisfying that any $t_{0} \in K$ is a $K_{p}$-value of one of the functions $f_{i}$ for almost all primes $p$ of $K$. We give strong necessary conditions on the shape of functions appearing in a minimal set with this property, as well as numerous concrete examples showing that these necessary conditions are in a way also close to sufficient. We connect the problem to well-studied concepts such as intersective polynomials and arithmetically exceptional functions. |
| title | On sets of rational functions which locally represent all of $\mathbb{Q}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.12630 |