On sets of rational functions which locally represent all of $\mathbb{Q}$

Benjamin Klahn; Joachim König

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arXiv · 2306.12630

On sets of rational functions which locally represent all of $\mathbb{Q}$

Benjamin Klahn·Joachim König

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Abstract

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.

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BibTeX RIS

Benjamin Klahn, Joachim König. 2024-08-16. On sets of rational functions which locally represent all of $\mathbb{Q}$. https://arxiv.org/abs/2306.12630

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