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arXiv · 2509.02290

Decidability of polynomial equations over function fields in positive characteristic

Nicolas Daans
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Abstract

Let $K$ be a field of positive characteristic with no algebraically closed subfield. Let $F$ be a function field over $K$ and $t \in F$ transcendental over $K$. Refining a result of Eisentr{ä}ger and Shlapentokh, we show that there is no algorithm which, on input a polynomial $f \in \mathbb{Z}[t][X_1, \ldots, X_n]$, determines whether $f$ has a zero in $F^n$. To this end, we revisit and partially extend several recent results from the literature on existential definability in function fields.

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Nicolas Daans. 2025-12-04. Decidability of polynomial equations over function fields in positive characteristic. https://arxiv.org/abs/2509.02290

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