Decidability of polynomial equations over function fields in positive characteristic
Nicolas Daans
TL;DR
This work proves that the positive-existential theory of polynomial equations over function fields F of positive characteristic p>0 (with no algebraically closed subfield) is undecidable when coefficients come from Z[t]. The authors develop an existential definition of the Frobenius orbit and construct a parameter-free valuation-like predicate definable in F, enabling a reduction from Hilbert’s tenth problem to solvability questions in F. A key innovation is handling both odd p and p=2 with careful genus-based and Artin–Schreier technical machinery, plus a descent framework for definability that preserves the needed parameters. The result sharpens the understanding of how decidability depends on the base ring of coefficients and provides a robust template for further (un)decidability results in function fields.
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.