Decidability of polynomial equations over function fields in positive characteristic

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 $f\in \mathbb{Z}[t][X_1,\dots ,X_n]$, 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.

Preprint, 20 pages. Lemma 2.1 rewritten under weaker hypotheses