Existential theories of henselian valued fields under a formal smoothness assumption

TL;DR

This paper investigates the existential theories of henselian valued fields with positive characteristic, relaxing previous assumptions by using formal smoothness and linking the theory to that of the residue field, with applications to function fields.

Contribution

It introduces a new approach to axiomatize the existential theory of henselian valued fields under weaker algebraic conditions, extending prior results.

Findings

Axiomatization of existential theories in terms of residue fields.

Decidability results for universal and existential sentences in function field completions.

Application to asymptotic theories of function fields over pseudo-algebraically closed fields.

Abstract

We study existential theories of henselian valued fields of positive characteristic with parameters from a trivially valued subfield. Compared to previous work, we relax perfectness and separability assumptions, and instead work with the weaker algebraic hypothesis of formal smoothness over the parameter field, which we discuss in detail in our setting. Assuming a weak consequence of resolution of singularities, which was already used in previous work, we obtain an axiomatisation of the existential theory of such a valued field in terms of the existential theory of the residue field, both over the same parameter field. This result has natural applications to asymptotic theories of completions of function fields of curves. We work these out in detail for the case of function fields over fairly general pseudo-algebraically closed fields, where we obtain decidability of the sets of universal/existential sentences holding in all completions or all but finitely many completions, respectively.