Undecidability of expansions of Laurent series fields by cyclic discrete subgroups

TL;DR

This paper extends the known undecidability results of Laurent series fields with a predicate for natural powers to fields generated by any element with positive valuation, broadening the scope of undecidability in valued fields.

Contribution

It generalizes Pheidas's 1987 undecidability result from the specific element t to any element with positive t-adic valuation in Laurent series fields.

Findings

Undecidability holds for fields with any element of positive valuation.

The result applies to a broader class of Laurent series fields.

The proof extends previous methods to more general elements.

Abstract

In 1987, Pheidas showed that the field of Laurent series $\mathbb{F}_q((t))$ with a constant for the indeterminate $t$ and a predicate for the natural powers $\{t^n \mid n > 0\}$ of $t$ is existentially undecidable. We show that the same result holds true if $t$ is replaced by any element $\alpha$ of positive $t$-adic valuation.