One-dimensional F-definable sets in F((t))

TL;DR

This paper characterizes one-dimensional definable sets in power series fields with perfect residue fields, showing they are unions of existentially definable sets and classifying the subfields generated by such sets in positive characteristic.

Contribution

It demonstrates that one-dimensional definable sets in these fields are unions of existentially definable sets and classifies the subfields they generate in positive characteristic.

Findings

Definable sets are unions of existentially definable sets.

Subfields generated by definable sets are either contained in F or are power series fields with p^n-th roots.

The results extend previous work on existential definability in henselian and large fields.

Abstract

In this note we study one-dimensional definable sets in power series fields with perfect residue fields. Using the description of automorphisms given by Schilling, in \cite{S44}, we show that such sets are unions of existentially definable in the language of rings, allowing parameters. We deduce that if $F$ is a perfect field of positive characteristic $p$, and $X$ is a subset of the $t$-adically valued $F((t))$ that is definable in the language of valued fields with parameters from $F$, then the subfield $(X)$ generated by $X$ is either contained in $F$ or equal to $F((t^{p^n}))$, for some $n\geq0$. The proof uses our earlier work on existentially definable subsets of henselian and large fields, of which power series fields are examples.