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

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.