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

Sylvy Anscombe

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.

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 is a perfect field of positive characteristic , and is a subset of the -adically valued that is definable in the language of valued fields with parameters from , then the subfield generated by is either contained in or equal to , for some . The proof uses our earlier work on existentially definable subsets of henselian and large fields, of which power series fields are examples.

Paper Structure

This paper contains 7 sections, 10 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:subfields_intro']}
  3. A simple application of Hensel's Lemma
  4. $F$-automorphisms of $F(\!(t)\!)$
  5. Suitable expansions
  6. Definability
  7. Orbits of one-dimensional tuples

Key Result

Theorem \oldthetheorem

Let $F$ be a perfect field of characteristic $p>0$, and let $X\subseteq F(\!(t)\!)$ be a subset definable in the language $\mathfrak{L}_{\mathrm{val}}(F)$ of valued fields, i.e. allowing parameters from $F$. Then either $X\subseteq F$ or there exists $n\in\mathbb{N}$ such that where $(X)$ denotes the subfield of $F(\!(t)\!)$ generated by $X$.

Theorems & Definitions (24)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 14 more