The existential decidability of power series rings over finite fields

Abstract

In this Master's Thesis we study two ways of answering the following question: is there an algorithm that decides the truth of existential statements about the power series ring over a finite field?

The first approach to the problem uses tools from algebraic geometry and is described in the article \emph{On the decidability of the existential theory of $\FF_p[[t]]$} by Jan Denef and Hans Schoutens (1999, \cite{denefschoutens}). It is shown that the truth of an existential statement in a ring $R$ corresponds to the existence of a rational point on a scheme over the spectrum of $R$. We study a form of Artin approximation that allows to related decidability in power series rings to decidability in the finite residue field. Finally, we look at the dependency of the results of Denef and Schoutens on the Resolutions of Singularities conjecture.

The other approach is the one described by Arno Fehm and Sylvy Anscombe in the article \emph{The existential theory of equicharacteristic henselian valued fields} (2016, \cite{ansfehm}). Here the problem is studied in the context of the model theory of valued fields. Using results by Franz-Viktor Kuhlmann on a special class of henselian valued fields, it can be shown that the truth of existential statements transfers for a larger class of henselian valued fields with finite residue field, of which the power series ring over a finite field is an example.

Besides trying to understand the details of the proofs in the two articles mentioned above, a short introduction to the relevant fields of mathematics is given.