# Decidability problems in number theory

## Victor Lisinski, Corpus Christi, Oxford

In its modern formulation, Hilbert’s tenth problem asks to find a general algorithm which decides the solvability of Diophantine equations. While this problem was shown to be unsolvable due to the combined work of Davis, Putnam, Robinson and Matiyasevich, similar question can be posed over domains other than the integers. Among the most important open questions in this area of research is if a version of Hilbert’s tenth problem for F_{p}((t)), the field of formal Laurent series over the finite field F_{p}, is solvable or not. The fact that this remains open stands in stark contrast to the fact that the first order theory of the much similar object Q_{p}, the field of p-adic numbers, is completely understood thanks to the work by Ax, Kochen and, independently, Ershov. In light of this dichotomy, I will present new decidability results obtained during my doctoral research on extensions of F_{p}((t)). This work is motivated by recent progress on Hilbert’s tenth problem for F_{p}((t)) by Anscombe and Fehm and builds on previous decidability results by Kuhlman.