Definability and decidability in global and local fields

• Applicant: Professor Dr. Arno Fehm
• Subject Area: Mathematics
• Term: from 2018 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 404427454

A mathematical problem is called decidable if, roughly speaking there exists an algorithm that computes the correct answer for each input. Questions on decidability in number theory have a long history going back at least to Hilbert's Tenth Problem on solvability of diophantine equations: Does there exist an algorithm that computes whether an integer polynomial has an integer zero? Nowadays this is a lively and highly interdisciplinary research area with influences from number theory, arithmetic geometry, Galois theory, model theory (mathematical logic) and computational complexity theory (computer science).Important open questions here are the decidability of the existential theory of the rational numbers (i.e. does there exist an algorithm that decides whether an algebraic variety has a rational point) and other global fields, and the decidability of the full first-order theory of local fields in positive characteristic. Very often, decidability problems in this area are closely linked to questions of definability: Which subsets of a ring or field are diophantine, that is, projection of the zero set of a polynomial, or, more generally, definable by a first-order formula in the language of rings?The aim of this research project is to contribute to questions of definability and decidability in fields of number theoretic interest, in particular in global fields, local fields and algebraic fields, by introducing new ideas and connecting the involved areas in novel ways. It mainly follows the path laid out by the results of the applicant and his coauthors on the decidability of the existential theory of local fields in positive characteristic and on diophantine henselian valuation rings, incorporating the recent breakthroughs in this area.Goals include in particular a closer investigation of a p-adic analogue of the Pythagoras number, a study of the theory of algebraic fields, the decidability of certain algebraic fields, and decidability of existential theories of fields. The methods employed are taken mainly from valuation theory and model theoretic algebra, but with a number theoretic flavor.

• DFG Programme: Research Grants
• International Connection: United Kingdom
• Cooperation Partners: Dr. Sylvy Anscombe; Dr. Philip Dittmann