Uniformities for Hilbert's Tenth Problem in henselian valued fields

Abstract: Hilbert's Tenth Problem in its original form asks for an algorithm to determine correctly whether -- or not -- a given multivariable polynomial equation with integer coefficients has integer solutions. The surprising resolution, by Davis, Putnam, Robinson, and finished by Matiyasevich in 1970, is that there is no such algorithm. In the terminology of mathematical logic, the existential theory of the ring of integers is undecidable. I am interested in the (un?)decidability of the existential theories of a variety of rings and fields, especially of "large" fields, for example those fields admitting a non-trivial valuation that satisfies "Hensel's Lemma", a weak form of completion. In this talk I'll describe work in this direction (joint with Dittmann, Fehm, Jahnke, and others, in various combinations), new "uniform" results, and some links with theories of function fields.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.