Hilbert's Tenth Problem for rational function fields over p-adic fields

TL;DR

This paper proves the undecidability of diophantine equations over rational function fields over p-adic fields by establishing quadratic reciprocity and isotropy results, extending classical number theory concepts to these fields.

Contribution

It introduces a quadratic reciprocity symbol for polynomials over p-adic fields and applies it to show undecidability of diophantine problems over K(t).

Findings

Established a quadratic reciprocity symbol for polynomials over K.

Proved isotropy of certain quadratic forms over K(t).

Demonstrated undecidability of diophantine equations over K(t).

Abstract

Let K be a p-adic field (a finite extension of some Q_p) and let K(t) be the field of rational functions over K. We define a kind of quadratic reciprocity symbol for polynomials over K and apply it to prove isotropy for a certain class of quadratic forms over K(t). Using this result, we give an existential definition for the predicate "v_t(x) >= 0" in K(t). This implies undecidability of diophantine equations over K(t).