Hilbert's Tenth Problem for rational function fields over p-adic fields
Logic
2011-06-27 v1 Number Theory
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).
Cite
@article{arxiv.1106.4912,
title = {Hilbert's Tenth Problem for rational function fields over p-adic fields},
author = {Claudia Degroote and Jeroen Demeyer},
journal= {arXiv preprint arXiv:1106.4912},
year = {2011}
}