Hilbertʼs Tenth Problem for rational function fields over p-adic fields

Let K be a p-adic field (a finite extension of some Qp) 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 “vt(x)⩾0” in K(t). This implies undecidability of diophantine equations over K(t).