  We study function fields of curves over a base field $K$ which is either a
global field or a large field having a separable field extension of degree
divisible by $4$. We show that, for any such function field, Hilbert's 10th
Problem has a negative answer, the valuation rings containing $K$ are uniformly
existentially definable, and finitely generated integrally closed
$K$-subalgebras are definable by a universal-existential formula. In order to
obtain these results, we develop further the usage of local-global principles
for quadratic forms in function fields to definability of certain subrings. We
include a first systematic presentation of this general method, without
restriction on the characteristic.
