Elementary equivalence versus Isomorphism

How does the first order language of fields encode birational invariants of varieties?... This question is related to rational points on varieties and effectiveness in algebraic/arithmetic geometry

First-order definitions in function fields over anti-Mordellic fields

A field k is called anti-Mordellic if every smooth curve over k with a k-point has infinitely many k-points. We prove that for a function field over an anti-Mordellic field, the subfield of constants is defined by a certain universal first order formula. Under additional hypotheses regarding 2-cohomological dimension we prove that algebraic dependence of an n-tuple of elements in such a function field can be described by a first order formula, for each n. We also give a result that lets one distinguish various classes of fields using first order sentences.Comment: 12 page