Quantifier elimination for the relative Frobenius

Abstract

Let (K, v) be a complete discretely valued field of characteristic zero with an algebraically closed residue field of positive characteristic. Let σ : K → K be a continuous automorphism of K inducing a Frobenius automorphism on the residue field. We prove quantifierelimination for (K, v, σ) in a language with angular component maps and in a language with predicates on leading terms. The proof passes through a generalization of the main Ax-Kochen-Ersov and quantifierelimination results of [6] to a wider class of D-henselian fields of char-