We give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism from Joyal.

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Résumé.On axiomatise la théorie du premier ordre de l'automorphisme de Frobenius du corps des vecteurs de Witt sur la clôture algébrique d'un corps fini. © 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS The Frobenius automorphism of Witt vectors Abstract.We give an axiomatization of the first-order theory of the field of Witt vectors over the algebraic closure of a finite field, with the lifting of the Frobenius map.

Abstract. We consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.