Submission date: 18 December 2018.

Abstract:

Following our first article, we continue to investigate ultrametic modules over a ring of twisted polynomials of the form [K;φ], where φ is a ring endomorphism of K. The main motivation comes from the the theory of valued difference fields (including characteristic p>0 valued fields equipped with the Frobenius endomorphism). We introduce the class of modules, that we call, affinely maximal and residually divisible and we prove (relative -) quantifier elimination results. Ax-Kochen & Erhov type theorems follows. As an application, we axiomatize, as a valued module, the ultraproduct of algebraically closed valued fields (𝔽_{p^n}(t)^{alg})_{n\in ℕ}, of fixed characteristic p>0, each equipped with the morphism x → x^{p^n} and with the t-adic valuation.

Mathematics Subject Classification: 03C60

Keywords and phrases:

Full text arXiv 1812.07333: pdf, ps.