arXiv subjects

Gönenç Onay

Publications and source records attributed to Gönenç Onay.

6 recordsLinked to original sources

arXiv2019

Valued Modules over Skew Polynomial Rings 2

Following our first article, we continue to investigate ultrametic modules over a ring of twisted polynomials of the form $[K;\vfi]$, where $\vfi$ 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, any ultraproduct of algebraically closed valued fields $(\mathbb{F}_{p^n}(t)^{alg})_{n\in \mathbb{N}}$, of fixed characteristic $p>0$, each equipped with the morphism $x\mapsto x^{p^n}$ and with the $t$-adic valuation.

math.LO↗

arXiv2019

Modules C-minimaux sur des anneaux de polynômes tordus

In this article we study modules endowed with a ultrametric, from the point of view of the geometric notion $C$-minimality. We give a complete characterization of $C$-minimal valued modules over non-commutative rings of skew polynomials of the form $R:=K[t;φ]$, where $K$ is a field, $φ$ an endomorphism of $K$ and $R$ is the $K$-algebra generated by $t$, such that $at=ta^φ$ for $a\in K$. We deduce for instance that the ring of Puiseux series over a finite field $\mathbb{F}$ of characteristic $p>0$, as a valued module over $\mathbb{F}[t;x\mapsto x^p]$ is $C$-minimal. Moreover, any ultraproduct $\mathcal{K}$, of algebraically closed valued fields $\mathcal{K}_{p^n}$ of characteristic $p>0$, endowed each with the morphism $x\mapsto x^{p^n}$, following a ultrafilter $U$ over $\{p^n\ |\ n\in \mathbb{N}, \, \text{et} \; p \; \text{prime}\}$, equipped with the {\it non-standart Frobenius}, i.e., the map $σ_{U}:=\lim_{U} x \mapsto x^{p^n}$, is $C$-minimal as a $\mathcal{K}[t;σ]$-valued module.

math.LO↗

arXiv2018

$\mathbb{F}_p((X))$ is decidable as a module over the ring of additive polynomials

Let $p$ be a prime number, $K$ be the henselization of the rational functions over the finite field $\mathbb{F}_p$ and $R$ be the ring of additive polynomials over K. We show that the field of Laurent series over $\mathbb{F}_p$ is decidable seen as an R-module. Moreover, we provide a recursively enumerable axiom system (satisfied by $K$) in the language of $R$-modules together with a unary predicate for the valuation ring, modulo which every positive primitive formula is equivalent to a universal formula. Consequently the $R$-module theory of the field of Laurent series is model-complete in this language and admits $K$ as its prime model.

math.LO↗

arXiv2017

Fields with automorphism and valuation

Abraham Robinson's method for finding model completions is refined and and generalized for model companions and is applied to the theory of fields equipped with both a valuation and an automorphism.

math.LO↗

arXiv2016

Valued modules over skew polynomial rings 1

We introduce a notion of valued module which is suitable to study valued fields of positive characteristic. Then we built-up a robust theory of henselianity in the language of valued modules and prove Ax-Kochen Ershov type results.

math.LO↗

arXiv2013

Quantifier Elimination for Valued Fields Equipped with an Automorphism

We provide axiomatization and relative quantifier elimination for valued fields equipped with an automorphism, in residue characteristic zero. Similar results are known under strong assumptions on the interaction between the automorphism and the valuation. We remove such assumptions and provide general treatment. As a consequence we obtain an axiomatization of the transseries field (as a valued field with an automorphism) equipped with the automorphism which sends f(x) to f(x+1).

math.LO↗