arXiv subjects

Mickael Matusinski

Publications and source records attributed to Mickael Matusinski.

5 recordsLinked to original sources

arXiv2016

The valuation difference rank of a quasi-ordered difference field

There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued {\it difference} field (that is, ordered or valued field endowed with a compatible field automorphism). We introduce the notion of {\it difference rank}. To treat simultaneously the cases of ordered and valued fields, we consider quasi-ordered fields. We characterize the difference rank as the quotient modulo the equivalence relation naturally induced by the automorphism (which encodes its growth rate). In analogy to the theory of convex valuations, we prove that any linearly ordered set can be realized as the difference rank of a maximally valued quasi-ordered difference field. As an application, we show that for every regular uncountable cardinal $κ$ such that $κ= κ^{< κ}$, there are $2^κ$ pairwise non-isomorphic quasi-ordered difference fields of cardinality $κ$, but all isomorphic as quasi-ordered fields.

math.LO↗

arXiv2013

A Note on Schanuel's Conjectures for Exponential Logarithmic Power Series Fields

In [1], J. Ax proved a transcendency theorem for certain differential fields of characteristic zero: the differential counterpart of the still open Schanuel's conjecture about the exponential function over the field of complex numbers [11, page 30]. In this article, we derive from Ax's theorem transcendency results in the context of differential valued exponential fields. In particular, we obtain results for exponential Hardy fields, Logarithmic-Exponential power series fields and Exponential-Logarithmic power series fields.

math.LO↗

arXiv2012

Hardy type derivations on generalized series fields

We consider the valued field $\mathds{K}:=\mathbb{R}((Γ))$ of generalized series (with real coefficients and monomials in a totally ordered multiplicative group $Γ$). We investigate how to endow $\mathds{K}$ with a series derivation, that is a derivation that satisfies some natural properties such as commuting with infinite sums (strong linearity) and (an infinite version of) Leibniz rule. We characterize when such a derivation is of Hardy type, that is, when it behaves like differentiation of germs of real valued functions in a Hardy field. We provide a necessary and sufficent condition for a series derivation of Hardy type to be surjective.

math.AC↗

arXiv2011

Hardy type derivations on fields of exponential logarithmic series

We consider the valued field $\mathds{K}:=\mathbb{R}((Γ))$ of formal series (with real coefficients and monomials in a totally ordered multiplicative group $Γ>$). We investigate how to endow $\mathds{K}$ with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on $\mathds{K}$. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure $\mathds{K}^{\rm{EL}}$ of $(\mathds{K},l)$. Here we show how to extend such a log-compatible derivation on $\mathds{K}$ to $\mathds{K}^{\rm{EL}}$.

math.AC↗

arXiv2010

Differential Puiseux theorem in generalized series fields of finite rank

We study differential equations $F(y,...,y^{(n)})=0$ where $F(Y_0,...,Y_n)$ is a formal series in $Y_0,...,Y_n$ with coefficients in some field of \emph{generalized power series} $\mathds{K}_r$ with finite rank $r\in\mathbb{N}^*$. Our purpose is to understand the connection between the set of exponents of the coefficients of the equation $\textrm{Supp} F$ and the set $\textrm{Supp} y_0$ of exponents of the elements $y_0\in\mathds{K}_r$ that are solutions.

math.CA↗