arXiv subjects

Michele Serra

Publications and source records attributed to Michele Serra.

6 recordsLinked to original sources

arXiv2026

Decomposing the automorphism group of the surreal numbers

We study the automorphism group of the field of surreal numbers. Our main structure theorem presents a decomposition of this group into a product of five significant factors. Using the representation of surreal numbers as generalized power series via their Conway normal form, we apply results on Hahn fields and groups from the literature in order to obtain this decomposition. Moreover, we provide explicit descriptions of the individual factors enabling us to construct automorphisms on the field of surreal numbers from simpler components. We then extend our study to strongly linear automorphisms in connection to derivations, as well as automorphisms that preserve further exponential structure on the surreals.

math.LO↗

arXiv2026

Automorphisms of valued Hahn groups

Hahn groups endowed with the canonical valuation play a fundamental role in the classification of valued abelian groups. In this paper we study the group of valuation (respectively order) preserving automorphisms of a Hahn group $G$. Under the assumption that $G$ satisfies some lifting property, we prove a structure theorem decomposing the automorphism group into a semidirect product of two notable subgroups. We characterise a class of Hahn groups satisfying the aforementioned lifting property. For some special cases we provide a matrix description of the automorphism group.

math.GR↗

arXiv2025

Automorphisms and derivations on algebras endowed with formal infinite sums

We establish a correspondence between automorphisms and derivations on certain algebras of generalised power series. In particular, we describe a Lie algebra of derivations on a field $k(\!(G)\!)$ of generalised power series, exploiting our knowledge of its group of valuation preserving automorphisms. The correspondence is given by the formal Taylor expansion of the exponential. In order to define the exponential map, we develop an appropriate notion of summability of infinite families in algebras. We show that there is a large class of algebras in which the exponential induces the above correspondence.

math.RA↗

arXiv2025

Generalised power series determined by linear recurrence relations

In 1882, Kronecker established that a given univariate formal Laurent series over a field can be expressed as a fraction of two univariate polynomials if and only if the coefficients of the series satisfy a linear recurrence relation. We introduce the notion of generalised linear recurrence relations for power series with exponents in an arbitrary ordered abelian group, and generalise Kronecker's original result. In particular, we obtain criteria for determining whether a multivariate formal Laurent series lies in the fraction field of the corresponding polynomial ring. Moreover, we study distinguished algebraic substructures of a power series field, which are determined by generalised linear recurrence relations. In particular, we identify generalised linear recurrence relations that determine power series fields satisfying additional properties which are essential for the study of their automorphism groups.

math.AC↗

arXiv2022

The automorphism group of a valued field of generalised formal power series

Let $ k $ be a field, $ G $ a totally ordered abelian group and $ \mathbb K = k((G)) $ the maximal field of generalised power series, endowed with the canonical valuation $ v $. We study the group $ v \mathrm{-Aut} K $ of valuation preserving automorphisms of a subfield $ k(G)\subseteq K\subseteq \mathbb K $, where $ k(G) $ is the fraction field of the group ring $ k[G] $. Under the assumption that $ K $ satisfies two lifting properties we are able to generalise and refine Hofberger's decomposition of $ v \mathrm{-Aut}\mathbb K $ and prove a structure theorem decomposing $ v\mathrm{-Aut} K $ into a 4-factor semi-direct product of notable subgroups. We then identify a large class of Hahn fields satisfying the two aforementioned lifting properties. Next we focus on the group of strongly additive automorphisms of $ K $. We give an explicit description of the group of strongly additive internal automorphisms in terms of the groups of homomorphisms $ \mathrm{Hom}(G,k^\times) $ of $ G $ into $ k^\times $ and $ \mathrm{Hom}(G,1+I_K) $ of $ G $ into the group of 1-units of the valuation ring of $ K $. Finally, we specialise our results to some relevant special cases. In particular, we extend the work of Schilling on the field of Laurent series and that of Deschamps on the field of Puiseux series.

math.AC↗

arXiv2021

On Rayner structures

In this note, we study substructures of generalised power series fields induced by families of well-ordered subsets of the group of exponents. We characterise the set-theoretic and algebraic properties of the induced substructures in terms of conditions on the families. We extend the work of Rayner by giving both \emph{necessary} and sufficient conditions to obtain truncation closed subgroups, subrings and subfields.

math.AC↗