Generalised power series determined by linear recurrence relations

TL;DR

This paper extends Kronecker's classical result on univariate Laurent series to multivariate power series with exponents in ordered abelian groups, characterizing their algebraic structure via generalized linear recurrence relations.

Contribution

It introduces generalized linear recurrence relations for multivariate power series and provides criteria for their membership in fraction fields, advancing the understanding of their algebraic properties.

Findings

Criteria for multivariate series to be in fraction fields

Identification of algebraic substructures via recurrence relations

Relations determining properties relevant to automorphism groups

Abstract

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.