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Generalised power series determined by linear recurrence relations

Lothar Sebastian Krapp, Salma Kuhlmann, Michele Serra

TL;DR

This work extends Kronecker's classical recurrence criterion to generalized power series with exponents in an arbitrary ordered abelian group by introducing generalized linear recurrence sequences (LRS) and their associated subspaces $iglraket{R}igr$. It develops a robust framework linking LRS to fields $k((G))$ and $k(G)$, and to multivariate settings, via generalized Hankel matrices and a universal description of when a series lies in a fraction field. The paper then connects these ideas to Rayner fields, $k$-hulls, and Hahn-field lifting properties, providing criteria for when fields determined by LRS enjoy canonical lifting properties and how these properties behave under automorphisms. Overall, the results yield constructive methods to identify, manipulate, and classify Hahn fields with prescribed algebraic and automorphism-structure properties, including explicit procedures to express such series as fractions of polynomials in several variables. These insights advance the understanding of automorphism groups of Hahn fields and illuminate the interplay between coefficient recurrences, field-theoretic structure, and geometric support conditions.

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.

Generalised power series determined by linear recurrence relations

TL;DR

This work extends Kronecker's classical recurrence criterion to generalized power series with exponents in an arbitrary ordered abelian group by introducing generalized linear recurrence sequences (LRS) and their associated subspaces . It develops a robust framework linking LRS to fields and , and to multivariate settings, via generalized Hankel matrices and a universal description of when a series lies in a fraction field. The paper then connects these ideas to Rayner fields, -hulls, and Hahn-field lifting properties, providing criteria for when fields determined by LRS enjoy canonical lifting properties and how these properties behave under automorphisms. Overall, the results yield constructive methods to identify, manipulate, and classify Hahn fields with prescribed algebraic and automorphism-structure properties, including explicit procedures to express such series as fractions of polynomials in several variables. These insights advance the understanding of automorphism groups of Hahn fields and illuminate the interplay between coefficient recurrences, field-theoretic structure, and geometric support conditions.

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.
Paper Structure (12 sections, 21 theorems, 93 equations)

This paper contains 12 sections, 21 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminology
  4. Kronecker's characterisation
  5. Generalised linear recurrence relations
  6. Linear recurrence Hahn fields
  7. Generalised Hankel matrices
  8. Relation to Rayner fields
  9. Power series fields in finitely many variables
  10. Lifting properties
  11. Canonical first lifting property
  12. Canonical second lifting property

Key Result

Corollary 2.4

Let Then $s\in k(t)$ if and only if the following holds: there exist $\ell\in \mathbb{N}$, $g_0,\ldots,g_\ell\in \mathbb{Z}$ with $g_0<\ldots<g_\ell$ and $r_0,\ldots,r_\ell\in k$, not all equal to $0$, such that for any $n\in \mathbb{Z}\setminus\{g_0,\ldots,g_\ell\}$ the following linear recurrence rela

Theorems & Definitions (77)

  • Definition 2.1
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Definition 3.1
  • Definition 3.3
  • Example 3.4
  • proof
  • Example 3.6
  • Proposition 3.7
  • ...and 67 more