Publikation: Generalised power series determined by linear recurrence relations
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2025
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Journal of Algebra. Elsevier. 2025, 681, S. 152-189. ISSN 0021-8693. eISSN 1090-266X. Verfügbar unter: doi: 10.1016/j.jalgebra.2025.05.012
Zusammenfassung
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.
Zusammenfassung in einer weiteren Sprache
Fachgebiet (DDC)
510 Mathematik
Schlagwörter
Generalised formal power series, Rational function fields, Hahn fields, Rayner fields, Linear recurrence, Lifiting properties, Automorphism groups
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KRAPP, Lothar Sebastian, Salma KUHLMANN, Michele SERRA, 2025. Generalised power series determined by linear recurrence relations. In: Journal of Algebra. Elsevier. 2025, 681, S. 152-189. ISSN 0021-8693. eISSN 1090-266X. Verfügbar unter: doi: 10.1016/j.jalgebra.2025.05.012BibTex
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title={Generalised power series determined by linear recurrence relations},
year={2025},
doi={10.1016/j.jalgebra.2025.05.012},
volume={681},
issn={0021-8693},
journal={Journal of Algebra},
pages={152--189},
author={Krapp, Lothar Sebastian and Kuhlmann, Salma and Serra, Michele}
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<dcterms: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.</dcterms:abstract>
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