Generalised power series determined by linear recurrence relations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866917976311070720 |
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| author | Krapp, Lothar Sebastian Kuhlmann, Salma Serra, Michele |
| author_facet | Krapp, Lothar Sebastian Kuhlmann, Salma Serra, Michele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_04126 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Generalised power series determined by linear recurrence relations Krapp, Lothar Sebastian Kuhlmann, Salma Serra, Michele Commutative Algebra Logic 13J05 (16W60 12J10 06F20) 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. |
| title | Generalised power series determined by linear recurrence relations |
| topic | Commutative Algebra Logic 13J05 (16W60 12J10 06F20) |
| url | https://arxiv.org/abs/2206.04126 |