The rational rank of the support of generalized power series solutions of differential and $q$-difference equations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909481385852928 |
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| author | Cano, J. Ayuso, P. Fortuny |
| author_facet | Cano, J. Ayuso, P. Fortuny |
| contents | Given a differential or $q$-difference equation $P$ of order $n$, we prove that the set of exponents of a generalized power series solution has its rational rank bounded by the rational rank of the support of $P$ plus $n$. We also prove that when the support of the solution has maximum rational rank, it is convergent. Using the Newton polygon technique, we show also that in the maximum rational rank case, an initial segment can always be completed to a true solution. The techniques are the same for the differential and the $q$-difference case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06115 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The rational rank of the support of generalized power series solutions of differential and $q$-difference equations Cano, J. Ayuso, P. Fortuny Classical Analysis and ODEs Complex Variables Dynamical Systems 34M25 (Primary), 30D05 (Secondary) Given a differential or $q$-difference equation $P$ of order $n$, we prove that the set of exponents of a generalized power series solution has its rational rank bounded by the rational rank of the support of $P$ plus $n$. We also prove that when the support of the solution has maximum rational rank, it is convergent. Using the Newton polygon technique, we show also that in the maximum rational rank case, an initial segment can always be completed to a true solution. The techniques are the same for the differential and the $q$-difference case. |
| title | The rational rank of the support of generalized power series solutions of differential and $q$-difference equations |
| topic | Classical Analysis and ODEs Complex Variables Dynamical Systems 34M25 (Primary), 30D05 (Secondary) |
| url | https://arxiv.org/abs/2406.06115 |