Twisted Mahler discrete residues
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910884567187456 |
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| author | Arreche, Carlos E. Zhang, Yi |
| author_facet | Arreche, Carlos E. Zhang, Yi |
| contents | Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p > 1$. Here we develop a notion of $λ$-twisted Mahler discrete residues for $λ\in\mathbb{Z}$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $p^λg(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_16765 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Twisted Mahler discrete residues Arreche, Carlos E. Zhang, Yi Number Theory Commutative Algebra 39A06(Primary), 12H10 (Secondary) I.1.2 Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p > 1$. Here we develop a notion of $λ$-twisted Mahler discrete residues for $λ\in\mathbb{Z}$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $p^λg(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations. |
| title | Twisted Mahler discrete residues |
| topic | Number Theory Commutative Algebra 39A06(Primary), 12H10 (Secondary) I.1.2 |
| url | https://arxiv.org/abs/2308.16765 |