Hermite's approach to Abelian integrals revisited
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918196062191616 |
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| author | Kawashima, Makoto |
| author_facet | Kawashima, Makoto |
| contents | In this article, we establish a new linear independence criterion for the values of certain {\it Lauricella hypergeometric series} $F_D$ with rational parameters, in both the complex and $p$-adic settings, over an algebraic number field. This result generalizes a theorem of C.~Hermite \cite{Hermite} on the linear independence of certain Abelian integrals. Our proof relies on explicit Padé type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Padé approximations for certain Abelian integrals in \cite{Hermite}. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Padé type approximants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07828 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hermite's approach to Abelian integrals revisited Kawashima, Makoto Number Theory In this article, we establish a new linear independence criterion for the values of certain {\it Lauricella hypergeometric series} $F_D$ with rational parameters, in both the complex and $p$-adic settings, over an algebraic number field. This result generalizes a theorem of C.~Hermite \cite{Hermite} on the linear independence of certain Abelian integrals. Our proof relies on explicit Padé type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Padé approximations for certain Abelian integrals in \cite{Hermite}. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Padé type approximants. |
| title | Hermite's approach to Abelian integrals revisited |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.07828 |