Hermite's approach to Abelian integrals revisited

Summary · qwen2.5:32b

The article presents a new criterion for the linear independence of values of Lauricella hypergeometric series $F_D$ with rational parameters, extending Hermite's work on Abelian integrals to both complex and $p$-adic settings. This extension is achieved through explicit Padé-type approximations to solutions of reducible Jordan-Pochhammer differential equations, with a key innovation being the proof of non-vanishing determinants associated with these approximants.

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Exploring Hermite's approach to Abelian integrals and its generalization for Lauricella hypergeometric series, providing new insights in algebra and number theory.

Excerpt

arXiv:2511.07828v2 Announce Type: replace Abstract: 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 Pad\'{e}-type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Pad\'{e} 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 Pad\'{e}-type approximants.