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arXiv · 2511.07828

Hermite's approach to Abelian integrals revisited

Makoto Kawashima
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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.

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Makoto Kawashima. 2025-11-11. Hermite's approach to Abelian integrals revisited. https://arxiv.org/abs/2511.07828

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