Hyperelliptic Integrals to Elliptic Integrals

International audience Consider a hyperelliptic integral , , with . When S is of degree ≤ 4, such integral can be calculated in terms of elementary functions and elliptic integrals of three kinds . When S is of higher degree, it is typically non elementary, but it is sometimes possible to obtain an...

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Bibliographic Details
Published in:Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation
Main Author: Combot, Thierry
Other Authors: Institut de Mathématiques de Bourgogne Dijon (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté COMUE (UBFC)-Centre National de la Recherche Scientifique (CNRS)
Format: Conference Object
Language:English
Published: HAL CCSD 2023
Subjects:
[MATH]Mathematics [math]
Norway
Tromsø
Online Access:https://hal.science/hal-04335021
https://doi.org/10.1145/3597066.3597111
Description
Summary:International audience Consider a hyperelliptic integral , , with . When S is of degree ≤ 4, such integral can be calculated in terms of elementary functions and elliptic integrals of three kinds . When S is of higher degree, it is typically non elementary, but it is sometimes possible to obtain an expression of I using also elliptic integrals when the Jacobian of y2 = S(x) has elliptic factors. We present an algorithm searching for elliptic factors and a modular criterion for their existence. Then, we present an algorithm for computing an expression of I using elliptic integrals, which always succeed in the completely decomposable Jacobian case.

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