2023/03/24 by Thierry Combot, Combot, Thierry
Computer Science · Mathematics · #11G30 #33F10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2303.14013
openalex publication_date 2023/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a hyperelliptic integral I=∫ P/(Q√(S)) dx, P,Q,S∈\mathbbK[x], with [\mathbbK:ℚ]<∞. When S is of degree ≤ 4, such integral can be calculated in terms of elementary functions and elliptic integrals of three kinds F,E,Π. 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.