Computing a hyperelliptic integral using arithmetic in the Jacobian of the curve
type: Item
instance of: scholarly article
title: Computing a hyperelliptic integral using arithmetic in the Jacobian of the curve
zbMATH Open document ID: 0838.12005
DOI: 10.1007/BF01235720
author: Laurent Bertrand
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1995-08-02T00:00:00Z
review text: In 1970 \textit{R. H. Risch} described the first algorithm for deciding whether an algebraic function has an elementary indefinite integral and for computing it if it exists; his algorithm was not suitable for implementation [Bull. Am. Math. Soc. 76, 605-608 (1970; Zbl 0196.06801)]. \textit{J. H. Davenport} [On the integration of algebraic functions, Springer Lect. Notes Comput. Sci. 102 (1981; Zbl 0471.14009)] and \textit{B. M. Trager} [Integration of algebraic functions, Ph. D. Thesis, Massachusetts Institute of Technology (1984)] gave the first general practical algorithms for computing indefinite integrals of algebraic functions. In this paper the author considers the special problem of computing hyperelliptic integrals, i.e. of integrals \[ \int {R_1 (x) + R_2 (x)y \over R_3 (x) + R_4 (x)y} dx \] where \(R_1, R_2, R_3, R_4\) are polynomials and where \(y\) is defined by an equation \(y^2 = f(x)\), \(f\) a squarefree polynomial of odd degree \(\geq 3\). His algorithm combines Trager's algorithm and the technique, introduced by \textit{D. G. Cantor} [Math. Comput. 48, 95-101 (1987; Zbl 0613.14022)], for computing the Jacobian of the hyperelliptic curve defined by the equation \(y^2 = f(x)\). The author implemented his algorithm in the computer algebra system Maple and compared it with Trager's general algorithm; some examples, which are discussed in detail, show that its computation time is significantly shorter.
reviewed by: Friedrich Schwarz
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 780889
zbMATH Keywords: integration in finite terms
zbMATH Keywords: Maple
zbMATH Keywords: algorithm
zbMATH Keywords: algebraic function
zbMATH Keywords: hyperelliptic integrals
zbMATH Keywords: hyperelliptic curve
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: Q5615009
cites work: Computing in the Jacobian of a Hyperelliptic Curve
cites work: Q5810622
cites work: Q5618470
cites work: On the integration of algebraic functions
cites work: Spécialisation de la suite de Sturm et sous-résultants (I)
cites work: The solution of the problem of integration in finite terms
cites work: Q3963106
description: scientific article; zbMATH DE number 780889