Computing a hyperelliptic integral using arithmetic in the Jacobian of the curve
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.
scientific article; zbMATH DE number 780889