Titre Integration of algebraic functions. Version française


BP 93, 06902 France


Manuel Bronstein


The Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands having only simple affine poles [1,2,4]. While it is very effective in the case of simple radicals of the form f(x, p(x)^(1/n)) dx, its use in more general algebraic extensions requires the precomputation of an integral basis, which makes that reduction impractical for either multiple algebraic extensions or complicated ground fields. We have developped a lazy version of that reduction that does not require an a priori computation of an integral basis [3], but there are no implementation or experimental results. The goal of this internship is to implement a prototype of the lazy reduction and to compare it to extant symbolic integrators.
It is possible to continue this work towards a doctoral thesis around the use of recent algorithms for computing in the Jacobians of algebraic curves in order to obtain an efficient and complete algorithm for integrating arbitrary algebraic functions.

[1] L.Bertrand, Computing a Hyperelliptic Integral using Arithmetic in the Jacobian of the Curve, Applicable Algebra in Engineering, Communication and Computing 6, 275-298 (1995).
[2] M.Bronstein, On the Integration of Elementary Functions, Journal of Symbolic Computation 9, 117-173 (1990).
[3] M.Bronstein,  The lazy Hermite reduction, INRIA Research Report RR-3562 (1998).
[4] B.Trager, On the integration of algebraic functions, PhD thesis, MIT, Computer Sciences (1984).


Unix workstation, Axiom computer algebra system or Aldor programming language.


3 - 4 months, possible prolongation as a doctoral thesis if desired.


February 1, 2001