Titre Integration of algebraic functions. |
Version
française |

**Location**

INRIA, Project
CAFÉ

BP 93, 06902 France

**Information**

**Description**

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).

**Tools**

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

**Duration**

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