|Title Computing the genus of an algebraic curve.||Version française|
BP 93, 06902 France
The genus of an algebraic curve is a fundamental quantity that several
algorithms in cryptography or computer algebra depend on. Computing it,
which is often a preliminary step for other algorithms, remains difficult
for general curves. The goal of this internship is to study and implement
a new method (unpublished work of B.Trager) based on Hurwitz' formula,
which expresses the genus in terms of local invariants at the singularities
of the curve. The original twist of this method consists in using the ordinary
linear differential equation satisfied by an algebraic function in order
to compute those local invariants. In a first part, this method will be
implemented using a scalar differential equation of the form
... + a2(x) y'' + a1(x) y' + a0(x) y = 0 and its Newton polygon, in
order to verify its practicability. Since that equation is costly to compute,
in a second part, a method based on a differential system of the form Y'
= A(x) Y will be implemented. It is conjectured that the algorithm
of produces the required local invariants.
It is possible to continue this work towards a doctoral thesis around using such a differential system for other computations with algebraic functions (for example Puiseux series expansions) as well as work on the above conjecture.
 S.Abramov et M.Bronstein (2001), On Solutions of Linear Functional Systems, manuscript submitted to ISSAC'2001.
Unix workstation, Axiom computer algebra system or Aldor programming language.
3 - 4 months, possible prolongation as a doctoral thesis if desired.