Titre Irreducible components of hypersurfaces. |

**Location**

INRIA, Project
CAFÉ

BP 93, 06902 France

**Information**

**Description**

A classical method to compute the number of irreducible components of
an hypersurface *P(x1,...,xn) = 0* is to factor the polynomial *P*
over an algebraically closed field. In the case of curves (*n=2*)
a new method was introduced in [1]. That method first computes a linear
ordinary differential operator *L(x,d/dx)* associated with the curve,
and shows that the number of irreducible components is equal to the dimension
of the space of rational solutions of *L(x,d/dx)(y) = 0*. The authors
also claim that a generalisation to hypersurfaces is possible, provided
that one uses an overdetermined system of linear *partial* differential
equations instead. The goal of this internship is to verify the proposed
generalisation on various examples using algorithms (existing and
under development) for computing the rational solutions of such systems,
then to check whether their genericity condition on *P* remains needed,
and finally to attempt to prove that the number of irreducible components
is indeed equal to the dimension of the rational solutions.

[1] O.Cormier, M.Singer, B.Trager and F.Ulmer (2001): *Linear
Differential Operators for Polynomial Equations,*
Journal of Symbolic Computation (to appear).

**Tools**

Unix workstation, the computer algebra system Maple.

**Duration**

2 - 3 months, possible prolongation as a doctoral thesis if desired.