|Titre Irreducible components of hypersurfaces.|
BP 93, 06902 France
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 . 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.
 O.Cormier, M.Singer, B.Trager and F.Ulmer (2001): Linear Differential Operators for Polynomial Equations, Journal of Symbolic Computation (to appear).
Unix workstation, the computer algebra system Maple.
2 - 3 months, possible prolongation as a doctoral thesis if desired.