Singer and Ulmer (1997) gave an algorithm to compute Liouvillian ("closed-form") solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semiinvariants and a theorem in Singer and Ulmer (1997) in such a way that, by computing one semi-invariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient. These coefficients come "for free" as a byproduct of our algorithm for computing semi-invariants. We specifically detail the algorithm in the cases of equations of order three (order two equations are handled by the algorithm of Kovacic, 1986, see also Ulmer and Weil, 1996 or Fakler, 1997).In the Appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm.

No abstract

We consider the set of polynomials in r indeterminates over a "nite "eld and with bounded degree. We give here a way to count the number of elements of some of its subsets, namely those sets de"ned by the multiplicities of their elements at some points of %P O . The number of polynomials having at least one zero in a given "nite "eld is computed as a particular applications.1999 Academic Press

Depuis 10 ans, l’IUT du Limousin a développé la VAE collective à travers un partenariat avec l’entreprise Orange sur la licence professionnelle Management de la Relation Client. Depuis, le processus s’est enrichi et a atteint une maturité suffisante pour que d’autres partenariats soient lancés sur cette licence et sur d’autres formations, contribuant ainsi au développement de la Formation Tout au Long de la Vie à l’université de Limoges.