Lifting procedures for ramified fibers and polynomial equation solving

AGUSTÍN BOMPADRE; GUILLERMO MATERA; ROSA WACHENCHAUZER; ARIEL WAISSBEIN

Buenos Aires

Conferencia; Jornadas Argentinas de Informática e Investigación Operativa; 2001

Sociedad Argentina de Informática e Investigación Operativa

In Heintz et al, J. Complexity 16(1), 2000, the following problem was solved: given a parametric polynomial equation system which represents a generically flat and unramified family of zero-dimensional algebraic varieties, and assuming that there is given a complete description of the solution of a particular unramified instance of the parametric system, produce efficiently a description of the parametric system. In this work we solve this problem under weaker hypotheses, namely admitting that the given zero-dimensional variety can be ramified. In this case, assuming that a complete description of the infinitesimal structure of the ramified instance is given, we generalize the techniques used in Kung-Traub (1978) in order to solve the problem. We also analyze the complexity of the underlying algorithm and show a few examples where our techniques allow us to solve some polynomial equations systems in a very efficient way.