Invariant Algebraic Sets and Symmetrization of Polynomial Systems

International audienceAssuming the variety of a polynomial set is invariant under a group action, we construct a set of invariants that define the same variety. Our construction can be seen as a generalization of the previously known construction for finite groups. The result though has to be understood outside an invariant variety which is independent of the polynomial set considered. We introduce the symmetrizations of a polynomial that are polynomials in a generating set of rational invariants. The generating set of rational invariants and the symmetrizations are constructed w.r.t. a section to the orbits of the group action

Properness Defects of Projection and Minimal Discriminant Variety

International audienceThe problem of describing the solutions of a polynomial system appears in many different fields such as robotic, control theory, etc. When the system depends on parameters, its minimal discriminant variety is the set of parameter values around which the roots of the system cannot be expressed as a continuous function of the parameters.In particular, an important component of the minimal discriminant variety is the set of properness defects. This article presents a method efficient in practice and in theory to compute the non-properness set of a projection mapping, by reducing the problem to a problem of variable elimination. We also present a reduction of the computation of the minimal discriminant variety to the computation of the non-properness set of a projection mapping. This result allows us to deduce a bound on the degree and the time computation of the minimal discriminant variety of a parametric system under some assumptions