Computing Parametric Geometric Resolutions

Abstract

 Given a polynomial system of n equations in n unknowns that depends on some parameters, we define the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters.

The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number d ⁿ, where d is a bound on the degrees of the input system. Then we present a probabilistic algorithm to compute a parametric resolution. Its complexity is polynomial in the size of the output and in the complexity of evaluation of the input system. The probability of success is controlled by a quantity polynomial in the Bézout number.

We present several applications of this process, notably to computa- tions in the Jacobian of hyperelliptic curves and to questions of real geometry.