Computing parametric geometric resolutions
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 Bezout 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 Bezout number. We present several applications of this process, notably to computations in the Jacobian of hyperelliptic curves and to questions of real geometry.
complexity polynomial systems with parameters theory of elimination symbolic newton operator