Rational Univariate Representation of Zero-Dimensional Ideals with Parameters

An algorithm for computing the rational univariate representation of zero-dimensional ideals with parameters is presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, by partitioning the parameter space we can ensure that under each branch the ideal has the same number of zeros. Subsequently based on the extended subresultant theorem for parametric cases, the separating element corresponding to each branch is chosen with the further partition of parameter space. Finally, with the help of parametric greatest common divisor theory a finite set of the rational univariate representation of zero-dimensional ideals with parameters can be obtained.