Two Algorithms for Computing Rational Univariate Representations of Zero-Dimensional Ideals with Parameters

Based on the partition of parameter space, two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Unlike 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, we first ensure that under each branch the ideal has the same number of zeros by partitioning the parameter space. Subsequently two ideas are given to choose and check the separating element. One idea is that by extending the subresultant theorem to parametric cases, we utilize the extended subresultant theorem to choose the separating element with the further partition of parameter space and then with the help of parametric greatest common divisor theory compute rational univariate representations. Another one is that we go straight to choose and check the separating element by the computation of parametric greatest common divisors, then immediately get the rational univariate representations. Based on these, we design two different algorithms for computing rational univariate representations of zero-dimensional ideals with parameters. Furthermore, the algorithms have been implemented on Singular and the performance comparison are presented.