  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.
