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.

翻译：本文基于参数空间的划分，提出了两种计算含参数零维理想有理单变量表示的算法。与不含参数的零维理想的有理单变量表示不同，含参数零维理想在不同参数特化下的零点个数各异，这导致选择和检验分离元——计算有理单变量表示的关键——变得困难。为了选取分离元，我们首先通过划分参数空间确保每个分支下理想具有相同数量的零点。随后，提出了两种选择和检验分离元的思路。一种思路是通过将子结式定理推广至参数情形，利用推广后的子结式定理，结合参数空间的进一步划分来选择分离元，进而借助参数最大公因子理论计算有理单变量表示。另一种思路是直接通过计算参数最大公因子来选择和检验分离元，随即得到有理单变量表示。基于此，我们设计了两种不同的算法来计算含参数零维理想的有理单变量表示。此外，算法已在Singular上实现，并给出了性能比较。