Two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are 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 with the help of the extended subresultant theorem for parametric cases, two ideas are given to conduct the further partition of parameter space for choosing and checking the separating element. Based on these, we give two algorithms for computing rational univariate representations of zero-dimensional ideals with parameters. Furthermore, the two algorithms have been implemented on the computer algebra system Singular. Experimental data show that the second algorithm has the better performance in contrast to the first one.

翻译：本文提出了两种计算含参数零维理想的有理单变量表示的算法。与不含参数零维理想的有理单变量表示不同，含参数零维理想在各种特化下的零点个数不同，导致选取和检验分离元——即计算有理单变量表示的关键步骤——变得困难。为筛选分离元，通过划分参数空间可确保在每个分支下理想具有相同的零点个数。进而借助参数情形下的拓展子结式定理，给出了两种对参数空间进行进一步划分以选取和检验分离元的思路。基于此，我们提出了两种计算含参数零维理想的有理单变量表示的算法。此外，这两种算法已在计算机代数系统Singular上实现。实验数据表明，与第一种算法相比，第二种算法具有更优的性能。