Two new efficient algorithms for computing greatest common divisors (gcds) of parametric multivariate polynomials over $k[U][X]$ are presented. The key idea of the first algorithm is that the gcd of two non-parametric multivariate polynomials can be obtained by dividing their product by the generator of the intersection of two principal ideals generated by the polynomials. The second algorithm is based on another simple insight that the gcd can be extracted using the generator of the ideal quotient of a polynomial with respect to the second polynomial. Since the ideal intersection and ideal quotient in these cases are also principal ideals, their generators can be obtained by computing minimal Gröbner bases of the ideal intersection and ideal quotient, respectively. To avoid introducing new variables which can adversely affect the efficiency, minimal Gröbner bases computations are performed on modules. Both of these constructions generalize to the parametric case as shown in the paper. Comprehensive Gröbner system constructions are used for the parametric ideal intersection and ideal quotient using the Kapur-Sun-Wang's algorithm. It is proved that whether in a minimal comprehensive Gröbner system of a parametric ideal intersection or in that of a parametric ideal quotient, each branch of the specializations corresponds to a principal parametric ideal with a single generator. Using this generator, the parametric gcd of that branch is obtained by division. For the case of more than two parametric polynomials, we can use the above two algorithms to compute gcds recursively, and get an extended algorithm by generalizing the idea of the second algorithm. Algorithms do not suffer from having to apply expensive steps such as ensuring whether parametric polynomials are primitive w.r.t. the main variable as used in both the algorithms proposed by Nagasaka (ISSAC, 2017). The resulting algorithms are not only conceptually simple to understand but are more efficient in practice. The proposed algorithms and both of Nagasaka's algorithms have been implemented in Singular, and their performance is compared on a number of examples.

两种新的高效算法，用于计算参数多元多项式的最大公因数（gcd） $ķ[ü][X]$被提出。第一种算法的关键思想是，可以通过将两个非参数多元多项式的乘积除以多项式生成的两个主要理想的交点的生成器来获得它们的乘积。第二种算法基于另一个简单的见解，即可以使用多项式相对于第二多项式的理想商的生成器来提取gcd。由于在这种情况下理想交点和理想商也是主要理想，因此可以分别通过计算理想交点和理想商的最小Gröbner基来获得它们的生成器。为了避免引入可能对效率产生不利影响的新变量，对模块进行最少的Gröbner基计算。如本文所示，这两种构造都可以推广到参数情况。使用Kapur-Sun-Wang算法，将完整的Gröbner系统构造用于参数理想交点和理想商。事实证明，无论是在参数理想交集的最小综合Gröbner系统中还是在参数理想商的最小Gröbner系统中，专业化的每个分支都对应于具有单个生成器的主要参数理想。使用该生成器，该分支的参数gcd通过除法获得。对于两个以上的参数多项式，我们可以使用以上两种算法递归计算gcds，并通过推广第二种算法的思想来获得扩展算法。算法无需经历昂贵的步骤，例如可以确保参数多项式是否是主要变量，就像Nagasaka提出的两种算法（ISSAC，2017年）中所使用的那样。所得的算法不仅在概念上易于理解，而且在实践中更为有效。所提出的算法和Nagasaka的两种算法均已在Singular中实现，并且在许多示例中对它们的性能进行了比较。