n元微分算子的因子分解
Factoring Differential Operators in n Variables
摘要：
本文给出了多项式n阶Weyl代数、多项式n阶移位代数和n阶q-Weyl代数中ZZ^n-分次多项式的一个新的分解算法和实验实现。最意想不到的结果是，这个非交换问题的分解偏微分算子可以有效地接近减少它的问题，解决系统的多项式方程的交换环。在给定多项式是ZZ^n-分次的情况下，我们可以将问题完全归结为对交换多元多项式环中的元素进行因式分解。Singular中的实现在广泛的多项式上是有效的，并且增加了计算机代数系统解决这一重要问题的能力。我们比较了我们的算法的性能和输出与其他实现商品计算机代数系统上的非平凡的例子。
英文摘要：
In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is ZZ^n-graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in commodity computer algebra systems on nontrivial examples.
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