基于2008年Zhou和Winkler给出的计算有限生成的差分-微分双滤模的希尔伯特多项式的算法,文章构造了差分-微分模上相对多个序的的Gr\"{o}bner基,并给出 和证明了计算这种Gr\"{o}bner基的算法.作为其应用, 给出了计算差分-微分模的多变量维数多项式的新算法.  推广了Zhou和Winkler (2008)所得结果,也推进了Levin (2007)所得结果.

In this paper we present a new algorithmic approach for computing the Gr¨obner bases with respect to several generalized term orders on Nm×Zn and on difference-differential modules. We define a special type of reduction for several generalized term orders in a free left module over a ring of difference-differential operators. This reduction is different from the reduction of Levin (2007). Then the concept of Gr¨obner bases with respect to several generalized
term orders is defined. An algorithm for constructing these Gr¨obner bases is presented and verified. Using the Gr¨obner bases, we are able to compute difference-differential dimension polynomials in several variables in the case of that the difference operators are inversive. So our results have developed the theorem of Levin (2007) to Laurent-Ore polynomial ring, while Levin considered difference-differential dimension polynomials in several variables for  odules
over Ore polynomial rings with non-inversive difference operators. Moreover, the result is a generalization of theories of Zhou and Winkler (2008).