We introduce the concept of difference-differential degree compatibility on generalized term orders. Then we prove that in the process of the algorithm the polynomials with higher and higher degree would not be produced, if the term orders ‘?’ and ‘?′’ are difference-differential degree compatibility. So we present a condition on the generalized orders and prove that under the condition the algorithm for computing relative Gr?bner bases will terminate. Also the relative Gr?bner bases exist under the condition. Finally, we prove the algorithm for computation of the bivariate dimension polynomials in difference-differential modules terminates.
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