On Computing Grobner Bases in Rings of Differential Operators with Coefficients in a Ring

Following the definition of Grobner bases in rings of differential operators given by Insa and Pauer (1998), we discuss some computational properties of Grobner bases arising when the coefficient set is a ring. First we give examples to show that the generalization of S-polynomials is necessary for computation of Grobner bases. Then we prove that under certain conditions the G-S-polynomials can be reduced to be simpler than the original one. Especially for some simple case it is enough to consider S-polynomials in the computation of Grobner bases. The algorithm for computation of Grobner bases can thus be simplified. Last we discuss the elimination property of Grobner bases in rings of differential operators and give some examples of solving PDE by elimination using Grobner bases.