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

Following the definition of Gr?bner bases in rings of differential operators given by Insa and Pauer (1998), we discuss some computational properties of Gr?bner 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 Gr?bner 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 Gr?bner bases. The algorithm for computation of Gr?bner bases can thus be simplified. Last we discuss the elimination property of Gr?bner bases in rings of differential operators and give some examples of solving PDE by elimination using Gr?bner bases. This work was supported by the NSFC project 60473019.     

Gr?bner bases in difference-differential modules with coefficients in a commutative ring

Insa and Pauer presented a basic theory of Grbner bases for differential operators with coefficients in a commutative ring and an improved version of this result was given by Ma et al.In this paper,we present an algorithmic approach for computing Grbner bases in difference-differential modules with coefficients in a commutative ring.We combine the generalized term order method of Zhou and Winkler with SPoly method of Insa and Pauer to deal with the problem.Our result is a generalization of theories of Insa and Pauer,Ma et al.,Zhou and Winkler and includes them as special cases.