STANDARD BASES OVER RINGS

Standard bases in mixed power series and polynomial rings over rings

In this paper we study standard bases for submodules of a mixed power series and polynomial ring RJt1, . . . , tmK[x1, . . . , xn] s respectively of their localization with respect to a t-local monomial ordering for a certain class of noetherian rings R. The main steps are to prove the existence of a division with remainder generalizing and combining the division theorems of Grauert–Hironaka an...

D-Bases for Polynomial Ideals over Commutative Noetherian Rings

Reduced Gröbner Bases in Polynomial Rings over a Polynomial Ring

We define reduced Gröbner bases in polynomial rings over a polynomial ring and introduce an algorithm for computing them. There exist some algorithms for computing Gröbner bases in polynomial rings over a polynomial ring. However, we cannot obtain the reduced Gröbner bases by these algorithms. In this paper we propose a new notion of reduced Gröbner bases in polynomial rings over a polynomial r...

On Signature-Based Gröbner Bases Over Euclidean Rings

In this paper we present first steps in using signature-based Gröbner basis algorithms like Faugère’s F5 or GVW for computation over Euclidean rings. We present problems appearing when having to deal with coefficients and zero divisors and give practical solution techniques. A hybrid algorithm is presented trying to combine the advantages of signature-based and non-signature-based Gröbner basis...

Projective modules over polynomial rings and dynamical Gröbner bases