Abstract:
The theory of standard bases in polynomial rings with coefficients in a ring A with respect
to local orderings is developed. A is a commutative Noetherian ring with 1 and we assume
that linear equations are solvable in A.
Then the generalization of Faug ́ere F4-algorithm for polynomial rings with coefficients
in Euclidean rings is given. This algorithm computes successively a Gr ̈obner basis replacing
the reduction of one single s-polynomial in Buchberger’s algorithm by the simultaneous
reduction of several polynomials.
And finally we present an algorithm to compute a primary decomposition of an ideal
in a polynomial ring over the integers. For this purpose we use algorithms for primary
decomposition in polynomial rings over the rationals resp. over finite fields, and the idea
of Shimoyama–Yokoyama resp. Eisenbud–Hunecke–Vasconcelos to extract primary ideals
from pseudo-primary ideals. A parallelized version of the algorithm is implemented in
Singular.
