standard Bases and Primary Decomposition in Polynomial Ring with Coefficients in Rings

Abstract

The theory of standard bases in polynomial rings with coe cients 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 coe cients in Euclidean rings is given.This algorithm computes successively a Grobner basis replacing the reduction of one single s-polynomial in Buchberger's algorithm by the simultaneous reduction of several polynomials. And nally 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 nite elds, 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.