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An algorithmic approach to local rings. (English) Zbl 0582.13002
Computer algebra, EUROCAL ’85, Proc. Eur. Conf., Linz/Austria 1985, Vol. 2, Lect. Notes Comput. Sci. 204, 518-525 (1985).
[For the entire collection see Zbl 0568.00019.]
Algorithms are presented for computing the form ring of a local ring A and a standard basis for any ideal I in A.
The basic ideas for the generalized Buchberger algorithm are given. It is shown how this algorithm allows to compute standard bases for ideals in localizations of polynomial rings at prime ideals. This technique is extended to rings of the kind \(A:=P_{{\mathfrak p}}/JP_{{\mathfrak p}}\), where P is a polynomial ring, \(J\subset {\mathfrak p}\) ideals, \({\mathfrak p}\) prime. Some applications related to Hilbert function and regularity are marked.
Reviewer: K.Peeva

13-04 Software, source code, etc. for problems pertaining to commutative algebra
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13H05 Regular local rings
68W30 Symbolic computation and algebraic computation