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La queste del Saint \(\text{Gr}_ a(\text{AL})\): A computational approach to local algebra. (English) Zbl 0752.13016
Let \(P=k[x_ 1,\ldots,x_ n]\) be the polynomial ring over a field \(k\) and \(<\) be an ordering of the set of monomials of \(P\) compatible with its semigroup structure. For a power series \(f\in\hat P=k[[x_ 1,\ldots,x_ n]]\) let \(L(f)\) be the leading monomial of \(f\) with respect to the ordering. For an ideal \(I\subseteq\text{Loc}(P):=\{(1+g)^{-1}\cdot f\mid L(g)<1,\;f,g\in P\}\) let \(L(I)\) be the ideal generated by all leading monomials of elements of \(I\). A set \(\{f_ 1,\ldots,f_ m\}\) of elements of \(I\) is called a standard base of \(I\) if \(\{L(f_ 1),\ldots,L(f_ m)\}\) generate \(L(I)\). If \(<\) is a well ordering then Buchberger’s algorithm computes a standard base in \(P\). — The author generalized this algorithm for a larger class of orderings. His tangent cone algorithm allows to compute standard bases in the local situation, i.e. for instance if \(\text{Loc}(P)=k[x_ 1,\ldots,x_ n]_{(x_ 1,\ldots,x_ n)}\), and it is therefore a very important tool for local algebraic geometry. A recent implementation of this algorithm is included in SINGULAR (developed at the Humboldt-University in Berlin and the University of Kaiserslautern).
The author gives a survey about the tangent cone algorithm and several applications in local algebra. He also describes a computational model for the ring of algebraic power series which gives effective versions of the Weierstraß preparation theorem and the Noether normalization lemma.
Reviewer: G.Pfister (Berlin)

MSC:
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14Q99 Computational aspects in algebraic geometry
Software:
SINGULAR
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