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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
##### Keywords:
standard base; tangent cone algorithm
SINGULAR
Full Text:
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