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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)

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
Full Text: DOI
[1] Alonso, M.E.; Luengo, I.; Raimondo, M., An algorithm on quasi-ordinary polynomials, () · Zbl 0692.13011
[2] Alonso, M.E.; Mora, T.; Raimondo, M., Computing with algebraic series, Proceedings ISSAC, 89, (1989)
[3] M.E. Alonso, T. Mora and M. Raimondo, A computational model for algebraic power series, J. Pure Appl. Algebra, to appear. · Zbl 0749.13017
[4] Bayer, D., The division algorithm and the Hilbert scheme, ()
[5] Bayer, D.; Stillman, M., Communication at COCOAII, (1989)
[6] Carrà, G., Some upper bounds for the multiplicity of an autoreduced subset of Nm and their application, () · Zbl 0614.68034
[7] Dickenstein, A.; Fitchas, N.; Giusti, M.; Sessa, A., The membership problem for unmixed polynomial ideals is solvable in single exponential time, Discrete appl. math., 33, 73-94, (1991) · Zbl 0747.13018
[8] Duval, D., Diverses questions relatives au calcul formel avec des nombres algébriques, ()
[9] Furukawa, A.; Kobayashi, H.; Sasaki, T., Gröbner bases of ideals of convergent power series, (1985)
[10] Galligo, A., A propos du théorème de préparation de Weierstrass, (), 543-579
[11] Galligo, A.; Traverso, C., Practical determination of the dimension of an algebraic variety, (), 46-52
[12] Gianni, P.; Trager, B.; Zacharias, G., Gröbner bases and primary decomposition of polynomial ideals, J. symbolic comput., 6, 149-168, (1988) · Zbl 0667.13008
[13] Grieco, M.; Zucchetti, B., Communication at AAECC-7, (1989)
[14] Gröbner, W., Algebraische geometrie, (1968), Bibliographiches Institut Mannheim · JFM 64.1320.02
[15] Kandrateva, M.V.; Pankratev, E.V., A recursive algorithm for computation of the Hilbert polynomial, (), 365-375
[16] Kredel, H.; Weispfenning, V., Computing dimension and independent sets for polynomial ideals, J. symbolic comput., 6, 231-248, (1988) · Zbl 0665.68024
[17] Lazard, D., Gröbner bases, Gaussian elimination and resolution of systems of algebraic equations, (), 146-156
[18] Logar, A., A computational proof of the Noether normalization lemma, (), 259-273
[19] Luengo, I.; Pfister, G., Normal forms and moduli spaces of curve singularities with semigroup 〈2p, 2q, 2pq+d〉, (1988), Universidad Complutense Madrid, Preprint
[20] Möller, H.M.; Mora, F., The computation of the Hilbert function, (), 157-167
[21] Möller, H.M.; Mora, F., New constructive methods in classical ideal theory, J. algebra, 100, (1986) · Zbl 0621.13007
[22] Mora, F., An algorithm to compute the equations of tangent cones, (), 158-165
[23] Mora, F., A constructive characterization of standard bases, Boll. un. mat. ital., D 2, 41-50, (1983) · Zbl 0619.13010
[24] Mora, F., An algorithmic approach to local rings, (), 518-525
[25] T. Mora, G. Pfister and C. Traverso, An introduction to the tangent cone algorithm, in: C. Hoffman, ed., Issues in Non-Linear Geometry and Robotics (JAI Press, Greenwidge, CT, to appear).
[26] Pfister, G.; Schönemann, H., Singularities with exact poincarécomplex but not quasihomogeneous, () · Zbl 0708.14018
[27] Robbiano, L., Coni tangenti a singolaritàrazionali, Atti conv. geom. alg. firenze, (1981)
[28] Robbiano, L., On the theory of graded structures, J. symbolic comput., 2, 139-170, (1986) · Zbl 0609.13007
[29] Spangher, W., On the computation of the Hilbert-Samuel series and multiplicity, (), 407-414
[30] Walker, R.J., Algebraic curves, (1978), Springer Berlin
[31] Zariski, O.; Samuel, P., Commutative algebra, (1985), Van Nostrand Reinhold New York · Zbl 0121.27901
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