Algorithms in Local Algebra

Abstract Let k be a field, S = k [ x v : v ϵ V ] be the polynomial ring over the finite set of variables ( x v : v ϵ V ), and m = ( x v : v ϵ V ) the ideal defining the origin of Spec S . It is theoretically known (see e.g. Alonso et el. , 1991) that the algorithmic ideas for the computation of ideal (and module) intersections, quotients, deciding radical membership etc. in S may be adopted not only for computations in the local ring S m but also for term orders of mixed type with standard bases replacing Grobner bases. Using the generalization of Mora's tangent cone algorithm to arbitrary term orders we give a detailed description of the necessary modifications and restrictions. In a second part we discuss a generalization of the deformation argument for standard bases and independent sets to term orders of mixed type. For local term orders these questions were investigated in Grabe (1991). The main algorithmic ideas described are implemented in the author's REDUCE package CALI (Grabe, 1993a).

[1]  Bruno Buchberger,et al.  Applications of Gro¨bner bases in non-linear computational geometry , 1988 .

[2]  Hans-Gert Gräbe,et al.  The tangent cone algorithm and homogenization , 1994 .

[3]  Bruno Buchberger,et al.  Applications of Gröbner Bases in Non-linear Computational Geometry , 1987, Trends in Computer Algebra.

[4]  Kei-ichi Watanabe,et al.  On graded rings, I , 1978 .

[5]  Patrizia M. Gianni,et al.  Gröbner Bases and Primary Decomposition of Polynomial Ideals , 1988, J. Symb. Comput..

[6]  Ferdinando Mora,et al.  An Algorithm to Compute the Equations of Tangent Cones , 1982, EUROCAM.

[7]  Daniel Lazard,et al.  Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations , 1983, EUROCAL.

[8]  Heinz Kredel,et al.  Computing Dimension and Independent Sets for Polynomial Ideals , 1988, J. Symb. Comput..

[9]  T. Becker.  Stability and buchberger criterion for standard bases in power series rings , 1990 .

[10]  Teo Mora,et al.  The Gröbner Fan of an Ideal , 1988, J. Symb. Comput..

[11]  D. Bayer.  The division algorithm and the hilbert scheme , 1982 .

[12]  Teo Mora,et al.  Local Decomposition Algorithms , 1990, AAECC.

[13]  Lorenzo Robbiano,et al.  On the Theory of Graded Structures , 1986, J. Symb. Comput..

[14]  Hans-Gert Gräbe,et al.  Two Remarks on Independent Sets , 1993 .

[15]  Teo Mora,et al.  Computing with algebraic series , 1989, ISSAC '89.

[16]  Heinz Kredel,et al.  Gröbner Bases: A Computational Approach to Commutative Algebra , 1993 .

[17]  Hans Schönemann,et al.  Standard bases, syzygies and their implementation in SINGULAR , 1994 .

[18]  Teo Mora,et al.  A computational model for algebraic power series , 1992 .