The generic Gröbner walk

The Grobner walk is an algorithm for conversion between Grobner bases for different term orders. It is based on the polyhedral geometry of the Grobner fan and involves tracking a line between cones representing the initial and target term order. An important parameter is the explicit numerical perturbation of this line. This usually involves both time and space, demanding arithmetic of integers much larger than the input numbers. In this paper we show how the explicit line may be replaced by a formal line using Robbiano's characterization of group orders on Q^n. This gives rise to the generic Grobner walk involving only Grobner basis conversion over facets and computations with marked polynomials. The infinite precision integer arithmetic is replaced by term order comparisons between (small) integral vectors. This makes it possible to compute with infinitesimal numbers and perturbations in a consistent way without introducing unnecessary long integers. The proposed technique is closely related to the lexicographic (symbolic) perturbation method used in optimization and computational geometry. We report on an implementation of our algorithm specifically tailored to computations with lattice ideals.

[1]  Quoc-Nam Tran.  A Fast Algorithm for Gröbner Basis Conversion and its Applications , 2000, J. Symb. Comput..

[2]  Thomas Yan,et al.  The Geobucket Data Structure for Polynomials , 1998, J. Symb. Comput..

[3]  Michael Kalkbrener,et al.  Converting Bases with the Gröbner Walk , 1997, J. Symb. Comput..

[4]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

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

[6]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[7]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[8]  Ian Morrison,et al.  Standard Bases and Geometric Invariant Theory I. Initial Ideals and State Polytopes , 1988, J. Symb. Comput..

[9]  Lorenzo Robbiano,et al.  Term Orderings on the Polynominal Ring , 1985, European Conference on Computer Algebra.

[10]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[11]  Wolfgang Küchlin,et al.  On the Walk , 1997, Theor. Comput. Sci..

[12]  Bruno Buchberger,et al.  Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal , 2006, J. Symb. Comput..

[13]  Carlo Traverso,et al.  Buchberger Algorithm and Integer Programming , 1991, AAECC.

[14]  Jean-Charles Faugère,et al.  Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering , 1993, J. Symb. Comput..

[15]  Arjen K. Lenstra,et al.  Hard Equality Constrained Integer Knapsacks , 2002, Math. Oper. Res..

[16]  Niels Lauritzen.  Homogeneous Buchberger algorithms and Sullivant's computational commutative algebra challenge , 2005 .