Journal of Symbolic Computation
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Theoretical Computer Science - Special volume on computer algebra
A fast algorithm for Gröbner basis conversion and its applications
Journal of Symbolic Computation - Special issue on applications of the Gröbner basis method
Buchberger Algorithm and Integer Programming
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Term Orderings on the Polynominal Ring
EUROCAL '85 Research Contributions from the European Conference on Computer Algebra-Volume 2
Hard Equality Constrained Integer Knapsacks
Mathematics of Operations Research
Gröbner bases: a sampler of recent developments
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Universal characteristic decomposition of radical differential ideals
Journal of Symbolic Computation
Converting subalgebra bases with the Sagbi walk
Journal of Symbolic Computation
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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.