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Published online by Cambridge University Press: 05 July 2011
Abstract
The Gröbner Walk is a method which converts a Gröbner basis of an arbitrary dimensional ideal I to a Gröbner basis of I with respect to another term order. The walk follows a path of intermediate Gröbner bases according to the Gröbner fan of I. One of the open problems in the walk algorithm is path finding in the Gröbner fan. In order to avoid intersection points in the fan, paths are perturbed up to a certain degree. The Fractal Walk allows us to perturb the path locally in each step rather than globally. Thus, it removes the difficulty of finding the globally best perturbation degree. Our implementation shows that we even obtain speedups over the best perturbation degree because of the “tunneling” effect of the Fractal Walk. In addition, the Fractal Walk is compared to other Gröbner basis conversion methods.
Introduction
It is well known that the term ordering strongly determines the complexity of the Gröbner basis computation. The choice of the term ordering usually depends on the type of problem we want to solve. Elimination orders such as lexicographic, which we need for polynomial system solving, are known to be slow term orders, that is, they lead to particularly long computations.
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