A Fast Algorithm for Gröbner Basis Conversion and its Applications

Abstract

The Gröbner walk method converts a Gröbner basis by partitioning the computation of the basis into several smaller computations following a path in the Gröbner fan of the ideal generated by the system of equations. The method works with ideals of zero-dimension as well as positive dimension. Typically, the target point of the walking path lies on the intersection of very many cones, which ends up with initial forms of a considerable number of terms. Therefore, it is crucial to the performance of the conversion to change the target point since we have to compute a Gröbner basis with respect to the elimination term order of such large initial forms. In contrast to heuristic methods found in the literature, in this paper the author presents a deterministic method to vary the target point in order to ensure the generality of the position, i.e. we always have just a few terms in the initial forms. It turns out that this theoretical result brings a dramatic speed-up in practice. We have implemented the Gröbner walk method together with the deterministic method for varying the target point in the kernel of Mathematica. Our experiments show the superlative performance of our improved Gröbner walk method in comparison with other known methods. Our best performance is 3 × 104times faster than the direct computation of the reduced Gröbner basis with respect to pure lexicographic term order (using the Buchberger algorithm and the sugar cube strategy). We also discuss the complexity of the conversion algorithm and prove a degree bound for polynomials in the target Gröbner basis. In the second part of the paper, we present some applications of the conversion method for implicitization and geometric reasoning. We compare the efficiency of the improved Gröbner walk method with other methods for elimination such as multivariate resultant methods.