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 zerodimension 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 × 10 4 times 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.

Computing the Radical of an Ideal in Positive Characteristic

“…Although the Gröbner basis can be computed with Buchberger's algorithm, we can use Gröbner basis conversion algorithms (Faugère et al, 1993;Traverso, 1996;Collart et al, 1997;Amrhein et al, 1997;Tran, 2000), because we can immediately derive a Gröbner basis for J with respect to an elimination order with the Y variables larger than the X variables from that for I as follows. Basis conversion algorithms are considered faster than Buchberger's algorithm in many cases.…”

The Legendre Symbol and the Modulo-2 Operator in Symmetric Schemes over Fnp

“…However, eliminating variables and finding the final solutions is in general more efficient for e.g. the lex term order [Tra00], which is why the second step usually consists of converting the Gröbner basis to a different monomial ordering. This can be done by algorithms such as FGLM [FGLM93].…”

The generic Gröbner walk

“…Computing an interior point in the target cone C ≺ 2 (I) is considerably more difficult, since we do not know the reduced Gröbner basis over ≺ 2 in advance. Tran [19] approached this problem using general degree bounds on polynomials in Gröbner bases. The general degree bounds in Tran's approach may lead to integral weight vectors with 10, 000digit entries in representing a lexicographic interior point in the case of polynomials of degree 10 in 10 variables.…”