Converting Bases with the Gröbner Walk

The objective of this note is the presentation of a procedure for converting a given Gröbner basis (Buchberger, 1965) of a polynomial ideal I to a Gröbner basis of I with respect to another term order. This procedure, which we call the Gröbner walk, is completely elementary and does not require any assumptions about the dimension or the number of variables of the ideal. The Gröbner walk breaks up the conversion problem into several simple steps between adjacent Gröbner bases following a path in the Gröbner fan. Since two term orders leading to adjacent Gröbner bases can be viewed as refinements of a common partial order, these simple transformations can be computed working just with the initial forms with respect to this partial order. Because the initial forms typically involve far fewer terms than the polynomials as a whole, the transformations can be computed cheaply. First experiments seem to indicate that the Gröbner walk performs rather well for large classes of examples. It is interesting to note that, although the theoretic concepts on which the algorithm is based are neither new nor complicated, it has not been considered before as a candidate for efficient change of basis, even though there has been some interest in Gröbner basis conversion for some time (see for instance Faugère et al., 1993; Traverso, 1993; Gianni et al., 1994; Faugère, 1994). The main reason for the interest in this question is the obvious demand for fast conversion algorithms. For instance, if for some polynomial ideal a Gröbner basis with respect to some elimination order is sought, it may well be more efficient to compute first a Gröbner basis with respect to a total degree order, and then to convert, since total degree bases are generally much faster to compute. More specialized applications which by nature involve basis conversions might for instance be the implicitization of varieties (Hoffmann, 1989; Licciardi and Mora, 1994; Kalkbrener, 1996) and the inversion of polynomial isomorphisms. The authors are indebted to several sources for their inspiration. The article by Faugère