A Gröbner walk is an algorithm that converts a Gröbner basis of a polynomial ideal for one monomial order into a Gröbner basis of the same ideal for another monomial order (Collart et al. 1997). It follows a path through the polyhedral cones of the Gröbner fan. At a shared face, a basis conversion for the corresponding initial ideal is lifted to a basis for the original ideal. This replaces one large conversion by a succession of smaller conversions. The method applies to ideals of positive Krull dimension as well as zero Krull dimension (Tran 2000).
A usual weight path has the form
 |
where 
and 
and 
represent the starting and target monomial
orders for the ideal. For example, for 
, the path 
crosses 
at 
. The leading term changes
from 
in the reduced Gröbner basis 
to 
in 
.
In practice, conversion can make it useful to compute a basis in an easier monomial order before converting to lexicographic order when solving polynomial equations (Trott 2006). This is a possible computational advantage, rather than a guarantee that every conversion is faster. The generic Gröbner walk replaces explicit numerical perturbations of the weight path by comparisons of vectors using monomial orders (Fukuda et al. 2007).
