  The Groebner walk is an algorithm for conversion between Groebner bases for
different term orders. It is based on the polyhedral geometry of the Groebner
fan and involves tracking a line between cones representing the initial and
target term order. An important parameter is explicit numerical perturbation of
this line. This usually involves both time and space demanding arithmetic of
integers much larger than the input numbers. In this paper we show how the
explicit line may be replaced by a formal line using Robbiano's
characterization of group orders on Q^n. This gives rise to the generic
Groebner walk involving only Groebner basis conversion over facets and
computations with marked polynomials. The proposed technique is closely related
to the lexicographic (symbolic) perturbation method used in optimization and
computational geometry. We report on computations with toric ideals, where a
version of our algorithm in certain cases computes test sets for hard integer
knapsack problems significantly faster than the Buchberger algorithm.
