ON THE WALK

The Grobner Walk is a basis conversion method proposed by Collart, Kal kbrener, and Mall. It converts a given Grobner basis G of a (possibly positive dimensional) polynomial ideal I to a Grobner basis G' of I wi th respect to another term order. The target Grobner basis is approach ed in several steps (the Walk), each performing a simpler Grobner basi s computation. We address a host of questions associated with this met hod: alternative ways of presenting the main algorithm, algorithmic va riations and refinements, implementation techniques, promising applica tions, and its practical performance, including a comparison with the FGLM conversion method. Our results show that the Walk has the potenti al to become a key tool for computing and manipulating ideal bases and solving systems of equations.