On the walk

AbstractThe Gröbner Walk is a basis conversion method proposed by Collart, Kalkbrener, and Mall. It converts a given Gröbner basis G of a (possibly positive dimensional) polynomial ideal I to a Gröbner basis G′ of I with respect to another term order. The target Gröbner basis is approached in several steps (the Walk), each performing a simpler Gröbner basis computation. We address a host of questions associated with this method: alternative ways of presenting the main algorithm, algorithmic variations and refinements, implementation techniques, promising applications, and its practical performance, including a comparison with the FGLM conversion method. Our results show that the Walk has the potential to become a key tool for computing and manipulating ideal bases and solving systems of equations