The Gröbner walk revisited

The Gröbner walk is an algorithm for Gröbner basis conversion of ideals in the polynomial ring I < k[x1, ..., xn]. It exploits the geometric properties of the Gröbner fan, which is a polyhedral fan in R^n, the maximal cones of which are in one-to-one correspondence with the marked Gröbner bases of I. In this text, we describe the functionality of the algorithm step by step and provide a formal proof of correctness. We then survey a series of modifications which have been proposed over the years, most notably path perturbation and the generic Gröbner walk. Finally we discuss the performance of the Gröbner walk in a variety of settings and compare it with other Gröbner basis conversion algorithms in both Macaulay2 and OSCAR. This is a master's thesis which was written at TU Berlin in 2024 under the supervision of Carlos Amendola.