Ed Mosteig
Using Valuations to Compute Gröbner Bases

Classical computations of Gröbner bases rely upon initially setting a term order that will be used in the division algorithm. Alternatively, Sweedler suggested in a paper (1986) another method of computing Gröbner bases by using valuations in place of term orders. This talk will address the use of valuations and some of the computational issues involved. As will be shown, all term orders can be described by appropriate valuations, but there are many well behaved valuations that have characteristics that are very different from term orders. In the setting of a polynomial ring with two variables, k[x,y], a description of some of the well-behaved valuations will be given when k has characteristic zero. Lastly, we will demonstrate some of the difficulties that arise when we shift our focus to fields of positive charcteristic.