Term orders on the polynomial ring and the Gro?bner fan of an ideal

by 1974- Tarrant, Dennis Wayne

Robbiano classified term orders by using ordered systems of vectors. Unfortunately his classification gives little information as to the intuitive “shape” of these spaces. We seek to understand the structure of the spaces of term orders by introducing a topology on them. We first consider the space of term orders in the bivariate case. We convert weight vectors into slopes and determine that rational slopes require the selection of a “tiebreaking” term order while irrational slopes represent term orders by themselves. By placing an order topology on this space of bivariate term orders, we show that this space has several topological properties. All of these topological properties imply that the space of bivariate term orders is homeomorphic to the Cantor set. We then consider the spaces of term orders in the general case. We set up a description of the space of term orders in n ? 2 variables as a subspace of a function space. When we consider the topological properties of this view of the term order space on n ? 2 variables, we find that it is homeomorphic to a compact subset of the Cantor set. These topological descriptions yield important facts about the spaces of term orders that are otherwise very difficult to see or prove. In particular the fact that the Gröbner fan of an ideal has finitely many cones is implied by the compactness of the space of term orders. This was shown previously, but the proof here is much simpler once the topological description of the spaces of term orders is determined. Finally some facts about the associated geometry are given. The realization of the term order spaces as compact subspaces of Cantor sets leads one to believe certain things about the Gröbner fan. We show the relations between the Gröbner fan and the Netwon polytopes of elements of the reduced Gröbner bases.