A valuation-theoretic approach to polynomial computations
Edward Mosteig, Moss Eisenberg Sweedler · 2000
The study of Grobner bases requires the use of term orders to perform various algorithms involving multivariate polynomials. Using techniques developed by Moss Sweedler and Lorenzo Robbiano, it is possible to perform similar computations in a more general setting. In this manuscript, we give an exposition of the general theory of Grobner bases and demonstrate how this theory specializes to the classical case in which term orders are utilized. Given a polynomial ring R, we define a general filtration to be a nested sequence of subsets of R. We can form a correspondence between valuations on the rational function field k( x1,…,xn) and filtrations on k[x1,…, xn] with special properties. The fundamental concept which allows Buchberger's classical reduction algorithms to be performed in finite time is the well-ordered property of term orders. In the spirit of the work laid out by Moss Sweedler and Lorenzo Robbiano, we consider methods of polynomial computations without the direct use of term orders. Instead, Grobner bases are constructed by using valuations having special properties. Although we present our computational theory in the framework of valuations, we could have just as well developed our theory in terms of valuation rings or filtrations with special additional properties. In this spirit, we formulate and discuss a correspondence between filtrations, valuations, and valuation rings in the intermediate chapters of the manuscript. In light of this correspondence, we give a criterion for the case when filtrations, valuations, and valuation rings arise from a term order in suitable variables. This characterization leads us to search for other well-ordered valuations which do not come from a term order in suitable variables. We close by giving the first examples of well-ordered, zero-dimensional valuations on k(x, y) with respect to the underlying polynomial ring k[x, y]. An infinite family of such valuations where k is a field of characteristic zero is given. Additionally, we give characteristic-free examples of well-ordered valuations, and in positive characteristic, we construct valuations which are not well-ordered.