Valuations and Filtrations

The classical theory of Gröbner bases, as developed by Bruno Buchberger, can be expanded to utilize objects more general than term orders. Each term order on the polynomial ring k [x] produces a filtration of k [x] and a valuation ring of the rational function fieldk (x). The algorithms developed by Buchberger can be performed by using directly the induced valuation or filtration in place of the term order. There are many valuations and filtrations that are suitable for this general computational framework that are not derived from term orders, even after a change of variables. Here we study how to translate between properties of filtrations and properties in valuation theory, and give a characterization of which valuations and filtrations are derived from a term order after a change of variables. This characterization illuminates the properties of valuations and filtrations that are desirable for use in a generalized Gröbner basis theory.