Rees valuations of monomial ideals

e establish a procedure to make the Rees valuations of an m-primary monomial ideal, where m is the maximal ideal generated by all the variables in a finite dimensional polynomial ring over a field, more effectively computable. This procedure involves using a correspondence between the Rees valuations of the ideal and the bounded facets of the Newton polyhedron which can be constructed from the ideal. We also introduce the notion of a minimal monomial reduction of a monomial ideal. It is shown that the vertex set of the Newton polyhedron of an m-primary monomial ideal can be read off from a presentation of any minimal monomial reduction of the ideal, and hence a minimal monomial reduction of such an ideal is unique.;We use the procedure for finding Rees valuations to generically compute the Rees valuations associated with a product of two one-fibered monomial ideals in a three-dimensional polynomial ring over a field, and establish numeric recipes for the Rees valuations of these products including a specific criterion for when excess valuations are produced. Finally we show how these formulas can be applied to questions of factorization and use these formulas to produce an explicit...