Homogenity, pseudo-homogenity, and Gröbner basis computations

Abstract

Let K[X] be a multivariate polynomial ring over a field K, and let U ⊑ X. We call f ∈ K[X] pseudo-homogeneous in U if either it contains no variable of U, or each of its terms does. U ⊑ X is called an H-set for F ⊑ K[X] if each f ∈ F is pseudo-homogeneous in U. We show that when computing an elimination ideal of some ideal (F) of K[X] w.r.t. V ⊑ X, one may disregard all those elements of F that contain variables from some H-set U ⊑ X/V for F. For given F and V, we compute a maximal H-set for F contained in V. Furthermore, we discuss how one can compute gradings by weighted total degree that make a given finite F ⊑ K[X] homogeneous. For certain limited purposes, one can then compute truncated Gröbner bases instead of full ones.