# 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.

## Keywords

Integer Linear Programming Finite Subset Critical Pair Finite Basis Integer Linear Programming Problem## Preview

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