TL;DR
This paper systematically explores the relationship between non-homogeneous and homogeneous Gr"obner bases using (de)homogenization techniques, providing a clearer method for computing Gr"obner bases through homogenized generators.
Contribution
It offers a systematic clarification of computing Gr"obner bases via homogenization, improving upon previous related works.
Findings
Established a general principle for computing Gr"obner bases using homogenization.
Strengthened previous results in the literature.
Clarified the relation between non-homogeneous and homogeneous Gr"obner bases.
Abstract
By employing the (de)homogenization technique in a relatively extensive setting, this note studies in detail the relation between non-homogeneous Gr\"obner bases and homogeneous Gr\"obner bases. As a consequence, a general principle of computing Gr\"obner bases (for an ideal and its homogenization ideal) by passing to homogenized generators is clarified systematically. The obtained results improve and strengthen the work of [LWZ], [Li1], [Li2], [Li3], and very recent [SL] concerning the same topic.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Polynomial and algebraic computation · Cancer Treatment and Pharmacology