On the Connection Between Ritt Characteristic Sets and Buchberger–Gröbner Bases

Abstract

For any polynomial ideal $$\mathcal {I}$$ , let the minimal triangular set contained in the reduced Buchberger–Gröbner basis of $$\mathcal {I}$$ with respect to the purely lexicographical term order be called the W-characteristic set of $$\mathcal {I}$$ . In this paper, we establish a strong connection between Ritt’s characteristic sets and Buchberger’s Gröbner bases of polynomial ideals by showing that the W-characteristic set $$\mathbb {C}$$ of $$\mathcal {I}$$ is a Ritt characteristic set of $$\mathcal {I}$$ whenever $$\mathbb {C}$$ is an ascending set, and a Ritt characteristic set of $$\mathcal {I}$$ can always be computed from $$\mathbb {C}$$ with simple pseudo-division when $$\mathbb {C}$$ is regular. We also prove that under certain variable ordering, either the W-characteristic set of $$\mathcal {I}$$ is normal, or irregularity occurs for the jth, but not the $$(j+1)$$ th, elimination ideal of $$\mathcal {I}$$ for some j. In the latter case, we provide explicit pseudo-divisibility relations, which lead to nontrivial factorizations of certain polynomials in the Buchberger–Gröbner basis and thus reveal the structure of such polynomials. The pseudo-divisibility relations may be used to devise an algorithm to decompose arbitrary polynomial sets into normal triangular sets based on Buchberger–Gröbner bases computation.