On the Connection Between Ritt Characteristic Sets and Buchberger-Gr"obner Bases

For any polynomial ideal $I$, let the minimal triangular set contained in the reduced Buchberger-Gr"obner basis of $I$ with respect to the purely lexicographical term order be called the W-characteristic set of $I$. In this paper, we establish a strong connection between Ritt’s characteristic sets and Buchberger’s Gr"obner bases of polynomial ideals by showing that the W-characteristic set $C$ of $I$ is a Ritt characteristic set of $I$ whenever $C$ is an ascending set, and a Ritt characteristic set of $I$ can always be computed from $C$ with simple pseudo-division when $C$ is regular. We also prove that under certain variable ordering, either the W-characteristic set of $I$ is normal, or irregularity occurs for the $j$th, but not the $(j+1)$th, elimination ideal of $I$ for some $j$. In the latter case, we provide explicit pseudo-divisibility relations, which lead to nontrivial factorizations of certain polynomials in the Buchberger-Gr"obner 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-Gr"obner bases computation.

💡 Research Summary

The paper introduces the notion of a W‑characteristic set for a polynomial ideal I, defined as the minimal triangular set contained in the reduced Buchberger‑Gröbner basis of I with respect to the pure lexicographic order. The authors first recall the necessary background on pseudo‑division, triangular and ascending sets, regular and normal chains, and the theory of Gröbner bases. They then prove a series of fundamental results that connect the W‑characteristic set with Ritt’s characteristic sets. The key theorem states that if the W‑characteristic set C is an ascending set—i.e., each element is pseudo‑reduced with respect to all preceding elements—then C itself is a Ritt characteristic set of the ideal. Moreover, when C is regular, a simple pseudo‑division process can transform C into a genuine Ritt characteristic set even if C is not initially ascending. This provides a constructive method for obtaining Ritt characteristic sets directly from Gröbner bases. The paper also investigates the effect of variable ordering. Under a suitable ordering, either the W‑characteristic set is normal (all initials are free of later variables) or irregularity appears at a specific elimination level j but not at level j + 1. In the irregular case the authors give explicit pseudo‑divisibility relations that reveal non‑trivial factorizations of certain basis elements, thereby exposing hidden structure in the Gröbner basis. These relations suggest an algorithmic scheme for decomposing arbitrary polynomial systems into normal triangular sets using Gröbner basis computation as a preprocessing step. The work thus bridges two historically independent computational frameworks, showing that the information encoded in a Gröbner basis is sufficient to reconstruct Ritt’s characteristic sets and to analyze the algebraic structure of the underlying ideal. The results have potential impact on both symbolic computation—by improving elimination and decomposition algorithms—and on theoretical algebraic geometry, where understanding the interplay between term‑order based reductions and variable‑order based triangular decompositions is essential.