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

For any polynomial ideal I, let the minimal triangular set contained in the reduced Buchberger–Gröbner 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 Gröbner 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 jth, 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–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.