An Attempt to Enhance Buchberger's Algorithm by Using Remainder Sequences and GCDs (II) (Computer Algebra - Theory and its Applications)

Let F = {F, , ..., Fm+1} ⊂ ℚ[x, u] be a given system, where m+l 2: 3, (x) = (x, , ..., xm) and (u) = (u, , ...，叫）， with ∀xi >-- ∀uj. Let GB(F) = {G₁, G₂, ・・・}, with G₁ --< G₂ --< ・・・, be the reduced Grabner basis of F w.r.t. the lexicographic order. In a previous paper [10], one of the authors proposed a method of enhancing Buchberger's algorithm for computing GB(F). His idea_is to compute a set g':= {G1 , G2, ... } ⊂ ℚ[x, u], such that each Gi is either O or as mall multiple of Gi, and apply Buchberger's algorithm to F ∨ g'. He proposed a scheme of computing G₁, G₂, ... by the PRSs (polynomial remainder sequences) and the GCDs in "G₁ ⇒ G₂ ⇒ ・・・" order, without computing Spolynomials. The scheme is supported by two new useful theorems and one proposition to remove the extraneous factor. In fact, for a simple but never toy example, his scheme has computed G₁ successfully (G₁ became G₁ by the proposition mentioned above). However, an unexpected difficulty occurred in computing G₂; it contained a pretty large extraneous factor which was not removed by the proposition. In this paper, we find a surprising phenomenon with which we can remove the above mentioned extraneous factor in G₂ and obtain G₂. As for G₃ and G₄, we obtain very good "body doubles" of them, by eliminating variables in leading coefficients of intermediate remainders of the PRSs computed for G₁. For systems of many sub-variables, n ≥ 3, our method introduces an extra factor in ℚ[u3, ..，un]， into the "LCto W" polynomial; see the text for the LCtoW polynomial. Furthermore, we present several techniques to enhance the computation