A Bridge between Euclid and Buchberger: (An Attempt to Enhance Gröbner Basis Algorithm by PRSs and GCDs)

This article surveys a very new method of enhancing Buchberger's Gröbner basis algorithm by the PRSs (polynomial remainder sequences) and the GCDs of multivariate polynomials. Let F = {F1,...,Fm+1} ⊂ K[x, u] be a given system, where (x) = (x1,...,xm) and (u) = (u1,...,un). Currently, we treat only such Fs that are "healthy" (see the text). Let [EQUATION], where [EQUATION], be the reduced Gröbner basis of ideal (F) w.r.t. the lexicographic order, to be abbreviated to GB(F). Let [EQUATION], be such that [EQUATION] is a small multiple of G1, and the leading monomial of [EQUATION], is a multiple (hopefully small) of the leading monomial of Gi. Our method computes [EQUATION] first, then computes [EQUATION]. Finally, we will apply Buchberger's method to system [EQUATION]. Four new theorems are given. The first and second ones are to compute the lowest-order element of the ideal generated by relatively prime G, H ∈ K[x, u], without computing any Spolynomial. The third theorem says that if F is healthy then [EQUATION]. We compute resultants in K[u], of F through different routes. Then, by Theorem 3, the resultants will be different multiples of G1. Hence, the GCD of them will be a small multiple of G1. In the elimination of x through different routes, we obtain sets of similar remainders such that the elements of each set have the same leading variable and nearly the same degrees. We call the leading coefficients of mutually similar remainders an "LCsystem". We eliminate the leading variables of suitably chosen LCsystems. The fourth theorem constructs a polynomial [EQUATION], such that the leading coefficient of [EQUATION] is the GCD of resultants of elements of an LCsystem chosen.