Using resultants for inductive Gröbner bases computation

Abstract

In his PhD thesis, B. Buchberger introduced the concept of Grobner basis and gave an algorithm to compute it [1]. Later on a number of inductive algorithms for computing Grobner bases appeared, which employ induction on the number of polynomials in the given basis of the ideal. For the slightly different but related problem of ideal membership, G. Hermann [3] proceeds by induction on the number of variables. In this work we are aiming to give an inductive approach to Grobner bases computation of a radical ideal with induction over the variables. To this end we employ resultants, which is an important tool in elimination theory [2]. Throughout this text we will use the following notation and conventions. K is an algebraically closed field of characteristic 0. We fix the lexicographic term order > with x1 > x2 > . . . > xn. I is a radical ideal of the polynomial ring K[x1, x2, . . . , xn], generated by F = {f1, f2, . . . , fs}. By Ii we denote the i-th elimination ideal of I, I ∩K[xi+1, . . . , xn]. Res(F ) denotes the set of the resultants of pairs of polynomials in F , {resx1(fi, fj)|1 ≤ i < j ≤ s}. Spol and NF will stand for s-polynomial and normal form. The main idea is to compute the reduced Grobner basis in two phases. In the first phase we recursively project the given ideal I into its elimination ideals I1, I2, . . . , Ik until we cannot project anymore. The following proposition gives us a method to do the projection.