Some bounds for the construction of Gröbner bases

Abstract

Let R=K[X1, ..., X n ] be a polynomial ring over a field. For any finite subset F of R, we put m=‖F‖, d=max(deg(F) : f e F), and we let s be the maximal size of the coefficients of all f e F. G=GB(F) denotes the unique reduced Grobner basis for the ideal (F) (see [B3]). We show that the number m′=‖G‖ of polynomials in G and their maximal degree d t as well as the length of the computation of G from F (with unit cost operations in K) are bounded recursively in (n, m, d). The same applies to the degrees of the polynomials occuring during the computation. Moreover, for fixed (n, m, d), G can be computed from F in polynomial time and linear space, when the operations of K can be performed in polynomial time and linear space; in addition, the vector space dimension of the residue ring R/(F) is computably stable under variation of the coefficients of polynomials in F. Corresponding facts hold for polynomial rings over commutative regular rings (see [We']) and non-commutative polynomial rings of solvable type over fields (see [KRW]). Our method does not apply to polynomial rings over Z or other Euclidean rings; in fact, we show that over Z, the length of the computation of G from F with unit cost operations in Z does depend on s.