Dimension-Dependent Upper Bounds for Gröbner Bases

Abstract

We improve certain degree bounds for Gröbner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension-(and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Gröbner basis (w.r.t. the degree reverse lexicographical ordering) of a homogeneous ideal in these positions.For the rest of this section, we thus assume that R is not Cohen-Macaulay, i. e. that D > λ. Corollary 3.15. If I is in strongly stable position and D > 1, then #F (I) ≤ d (n−D)2 D−λ−1 and deg(I, ≺) ≤ 2d (n−D)2 D−λ−1 . The maximal degree of an element of the Pommaret basis of an ideal in quasi stable position equals the Castelnuovo-Mumford regularity [34, Cor. 9.5]. If the ideal is even in stable position, then the Pommaret basis coincides with the reduced Gröbner basis [26, Thm. 2.15]. These considerations imply now immediately the following two results.Corollary 3.16. If the ideal I is in strongly stable position and D > 1, then reg(I) ≤ 2d (n−D)2 D−λ−1 .Corollary 3.17. Let the ideal I be in quasi stable position, H its Pommaret basis and D > 1. If we write deg(H) for the maximal degree of an element of H, then deg (I, ≺) ≤ deg (H) ≤ 2d (n−D)2 D−λ−1 .