Degree bounds for Gröbner bases of modules

Abstract

Let F be a non-negatively graded free module over a polynomial ring K[x1,…,xn] generated by m basis elements. Let M be a submodule of F generated by elements with degrees bounded by D and dim F/M=r. We prove that if M is graded, the degree of the reduced Gröbner basis of M for any term order is bounded by 2[1/2((Dm)n−rm+D)]2r−1. If M is not graded, the bound is 2[1/2((Dm)(n−r)2m+D)]2r. This is a generalization of Dubé (1990) and Mayr-Ritscher (2013)'s bounds for ideals in a polynomial ring.