Degree bounds for Gröbner bases of modules. (July 2022) - Liang, Yihui
Abstract: Let F be a non-negatively graded free module over a polynomial ring K [ x 1, …, x n ] 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 ( ( D m ) n − r m + D ) ] 2 r − 1 . If M is not graded, the bound is 2 [ 1 / 2 ( ( D m ) ( n − r ) 2 m + D ) ] 2 r . This is a generalization of Dubé (1990) and Mayr-Ritscher (2013)'s bounds for ideals in a polynomial ring.