Degree bounds for Gröbner bases of modules. (July 2022)

• Record Type: Journal Article
• Authors: Liang, Yihui
• Is Part Of: Journal of symbolic computation. Volume 111(2022)
• Page Start: 27
• Page End: 43
• Publication Date: 2022-07
• Subjects: Gröbner bases -- Degree bound -- Cone decompositions -- Hilbert functions; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2021.11.003
• Languages: English
• ISSNs: 0747-7171

Abstract

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.