Computation of Macaulay constants and degree bounds for Gröbner bases. (July 2022)

• Record Type: Journal Article
• Authors: Hashemi, Amir; Parnian, Hossein; Seiler, Werner M.
• Is Part Of: Journal of symbolic computation. Volume 111(2022)
• Page Start: 44
• Page End: 60
• Publication Date: 2022-07
• Subjects: Polynomial ideals -- Gröbner bases -- Degree upper bounds -- Hilbert series -- Cone decompositions -- Macaulay constants; 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.2021.11.004
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper, following the approach by Dubé (1990) and by applying the Hilbert series method, we provide an efficient algorithm to compute the Macaulay constants of the quotient ring of a monomial ideal without computing any exact cone decomposition for the quotient ring . Then, based on this construction and the method proposed by Mayr and Ritscher (2013), a new upper bound for the maximum degree of the elements of any reduced Gröbner basis of an ideal generated by a set of homogeneous polynomials is given. The new bound depends on the Krull dimension and the maximum degree of the generating set of the ideal. Finally, we show that the presented upper bound is sharper than the bounds proposed by Dubé (1990) and Mayr and Ritscher (2013) .