Border bases for lattice ideals. (March 2017)

• Record Type: Journal Article
• Authors: Boffi, Giandomenico; Logar, Alessandro
• Is Part Of: Journal of symbolic computation. Volume 79(2017)Part 1
• Page Start: 43
• Page End: 56
• Publication Date: 2017-03
• Subjects: Border basis -- Gröbner basis -- Lattice ideal -- Maximal clique -- Maximum clique -- Order ideal; 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.2016.08.005
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: The main ingredient to construct an O -border basis of an ideal I ⊆ K [ x 1, …, x n ] is the order ideal O, which is a basis of the K -vector space K [ x 1, …, x n ] / I . In this paper we give a procedure to find all the possible order ideals associated with a lattice ideal I M (where M is a lattice of Z n ). The construction can be applied to ideals of any dimension (not only zero-dimensional) and shows that the possible order ideals are always in a finite number. For lattice ideals of positive dimension we also show that, although a border basis is infinite, it can be defined in finite terms. Furthermore we give an example which proves that not all border bases of a lattice ideal come from Gröbner bases. Finally, we give a complete and explicit description of all the border bases for ideals I M in case M is a 2-dimensional lattice contained in Z 2 .