Term-ordering free involutive bases. (May 2015)

• Record Type: Journal Article
• Authors: Ceria, Michela; Mora, Teo; Roggero, Margherita
• Is Part Of: Journal of symbolic computation. Volume 68:Part 2(2015)
• Page Start: 87
• Page End: 108
• Publication Date: 2015-05
• Subjects: 14C05 -- 14Q20 -- 13P10; Involutive bases -- Quasi-stable ideals; 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.2014.09.005
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper, we consider a monomial ideal J ◃ P : = A [ x 1, …, x n ], over a commutative ring A, and we face the problem of the characterization for the family M f ( J ) of all homogeneous ideals I ◃ P such that the A -module P / I is free with basis given by the set of terms in the Gröbner escalier N ( J ) of J . This family is in general wider than that of the ideals having J as initial ideal w.r.t. any term-ordering, hence more suited to a computational approach to the study of Hilbert schemes. For this purpose, we exploit and enhance the concepts of multiplicative variables, complete sets and involutive bases introduced byRiquier (1893, 1899, 1910) and inJanet (1920, 1924, 1927) and we generalize the construction of J -marked bases and term-ordering free reduction process introduced and deeply studied inBertone et al. (2013a), Cioffi and Roggero (2011) for the special case of a strongly stable monomial ideal J . Here, we introduce and characterize for every monomial ideal J a particular complete set of generators F ( J ), called stably complete, that allows an explicit description of the family M f ( J ) . We obtain stronger results if J is quasi-stable, proving that F ( J ) is a Pommaret basis and M f ( J ) has a natural structure of affine scheme. The final section presents a detailed analysis of the origin and the historical evolution of the main notions we refer to.