Triangular bases of integral closures. (July 2018)

• Record Type: Journal Article
• Authors: Stainsby, Hayden D.
• Is Part Of: Journal of symbolic computation. Volume 87(2018)
• Page Start: 140
• Page End: 175
• Publication Date: 2018-07
• Subjects: Fractional ideal -- Local field -- Montes algorithm -- Integral basis -- OM representation -- Discrete valuation; 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.2017.07.012
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings. Let ( K, v ) be a discrete valued field with valuation ring O and let m be the maximal ideal. We take f ∈ O [ x ], a monic irreducible polynomial of degree n and consider the extension L = K [ x ] / ( f ( x ) ) as well as O L the integral closure of O in L, which we suppose to be finitely generated as an O -module. The algorithm MaxMin, presented in this paper, computes triangular bases of fractional ideals of O L . The theoretical complexity is equivalent to current state of the art methods and in practice is almost always faster. It is also considerably faster than the routines found in standard computer algebra systems, excepting some cases involving very small field extensions.