Triangular bases of integral closures

Abstract

Abstract. In this work, we consider the problem of computing triangular bases of integral closures of one-dimensional local rings.Let pK, vq be a discrete valued field with valuation ring O and let m be the maximal ideal. We take f P Orxs, a monic irreducible polynomial of degree n and consider the extension L " Krxs{pf pxqq 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.

Reduced normal form of local integral bases

“…Bauch [1] and Stainsby [14], found independent algorithms, called multipliers and MaxMin respectively, which compute reduced integral bases as an application of the Montes algorithm in combination with the Single Factor Lifting algorithm (SFL) [7]. The MaxMin algorithm has the advantage of computing directly triangular reduced integral bases.…”

“…Therefore, in order to compare the computational performance of the two methods, we must compare the cost of the triangulation routine (Q3) with the extra tasks of MaxMin: computation of the Okutsu approximations (part of (MM1)) and their improvements (MM3). Now, the complexity of the steps (MM1)-(MM4) [14,Thm. 3.5] is lower than the complexity of Gaussian elimination, which requires O(n 3 ) multiplications in A.…”

“…2.6] these polynomials may be obtained as adequate products of φ-polynomials and Okutsu approximations. After a very simple search [14,Thm. 3.3], we take: (9) g 0 = 1, g 3 = x φ p 3 , g 6 = x 2 (x 2 + 2x + 2) φ p 3 , g 1 = x, g 4 = (x 2 + 2x + 2) φ p 3 , g 7 = x φ p 1 φ p 3 .…”

Computation of triangular integral bases

“…Our algorithm follows the approach from [16] and is based on simple linear algebra after a p-adic initialization step.…”