Single-factor lifting and factorization of polynomials over local fields

Abstract

Let f (x) be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of f (x), with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this ‘‘single-factor lifting’’ routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is requiredPostprint (published version

Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields

“…In the paper [9] we prove that the f -complete and optimal types computed by the Montes algorithm determine certain canonical invariants of each of the irreducible p-adic factors of the input polynomial f (x). Also, in the paper [11] we find upper bounds for the total number of refinement steps, based on these invariants. Some of these canonical invariants had been introduced by K. Okutsu in [13].…”

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.…”

Okutsu-Montes representations of prime ideals of one-dimensional integral closures

“…In a recent work with J. Guàrdia and S. Pauli [GNP10], we develop a single-factor lifting algorithm that improves each one of these approximations up to a prescribed precision. This algorithm has quadratic convergence.…”

“…This computation requires to improve the approximations φ p till v p (φ p (θ)) has a sufficiently large value. As mentioned at the end of the last section, this can be carried out with the single-factor lifting algorithm [GNP10]. The polynomials Q i,1 (x), .…”