Single-factor lifting and factorization of polynomials over local fields

TL;DR

This paper introduces a single-factor lifting algorithm that refines polynomial factor approximations over local fields, enhancing polynomial factorization and global arithmetic computations.

Contribution

It develops a novel algorithm to improve individual polynomial factor approximations, integrating with Montes algorithm for faster factorization and broader arithmetic applications.

Findings

Algorithm achieves prescribed precision in factor approximation

Enhances efficiency of polynomial factorization over local fields

Accelerates solutions to global arithmetic problems

Abstract

Let $f(x)$ be a separable polynomial over a local field. 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 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 required.