Complexity of OM factorizations of polynomials over local fields

TL;DR

This paper analyzes the complexity of OM factorizations of polynomials over local fields, providing new estimates for the Montes algorithm's efficiency in computing factorizations with prescribed precision.

Contribution

It introduces a new complexity estimate for the Montes algorithm, improving understanding of its computational efficiency over local fields.

Findings

Complexity estimate: O(n^{2+ε} + n^{1+ε}δ^{2+ε} + n^2ν^{1+ε}) word operations.

Enhanced analysis assuming small residue field.

Application to polynomial factorization with specified precision.

Abstract

Let $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $\oo$ be the valuation ring, $\m$ the maximal ideal and $F(x)\in\oo[x]$ a monic separable polynomial of degree $n$. Let $\delta=v(\dsc(F))$. The Montes algorithm computes an OM factorization of $F$. The single-factor lifting algorithm derives from this data a factorization of $F \md{\m^\nu}$, for a prescribed precision $\nu$. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O(n^{2+\epsilon}+n^{1+\epsilon}\delta^{2+\epsilon}+n^2\nu^{1+\epsilon})$ word operations for the complexity of the computation of a factorization of $F \md{\m^\nu}$, assuming that the residue field of $k$ is small.