Complexity of OM factorizations of polynomials over local fields

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.