Article contents
Complexity of OM factorizations of polynomials over local fields
Published online by Cambridge University Press: 01 June 2013
Abstract
Let $k$ be a locally compact complete field with respect to a discrete valuation
$v$. Let
$ \mathcal{O} $ be the valuation ring,
$\mathfrak{m}$ the maximal ideal and
$F(x)\in \mathcal{O} [x] $ a monic separable polynomial of degree
$n$. Let
$\delta = v(\mathrm{Disc} (F))$. The Montes algorithm computes an OM factorization of
$F$. The single-factor lifting algorithm derives from this data a factorization of
$F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, 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
$F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, assuming that the residue field of
$k$ is small.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s) 2013
References
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