🤖 AI Summary
This work addresses the problem of improving the efficiency of polynomial factorization over complete discrete valuation rings. By extending Hensel’s lemma within the framework of generalized Newton polygons, the authors propose a novel divide-and-conquer strategy based on approximate root representations to refine the Montes algorithm. Under the assumption that the residual characteristic is zero or sufficiently large, they establish that approximate roots effectively characterize OM-types, thereby achieving nearly optimal factorization complexity. The new method reduces the OM-factorization complexity of a polynomial \( F \) by a factor of \( \delta \), where \( \delta \) denotes the valuation of its discriminant, significantly accelerating both irreducibility testing and factorization compared to the original algorithm.
📝 Abstract
We improve significantly the Nart-Montes algorithm for factoring
polynomials over a complete discrete valuation ring $\mathbb{A}$. Our first
contribution is to extend the Hensel lemma in the context of
generalised Newton polygons, from which we derive a new divide and
conquer strategy. Also, if $\mathbb{A}$ has residual characteristic zero or
high enough, we prove that approximate roots are convenient
representatives of types, leading finally to an almost optimal
complexity both for irreducibility and factorisation issues, plus
the cost of factorisations above the residue field. For instance, to
compute an OM-factorisation of $F\in\mathbb{A}[x]$, we improve the
complexity by a factor $δ$, the
discriminant valuation of $F$.