Local polynomial factorisation: improving the Montes algorithm

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 $\delta$, the discriminant valuation of $F$.