We propose three floating-point validated algorithms to compute respectively fast inversion, Euclidean division and Hensel lifting over \(\mathbb {C}[[x]][y]\) . This is the second step (after Bréhard, Poteaux and Soudant in ISSAC 2023) towards a validated numerical Newton–Puiseux algorithm, and will also be useful towards a validated OM-algorithm over \(\mathbb {C}[[x]][y]\) . Our strategy is simply to first compute a floating-point approximation using the classical algorithm, then to a posteriori validate the result using a Newton-like fixed-point operator. We also provide a prototype Julia implementation of these algorithms and several examples.
