2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69999Building upon work of Villamayor and Bierstone-Milman we give a proof of the canonical Hironaka principalization and desingularization. The idea of "homogenized ideals" introduced in the paper gives {\it a priori} the canonicity of algorithm and radically simplifies the proof.28 pagesAlgebraic GeometryComplex Variables14E15; 14B05; 14M25; 32S05; 32S45text