2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139952We establish an improvement of Philippon's zero estimates primarily in the multiplicity setting. The improvement is made possible by a more geometric approach and in particular the use of Seshadri constants.17 pages, final version, accepted for publication in Mathematische ZeitschriftNumber TheoryAlgebraic Geometrytext