2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65617The main result of this paper shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring is effectively bounded by the dimension and any extended degree. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.15 pagesCommutative AlgebraAlgebraic Geometry13A30, 13D40text