2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76013We establish bounds for the coefficient $\bar{e}_1(I)$ of the Hilbert function of the integral closure filtration of equimultiple ideals. These values are shown to help control all algorithmic processes of normalization that make use of extensions satisfying the condition $S_2$ of Serre.16 pages. to appear in Math. Res. LettersCommutative AlgebraPrimary 13A30; Secondary 13B22, 13H10, 13H15text