Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Upper bounds of Hilbert coefficients and Hilbert functions


a1 Departament d'Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain. e-mail:


Let (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I xs2282 R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.

(Received December 19 2006)

(Revised June 01 2007)