The positivity of the first coefficients of normal Hilbert polynomials

Abstract

Let R be an analytically unramified local ring with maximal ideal m and d = dim R > 0. If R is unmixed, then e 1 I (R) ≥ 0 for every m-primary ideal I in R, where e 1 I (R) denotes the first coefficient of the normal Hilbert polynomial of R with respect to I. Thus the positivity conjecture on e 1 I (R) posed by Wolmer V. Vasconcelos is settled affirmatively.