The positivity of the first coefficients of normal Hilbert polynomials - Goto, Shiro; Hong, Jooyoun; Mandal, Mousumi (Proceedings of the American Mathematical Society, 2010)
Let R be an analytically unramified local ring with maximal ideal m and d = dimR > 0. If R is unmixed, then e1I (R) = 0 for every m-primary ideal I in R, where e1I (R) denotes the first coefficient of the normal Hilbert polynomial of R with respect to I. Thus the positivity conjecture on e1I(R) posed by Wolmer V. Vasconcelos is settled affirmatively. © 2010 American Mathematical Society.
Keywords: Associated graded ring; Normal Hilbert polynomial; Normal ideal; Rees algebra; dept.mat

Cohen–Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals - Ghezzi, Laura; Goto, Shiro; Hong, Jooyoun; Ozeki, Kazuho; Phuong, Tran Thi; Vasconcelos, Wolmer V (Journal of The London Mathematical Society-second Series, 2010)
The conjecture of Wolmer Vasconcelos on the vanishing of the first Hilbert coefficient e1(Q) is solved affirmatively, where Q is a parameter ideal in a Noetherian local ring. Basic properties of the rings for which e 1(Q) vanishes are derived. The invariance of e1(Q) for parameter ideals Q and its relationship to Buchsbaum rings are studied. © 2010 London Mathematical Society.
Keywords: dept.mat