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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The positivity of the first coefficients of normal Hilbert polynomials


Authors: Shiro Goto, Jooyoun Hong and Mousumi Mandal
Journal: Proc. Amer. Math. Soc. 139 (2011), 2399-2406
MSC (2010): Primary 13H10; Secondary 13A30, 13B22, 13H15
Published electronically: December 16, 2010
MathSciNet review: 2784804
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Abstract: Let $ R$ be an analytically unramified local ring with maximal ideal $ \mathfrak{m}$ and $ d = \dim R > 0$. If $ R$ is unmixed, then $ \overline{\mathrm{e}}^{1}_I(R) \geq 0 $ for every $ \mathfrak{m}$-primary ideal $ I$ in $ R$, where $ \overline{\mathrm{e}}_I^1(R)$ denotes the first coefficient of the normal Hilbert polynomial of $ R$ with respect to $ I$. Thus the positivity conjecture on $ \overline{\mathrm{e}}_I^1(R)$ posed by Wolmer V. Vasconcelos is settled affirmatively.


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Additional Information

Shiro Goto
Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
Email: goto@math.meiji.ac.jp

Jooyoun Hong
Affiliation: Department of Mathematics, Southern Connecticut State University, 501 Crescent Street, New Haven, Connecticut 06515-1533
Email: hongj2@southernct.edu

Mousumi Mandal
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
Email: mousumi@math.iitb.ac.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10710-4
PII: S 0002-9939(2010)10710-4
Keywords: Associated graded ring, Rees algebra, normal ideal, normal Hilbert polynomial
Received by editor(s): July 1, 2010
Published electronically: December 16, 2010
Communicated by: Irena Peeva
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.