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Hilbert functions of Gorenstein monomial curves

Authors: Feza Arslan and Pinar Mete
Journal: Proc. Amer. Math. Soc. 135 (2007), 1993-2002
MSC (2000): Primary 13H10, 14H20; Secondary 13P10
Published electronically: March 2, 2007
MathSciNet review: 2299471
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Abstract: It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their defining ideals in the non-complete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non-decreasing Hilbert functions.

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Additional Information

Feza Arslan
Affiliation: Department of Mathematics, Middle East Technical University, Ankara, 06531 Turkey

Pinar Mete
Affiliation: Department of Mathematics, Middle East Technical University, Ankara, 06531 Turkey
Address at time of publication: Department of Mathematics, Balıkesir University, Balıkesir, 10145 Turkey

Keywords: Gorenstein local ring, Hilbert function of a local ring, tangent cone, monomial curve, numerical semigroup, standard basis.
Received by editor(s): December 17, 2005
Received by editor(s) in revised form: April 1, 2006
Published electronically: March 2, 2007
Additional Notes: The second author was supported by TÜBİTAK with grant no. TBAG-HD/108(105T543).
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2007 American Mathematical Society

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