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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Combinatorics and geometry of power ideals


Authors: Federico Ardila and Alexander Postnikov
Journal: Trans. Amer. Math. Soc. 362 (2010), 4357-4384
MSC (2000): Primary 05A15, 05B35, 13P99, 41A15, 52C35
Published electronically: April 1, 2010
MathSciNet review: 2608410
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Abstract: We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines.

We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement $ \mathcal{A}$. We prove that their Hilbert series are determined by the combinatorics of $ \mathcal{A}$ and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings.

Our work unifies and generalizes results due to Dahmen-Micchelli, Holtz-Ron, Postnikov-Shapiro-Shapiro, and Sturmfels-Xu, among others. It also settles a conjecture of Holtz-Ron on the spline interpolation of functions on the lattice points of a zonotope.


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Additional Information

Federico Ardila
Affiliation: Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94110
Email: federico@math.sfsu.edu

Alexander Postnikov
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139
Email: apost@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05018-X
PII: S 0002-9947(10)05018-X
Received by editor(s): October 31, 2008
Received by editor(s) in revised form: February 11, 2009
Published electronically: April 1, 2010
Additional Notes: The first author was supported in part by NSF Award DMS-0801075.
The second author was supported in part by NSF CAREER Award DMS-0504629.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.