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Combinatorics and geometry of power ideals
Author(s):
Federico
Ardila;
Alexander
Postnikov
Journal:
Trans. Amer. Math. Soc.
362
(2010),
4357-4384.
MSC (2000):
Primary 05A15, 05B35, 13P99, 41A15, 52C35
Posted:
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  . We prove that their Hilbert series are determined by the combinatorics of  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 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email:
apost@math.mit.edu
DOI:
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
Posted:
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.
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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