Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 362, Number 8, August 2010, Pages 43574384
Article electronically published on April 1, 2010
COMBINATORICS AND GEOMETRY OF POWER IDEALS
FEDERICO ARDILA AND ALEXANDER POSTNIKOV
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 A. We prove that their Hilbert series are
determined by the combinatorics of 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.