# Mathematics Colloquia and Seminars

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### Combinatorics and geometry of power ideals

**Algebra & Discrete Mathematics**

Speaker: | Federico Ardila, San Francisco State University |

Location: | 2112 MSB |

Start time: | Fri, Jan 16 2009, 2:10PM |

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 the resulting fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results on power ideals obtained by Dahmen-Micchelli, de Boor-Ron, 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. This is joint work with Alex Postnikov from MIT.