Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 51, No. 2, pp. 493-507 (2010)

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Positivity theorems for solid-angle polynomials

Matthias Beck, Sinai Robins and Steven V. Sam

Department of Mathematics, San Francisco State University, San Francisco, CA 94132, U.S.A. e-mail:; Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371, e-mail:; Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e-mail:

Abstract: For a lattice polytope $\P$, define $A_{\P}(t)$ as the sum of the solid angles of all the integer points in the dilate $t\P$. Ehrhart and Macdonald proved that $A_{\P}(t)$ is a polynomial in the positive integer variable $t$. We study the numerator polynomial of the solid-angle series $\sum_{ t \ge 0 } A_\P(t) z^t$. In particular, we examine nonnegativity of its coefficients, monotonicity and unimodality questions, and study extremal behavior of the sum of solid angles at vertices of simplices. Some of our results extend to more general valuations.

Keywords: solid angle, lattice polytope, Ehrhart polynomial, lattice points

Classification (MSC2000): 28A75; 05A15, 52C07

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Electronic version published on: 24 Jun 2010. This page was last modified: 8 Sep 2010.

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