Positivity Theorems for Solid-Angle Polynomials

Details Positivity Theorems for Solid-Angle Polynomials Positivity Theorems for Solid-Angle Polynomials

2009-06-22

arxiv.org/abs/0906.4031v2

Beitr. Algebra Geom. 51, no. 2 (2010), 493-507

Mathematics

Combinatorics

10 pages; v2: fixed errors in Section 4

Scientific paper

For a lattice polytope P, define A_P(t) as the sum of the solid angles of all the integer points in the dilate tP. 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 >= 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.

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