Positivity theorems for solid-angle polynomials
Combinatorics
2015-06-29 v3 Metric Geometry
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 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.
Keywords
Cite
@article{arxiv.0906.4031,
title = {Positivity theorems for solid-angle polynomials},
author = {Matthias Beck and Sinai Robins and Steven V Sam},
journal= {arXiv preprint arXiv:0906.4031},
year = {2015}
}
Comments
10 pages; v2: fixed errors in Section 4; v3: Removed Theorems 3 and 4 due to a mistake, see Section 6 or corrigendum in ancillary files for details. Numbering kept consistent