19 research outputs found
Roots of Ehrhart Polynomials of Smooth Fano Polytopes
V. Golyshev conjectured that for any smooth polytope P of dimension at most
five, the roots z\in\C of the Ehrhart polynomial for P have real part equal
to -1/2. An elementary proof is given, and in each dimension the roots are
described explicitly. We also present examples which demonstrate that this
result cannot be extended to dimension six.Comment: 10 page
Roots of the Ehrhart polynomial of hypersimplices
The Ehrhart polynomial of the -th hypersimplex of order
is studied. By computational experiments and a known result for , we
conjecture that the real part of every roots of the Ehrhart polynomial of
is negative and larger than if . In
this paper, we show that the conjecture is true when and that every root
of the Ehrhart polynomial of satisfies if .Comment: 18 pages, 8 figure
Periods of Ehrhart Coefficients of Rational Polytopes
Let P⊂R^n be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the kth dilate of P (k a positive integer) is a quasi-polynomial function of k — that is, a "polynomial" in which the coefficients are themselves periodic functions of k. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values