We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacings polynomials we provide an explicit formula for the h * -polynomial in case of complete bipartite graphs. In particular, we show that the h * -polynomial is γ-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011).2010 Mathematics Subject Classification. 05A15, 52B12 (primary); 13P10, 26C10, 52B15, 52B20 (secondary).

We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart h * -vectors. We give a surprisingly simple characterization of combinatorially positive valuations that implies Stanley's nonnegativity and monotonicity of h * -vectors and generalizes work of Beck et al. (2010) from solid-angle polynomials to all translation-invariant simple valuations. For general polytopes, this yields a new characterization of the volume as the unique combinatorially positive valuation up to scaling. For lattice polytopes our results extend work of Betke-Kneser (1985) and give a discrete Hadwiger theorem: There is essentially a unique combinatorially-positive basis for the space of lattice-invariant valuations. As byproducts of our investigations, we prove a multivariate Ehrhart-Macdonald reciprocity and we show universality of weight valuations studied in Beck et al. (2010).

The Ehrhart polynomial of a lattice polytope P encodes information about the number of integer lattice points in positive integral dilates of P . The h * -polynomial of P is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the h *polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the h * -polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the h * -polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley (1991) on the Ehrhart polynomial. Our results hold not only for h * -polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all h * -polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.

We study real sequences {an} n∈N that eventually agree with a polynomial. We show that if the numerator polynomial of its rational generating series is of degree s and has only nonnegative coefficients, then the numerator polynomial of the subsequence {arn+i} n∈N , 0 ≤ i < r, has only nonpositive, real roots for all r ≥ s − i. We apply our results to combinatorially positive valuations on polytopes and to Hilbert functions of Veronese submodules of graded Cohen-Macaulay algebras. In particular, we prove that the Ehrhart h * -polynomial of the r-th dilate of a d-dimensional polytope has only distinct, negative, real roots if r ≥ min{s + 1, d}. This proves a conjecture of Beck and Stapledon (2010).

For a pair of posets A ⊆ P and an order preserving map λ : A → R, the marked order polytope parametrizes the order preserving extensions of λ to P . We show that the function counting integral-valued extensions is a piecewise polynomial in λ and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).