Ehrhart polynomials and stringy Betti numbers

Characteristic and Ehrhart Polynomials

Let A be a subspace arrangement and let (A; t) be the characteristic polynomial of its intersection lattice L(A). We show that if the subspaces in A are taken from L(Bn), where Bn is the type B Weyl arrangement, then (A; t) counts a certain set of lattice points. This is the only known combinatorial interpretation of this polynomial in the subspace case. One can use this result to study the par...

-betti Numbers

The Atiyah conjecture predicts that the L-Betti numbers of a finite CW -complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator g...

Generalized Ehrhart Polynomials

Let P be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations P (n) = nP is a quasi-polynomial in n. We generalize this theorem by allowing the vertices of P (n) to be arbitrary rational functions in n. In this case we prove that the number of lattice points in P (n) is a quasi-polynomial for n sufficiently large. Our work w...

Mixed Ehrhart polynomials

For lattice polytopes P1, . . . , Pk ⊆ Rd, Bihan (2016) introduced the discrete mixed volume DMV(P1, . . . , Pk) in analogy to the classical mixed volume. In this note we study the associated mixed Ehrhart polynomial MEP1,...,Pk(n) = DMV(nP1, . . . , nPk). We provide a characterization of all mixed Ehrhart coefficients in terms of the classical multivariate Ehrhart polynomial. Bihan (2016) show...

Coverings and Betti Numbers