On Ehrhart Polynomials of Lattice Triangles

The Ehrhart polynomial of a lattice polygon PP is completely determined by the pair (b(P),i(P))(b(P),i(P)) where b(P)b(P) equals the number of lattice points on the boundary and i(P)i(P) equals the number of interior lattice points. All possible pairs (b(P),i(P))(b(P),i(P)) are completely described by a theorem due to Scott. In this note, we describe the shape of the set of pairs (b(T),i(T))(b(T),i(T)) for lattice triangles TT by finding infinitely many new Scott-type inequalities

Rademacher-Carlitz Polynomials

We introduce and study the \emph{Rademacher-Carlitz polynomial} \RC(u, v, s, t, a, b) := \sum_{k = \lceil s \rceil}^{\lceil s \rceil + b - 1} u^{\fl{\frac{ka + t}{b}}} v^k where $a, b \in \Z_{>0}$, $s, t \in \R$, and $u$ and $v$ are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view \RC(u, v, s, t, a, b) as a polynomial analogue (in the sense of Carlitz) of the \emph{Dedekind-Rademacher sum} \r_t(a,b) := \sum_{k=0}^{b-1}\left(\left(\frac{ka+t}{b} \right)\right) \left(\left(\frac{k}{b} \right)\right), which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms $$\sigma(x,y):=\sum_{(j,k) \in \mathcal{P}\cap \Z^2} x^j y^k$$ of any rational polyhedron $\mathcal{P}$, and we derive a novel reciprocity theorem for Dedekind-Rademacher sums, which follows naturally from our setup

Rational Ehrhart quasi-polynomials

Ehrhart's famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors and it turns out that the number of integral points can still be written as a rational quasi-polynomial. Furthermore the coefficients of this rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation.Comment: 15 pages, several changes in the expositio

A note on lattice-face polytopes and their Ehrhart polynomials

We give a new definition of lattice-face polytopes by removing an unnecessary restriction in the paper "Ehrhart polynomials of lattice-face polytopes", and show that with the new definition, the Ehrhart polynomial of a lattice-face polytope still has the property that each coefficient is the normalized volume of a projection of the original polytope. Furthermore, we show that the new family of lattice-face polytopes contains all possible combinatorial types of rational polytopes.Comment: 11 page