Generalized Ehrhart polynomials

Abstract

Abstract. 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 was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in n, and we explain how these two problems are related.

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

“…Property 1 will follow from Property 3 trivially. Property 2 follows directly from the Chen, Li, Sam result [3], as long as our polyhedron is bounded. Since we know whether our polyhedron has non-trivial recession cone and lineality space, we know whether it is bounded.…”

“…As a concrete example: Example 1.5. Let P be the triangle with vertices (0, 0), More recently, Chen, Li, and Sam proved [3] that |S t | is still an EQP, even if the normal vectors in the linear inequalities are allowed to vary with t, that is, if they are of the form a(t) · x ≤ b(t), where…”

“…Then we will apply a series of logical equivalences and affine reductions as follows: At each stage, the reductions preserve all the properties we are interested in, until we finally reduce to the case of the parametric polyhedra defined by ϕ 5 . In Step 6, we deal with these polyhedra by applying previously-established combinatorial techniques from Chen-Li-Sam [3], Woods [19], and Shen [17].…”

“…Now we have S t written as the set of integer points in a "parametric polyhedron", that is, defined by linear inequalities that depend on t, as above. If this polyhedron is bounded, then it immediately follows from [3] that |S t | ∈ EQP. We must detect when S t is unbounded.…”

“…In the output of Proposition 4.12, any polyhedron P with r = s = 0 is bounded, and we may apply the results from [3] and from [17]. The following lemma helps us deal with unbounded polyhedra.…”