In 1967, Ehrhart proved that the number of lattice points inside a dilated polytope mP (by positive integer m) is a polynomial function L(P, m) of m. Moreover, he showed that the following reciprocity law of L: ^{0 dimP dimP}...[ Read more ]

In 1967, Ehrhart proved that the number of lattice points inside a dilated polytope mP (by positive integer m) is a polynomial function L(P, m) of m. Moreover, he showed that the following reciprocity law of L:

L(P⁰, -t) = (-1)^dimPL(P,t),

similar to the reciprocity law of binomial coefficients. He conjectured that the reciprocity law was also true for lattice manifold M with boundary ∂M:

L(M, -t) = (-1)^dimPL(M - ∂M,t).

The conjecture was proved by Macdonald in 1971. Afterwards, Brion and Vergne counted the lattice points evaluated by a polynomial function inside a lattice polynomial, and extended the reciprocity law to Ehrhart Polynomial with polynomial weight. In 2004, B. Chen generalized their results the reciprocity law to polyhedral functions with polynomial function weight. In this thesis, I will further extend the Ehrhart Polynomial L(P, h; t) of polytope P with polynomial weight h to entire function of polytope P, with entire function weight φ, and prove the reciprocity law:

L(P,φ, Λ; -m) = (-1))^dimPE(P⁰, φ̅,Λ; m),

where φ̅ is defined by φ̅(x) = φ(-x).