Emergent Features in Rational Ehrhart Quasi-Polynomials (2022)
Ehrhart Theory presents a tool for the study of integer dilations of integral polytopes (convex polytopes whose vertices all have integer coordinates) via an associated polynomial in the dilation factor. This is then generalized to rational polytopes (allowing vertices to have rational coordinates) yielding a quasi-polynomial in the dilation factor. Linke presented in 2011 a further generalization by allowing the dilation to be rational which again results in a quasi-polynomial in the dilation factor whose coefficients are piecewise polynomial. We herein present our first foray into this field, beginning with a careful review of the literature and all the necessary concepts before tying it all up with a fleshed-out example and the beginnings of an exploration into the form a particular rational Ehrhart quasi-polynomial takes and the information about its associated polytope that may be gleaned from it.