Parametrizing an integer linear program by an integer

We consider a family of integer linear programs in which the coefficients of the constraints and objective function are polynomials of an integer parameter $t\mathrm{.}$ For $\mathrm{\ell }$ in ${\mathbb{Z}}_{+},$ we define $f_{\mathrm{\ell }}(t)$ to be the ${\mathrm{\ell }}^{\text{th}}$ largest value of the objective function with multiplicity for the integer linear program at $t\mathrm{.}$ We prove that for all $\mathrm{\ell },$ $f_{\mathrm{\ell }}$ is eventually quasi-polynomial; that is, there exists $d$ and polynomials $P_0, \ldots, P_{d-1}$suchthatforsufficientlylarge$t,$ $f_\ell(t)=P_{d \pmod{t}}(t).$Closelyrelatedtofindingthe$\ell^{$\text{th}$}$ largest value is describing the vertices of the convex hull of the feasible set. Calegari and Walker showed that if $R(t)$ is the convex hull of $${\mathbf{v}}_{\mathbf{1}}$(t), \ldots, ${\mathbf{v}}_{\mathbf{k}}$(t)$where$${\mathbf{v}}_{\mathbf{i}}$$isavectorwhosecoordinatesarein$$\mathbb{Q}$(u)$andofsize$O(u),$ then the vertices of the convex hull of the set of lattice points in $R(t)$ has eventually quasi-polynomial structure. We prove this without the $O(u)$ assumption.