Counting Lattice Points with Ehrhart Polynomials

Question

Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the origin and the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-polynomial $L_{\mathcal{P}}(t) = |t \bar{\mathcal{P}} \cap \mathbb{Z}^{n}|$, which has the form: $\sum c_{k}(t) \ t^{k}$ with periodic functions $c_{k}(t)$.

Question 1: When is the Ehrhart quasi-polynomial a polynomial, i.e., the functions $c_{k}(t)$ are constants? Does this phenomenon occur only when vertices are at integral lattice points (i.e., $b_{i}$ are positive integers)?

Question 2: Suppose I have an Ehrhart polynomial in an indeterminate $t$. What is the significance of the value of the function at rational (non-integral) $t$?

Question 3: Suppose I'd like to count positive solutions (instead of non-negative solutions) of $\sum \frac{x_i}{b_i} \leq t$ with $t$ positive and fixed. Assuming that $b_{i}$ are positive integers, what is the corresponding "Ehrhart-like" polynomial in $t$ which enumerates the (positive) integral points in the $t$-dilate $t\bar{\mathcal{P}}$? Does it follow from a simple variable change in $t$ or $b_{i}$?

(Update) Here is an example of what I'm trying to do. Suppose I'd like to calculate the non-negative integer solutions of \begin{eqnarray} 21 x_{1} + 14 x_{2} + 6 x_{3} \leq 1 \end{eqnarray} (corresponding to the number of positive integer solutions of $21 x_{1} + 14 x_{2} + 6 x_{3} \leq 42$). Equivalently, by dividing through by the product $6 \cdot 14 \cdot 21 = 1764$, we can consider \begin{eqnarray} \frac{x_{1}}{84} + \frac{x_2}{126} + \frac{x_3}{294} \leq \frac{1}{1764}. \end{eqnarray} Here, $\mathbf{b} = (84, 126,294)$, so the corresponding polytope is integral. The Ehrhart polynomial for $t$-dilates is \begin{eqnarray} L_{\bar{\mathcal{P}}}(t) = 1 + 231 t + 18522 t^{2} + 518616 t^{3}, \end{eqnarray} but setting $t = \frac{1}{1764}$ gives a meaningless answer. My initial impression is that along with violating the requirement that $t$ must be an integer, what I am actually calculating is the number of lattice points in the $t$-dilate of the polytope defined by $\frac{1}{1764}$ replaced with $1$. Is there an interpolation scheme to correctly calculate the number of non-negative solutions of the first equation by finding the values of $L_{\bar{\mathcal{P}}}(0)$ and $L_{\bar{\mathcal{P}}}(1)$? Thoughts?

Thanks!

Answer 1

For question 3, by the reciprocity law $(-1)^{n}P(-t)$ counts positive integer solutions of $\sum x_i/b_i<t$ where $P$ is the Ehrhart plynomial of the $n$-simplex with vertices $(0,\ldots,0)$ and the $(0,\ldots,b_i,\ldots,0)$. This is short of your target by the number of positive integer solutions of $\sum x_i/b_i=t$. But that is $(-1)^{n-1}Q(-t)$ where $Q$ is the Ehrhart polynomial of the $(n-1)$-simplex with vertices the $(0,\ldots,b_i,\ldots,0)$.

Answer 2

(Partial Answer to Question 1) No, this phenomenon isn't restricted to integral convex polytopes. There are examples of non-integral rational convex polytopes with Ehrhart quasi-polynomials that are polynomials. See this reference.