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Let $\bar{\mathcal{P}}$ denote the closed $n$-dimensional convex polytope subtended by the origin and the lattice points {$b_{i} \textbf{e} _ {i}$}, where {$\mathbf{e}_{i}$} is the standard basis of $\mathbb{R}^{n}$. Define the Ehrhart function $L _{\bar{\mathcal{P}}}(t) = | t \bar{\mathcal{P}} \cap \mathbb{Z} ^{n}|$, where $t \bar{ \mathcal{P} }$ denotes the $t$-dilate of $\bar{ \mathcal{P} }$.

It is known that $L_{\bar{ \mathcal{P} } }(t)$ is a polynomial of degree $n$ in $t \in \mathbb{N}$ if $\bar{\mathcal{P}}$ is integral, i.e., {$b_{i}$} are positive integers. My question is about the meaning of $L_{\bar{ \mathcal{P} } }(t)$ when $t$ is rational.

Question: Suppose I'd like to calculate the number of non-negative integer solutions of \begin{eqnarray} \frac{x_1}{b_1} + \cdots + \frac{x_n}{b_n} \leq t \end{eqnarray} for some positive rational $t$. Is there a way to compute this number from the aforementioned Ehrhart polynomial (via some interpolation method, etc.) or is some other machinery necessary?

Thanks in advance!

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up vote 5 down vote accepted

R. Diaz, S. Robins, and I studied your question for the inequality $b_1 x_1 + \dots + b_d x_d \le t$ for integral $t$ (which gives a rational polytope) in http://front.math.ucdavis.edu/math.NT/0204035. The case where $t$ is truly a rational variable is more complicated. http://front.math.ucdavis.edu/1006.5612 is a starting point...

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@Matthias: Thanks for the post. Is the exact counting function for real (or even integral) dilates of real $n$-polytopes of this type known? It seems that such a formula would be a major advance in combinatorics and number theory, no? –  user02138 Mar 2 '11 at 19:28
    
It depends what you mean by "exact"... the two papers I mentioned are the state of the art, as far as I know. –  matthias beck Mar 3 '11 at 6:56
    
@Matthias: Thanks. Let me clarify my question. Suppose one has an arbitrary simplicial $n$-polytope $\mathcal{P}$ of the form $\text{conv}(\mathbf{0}, a_1 \mathbf{e}_{1}, \dots, a_{n} \mathbf{e}_{n})$, where $a_{i} \in \mathbb{R}_{>0}$. Are there simple formulas to compute the exact number of lattice points intersecting the real dilate $t \mathcal{P}$? I gather, no, right? –  user02138 Mar 7 '11 at 3:18
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If your parameters are real, there are no simple formulas. The best results in this case are bounds (for the exact counting problem) given in a series of papers by Stephen Yau (UIC) and coauthors. –  matthias beck Mar 8 '11 at 0:58
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in your case we have a generating function $$ \prod (1-t^{1/b_i})^{-1}, $$ and its coefficient in $t^r$ is the number of solutions of equation $\sum x_i/b_i=r$. Knowing such coefficient for each $r$ means knowing generating function itself, hence knowing set of $b$'s. But knowing only sum of coefficients between consecutive integers looks like a significantly less information (it is symmetrization of $f$ by multiplying variable to roots of unity of degree $\prod b_i$). So, my guess is that even if such reconstruction exist, reasons are quite special...

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