The integer point transform as a complete invariant

A polytope $\mathcal{P}\subset\mathbb{R}^{d}$ is called an integer polytope (respectively, a rational polytope) if the coordinates of all its vertices are integers (respectively, rationals). Given a rational polytope $\mathcal{P}\subset\mathbb{R}^{d}$, we define its integer point transform by

for all $\xi\in\mathbb{R}^{d}$. Throughout, the word polytope refers to a convex set, by definition. We observe that more generally, Definition (1) makes sense for an arbitrary finite set of integer points $S\subset\mathbb{Z}^{d}$:

Indeed, we will often develop general principles for any finite set of integer points in $\mathbb{R}^{d}$, and then later pass to polytopes, using the assumption of convexity.

One of the main utilities of the integer point transform is the special evaluation at the origin:

the number of integer points in $\mathcal{P}$. So we see that if $\mathcal{P}$ is a polytope, then the integer point transform discretizes the volume of $\mathcal{P}$ in this sense. In fact, the integer point transform was so named because, by using a lattice, it discretizes the Fourier transform of $\mathcal{P}$, which is defined by $\int_{\mathcal{P}}e^{2\pi i\langle\xi,x\rangle}dx$ (see [7], for example). We already know from (3) that if we have any two rational polytopes $\mathcal{P},\mathcal{Q}\subset\mathbb{R}^{d}$, then $\sigma_{\mathcal{P}}(0)=\sigma_{\mathcal{Q}}(0)\implies|\mathcal{P}\cap\mathbb{Z}^{d}|=|\mathcal{Q}\cap\mathbb{Z}^{d}|$. But is it possible that knowledge of $\sigma_{\mathcal{P}}(\xi)$, for some finite collection of points $\xi\in\mathbb{R}^{d}$, might help us to uniquely identify $\mathcal{P}$? This is our main motivating question.

Historically, the importance of the integer point transform $\sigma_{\mathcal{P}}$ for a rational polytope $\mathcal{P}$ surfaced naturally in combinatorial geometry, namely in Ehrhart’s theory of integer point enumeration in polytopes ([2], [7]). In particular, Ehrhart’s main theorem follows quickly from Brion’s theorem, which enables us to write $\sigma_{\mathcal{P}}$ as a finite linear combination of exponential-rational functions [7] (see Remark 4). The work of Fink, Mészáros, and Dizier [4] gives applications of integer point transforms to Schubert polynomials. The recent work of Katharina Jochemko [6] shows that the sequence of integer point transforms ${\sigma_{kP+Q}}\big{|}_{k\geq 0}$ satisfies a multivariate linear recursion, where $\mathcal{P}$ is an integer polytope, and $Q$ is any polytope. Here we do not assume knowledge of Ehrhart theory, and rather proceed from first principles.

The integer point transform (1) clearly lives on the torus - in other words, $\sigma_{\mathcal{P}}$ is periodic on $\mathbb{R}^{d}$, with a fundamental domain $[0,1)^{d}$. We will therefore often restrict attention to the torus, or equivalently to the half-open cube $[0,1)^{d}$.

It is elementary that given any positive integer $N$, there are infinitely many distinct $d$-dimensional integer polytopes $\mathcal{P}$ with $\left|\mathbb{Z}^{d}\cap\mathcal{P}\right|=N$. Even after we mod out by the action of the modular group $GL_{d}(\mathbb{Z})$, there are in general many (though finitely many) distinct integer polytopes $\mathcal{P}$ with $\left|\mathbb{Z}^{d}\cap\mathcal{P}\right|=N$. It is very natural to ask: “What extra information do we need in order to uniquely determine $\mathcal{P}$?” To make this question more rigorous, we formulate it as follows.

We will answer both of these Questions in the affirmative. Somewhat surprisingly, it turns out that just one point suffices for both questions. We also prove a similar, but slightly weaker result, for rational polytopes (Theorem 1.1, part 3 below). Because the direction $(\Longleftarrow)$ is trivial, it suffices to always prove the $(\Longrightarrow)$ direction. Throughout the paper, we will use the special point

where we have picked the first $d$ primes $2,3,5,7,\dots p_{d}$ to ensure that $\sqrt{2},\dots,\sqrt{p}_{d}$ are algebraically independent over $\mathbb{Q}$, and therefore all integer linear combinations of the coordinates of $\xi^{*}$ are distinct.

We note that no assumption had to be made about the dimensions of $\mathcal{P}$ or $\mathcal{Q}$. Furthermore, it is important to remark that the proof shows that more is true: we may pick any algebraic numbers in the exponents of our exponentials in equation (8), so in choosing $\xi^{*}$, there is clearly a dense set of vectors to choose from.

Let us start with a non-example, which will show the importance of convexity when passing from arbitrary sets of integer points in $\mathbb{Z}^{d}$ to the family of integer polytopes. Suppose we have two non-convex polytopes $\mathcal{P},\mathcal{Q}\subset\mathbb{R}^{d}$. It would be nice to say that if $\mathcal{P}\cap\mathbb{Z}^{d}=\mathcal{Q}\cap\mathbb{Z}^{d}$, then $\mathcal{P}=\mathcal{Q}$. Unfortunately, even in $\mathbb{R}^{2}$ we have simple counterexamples, as Figure 1 shows.

Fortunately, we have the very easy fact (Lemma 2.1 below) that in the context of convex polytopes we do have such an implication, as follows.

We will use Lemma 2.1 repeatedly in the sequel, when moving from a formal result about finite sets of integer points to results about polyhedra (and in particular polytopes).

Here we extend Theorem 1.1 to show that the Fourier transform of a rational polytope $\mathcal{P}$ uniquely determines $\mathcal{P}$ by evaluating the Fourier transform at a single algebraic point. Given a $d$-dimensional compact set $S\subset\mathbb{R}^{d}$, we define the continuous Fourier transform of $S$ by

for each $\xi\in\mathbb{R}^{d}$. The present notation $\mathcal{F}$ is used to distinguish the Fourier transform (9) from the notation that we used for the Fourier transform over a finite abelian group, namely $\mathcal{F}_{G}$.

Let $\mathcal{P}\subset\mathbb{R}^{d}$ be an integer polytope. We define

to be the integer sublattice of $\mathbb{Z}^{d}$ that is generated by the integer span of all the points in $|\mathbb{Z}^{d}\cap\mathcal{P}|$. We’ll call $\mathcal{L}_{\mathcal{P}}$ the spanning lattice of the set $\mathcal{P}\cap\mathbb{Z}^{d}$. It is a trivial fact—perhaps as old as the hills themselves—that for $2$-dimensional integer polygons, we always have $\mathcal{L}_{\mathcal{P}}=\mathbb{Z}^{2}$. The proof is easy: we can always triangulate any integer polygon into unimodular triangles, and the vertices of any such unimodular triangle already generate $\mathbb{Z}^{2}$.

Whenever $\mathcal{L}_{\mathcal{P}}=\mathbb{Z}^{d}$, $\mathcal{P}$ is called a spanning polytope. In dimensions $d\geq 3$, there are integer polytopes that are not spanning polytopes, because a unimodular triangulation is not always available for an arbitrary integer polytope.

In the recent work [5], Hofscheier, Katthän, and Nill develop an Ehrhart-type theory for spanning lattice polytopes. Here we may ask another natural question about the integer point transform:

It turns out that the dual lattice $\mathcal{L}_{\mathcal{P}}^{*}$ comes in naturally here, and it satisfactorily answers Question 5.2.

We notice that if replace the polytope $\mathcal{P}$ by any finite subset of integer points $S\subset\mathbb{Z}^{d}$, then the proof of Theorem 5.3 remains valid, with the following definition. We define $\mathcal{L}_{S}\subset\mathbb{Z}^{d}$ to be the lattice generated by the integer span of all $n\in S$. We obtain the following immediate consequence from the proof of Theorem 5.3.

Using the definition of a spanning polytope, together with Theorem 5.3, we immediately obtain the following consequence as well.

Given the nice structure of the aboslute maxima given by Theorem 5.3, it is also natural to ask:

We have the sublattice containments $\mathcal{L}_{\mathcal{P}}\subseteq\mathbb{Z}^{d}\subseteq\mathcal{L}_{\mathcal{P}}^{*}$. Question 5.7 asks for the value of $[\mathcal{L}_{\mathcal{P}}^{*}:\mathbb{Z}^{d}]$, which has a simple answer:

The finite sum of exponentials that defines $\sigma_{\mathcal{P}}(\xi)$ lends the feeling that we should be studying a connection to the finite Fourier transform of some finite abelian group. To make this feeling rigorous, we develop this connection here. Although there are many possible choices for our finite abelian group, we first choose a box

and then consider the set of integers points in it, namely $\mathbb{Z}^{d}\cap B$. Clearly, any integer polytope $\mathcal{P}$ is contained in the interior of the box $B$, for some appropriately chosen integers $k_{1},\dots,k_{d}$. To avoid ambiguities when we take the quotient, we assume that each vertex of $\mathcal{P}$ is contained in the slightly smaller box $[-\frac{k_{1}}{2}+\frac{1}{2},\frac{k_{1}}{2}-\frac{1}{2}]\times\cdots\times[-\frac{k_{d}}{2}+\frac{1}{2},\frac{k_{d}}{2}-\frac{1}{2}]$.

Now we consider the finite abelian group $G:=\mathbb{Z}/k_{1}\mathbb{Z}\times\cdots\mathbb{Z}/k_{d}\mathbb{Z}$, which we identify with the integer points in the torus $\mathbb{R}/k_{1}\mathbb{Z}\times\cdots\times\mathbb{R}/k_{d}\mathbb{Z}$, and which can also be thought of as the box $B$ after identifying its opposite facets. Although the choice of integers $k_{1},\dots k_{d}$ is not canonical, such an embedding of the integer points of $\mathcal{P}$ into a finite abelian group will prove to be worthwhile. In other words, we have, by definition:

We will freely use the usual fact that the Pontryagin dual $\hat{G}$ (simply the group of characters of $G$) is in this case isomorphic to $G$. Even though the full geometry of $\mathcal{P}$ may not be immediately apparent in this discrete setting, we will be able to shed some additional light on the integer points $\mathcal{P}\cap\mathbb{Z}^{d}\subset B.$ It is now natural to consider the indicator function $1_{\mathcal{P}}$ as a function on $G$, and therefore expand it into its finite Fourier series. Precisely, each element $\xi\in G$ gives us a character $\chi_{\xi}:G\rightarrow S^{1}$ defined by

The first theorem of finite Fourier analysis gives us:

where the (finite) Fourier coefficients have the form

and where we have used the isomorphism $G\cong\hat{G}$. So by definition we have the finite Fourier transform

Let us “massage” $c_{\xi}$ a bit. For each $\xi\in G$, we have:

by definition of the integer point transform. We will also use the latter identification in Section 7 below, by giving an equivalent condition for central symmetry in terms of finite Fourier transforms—or equivalently the integer point transform.

We now notice that we have never required $\mathcal{P}$ to be a polytope in the theory above, but merely that we have:

One reason for initially using the integer points that belong to a polytope, as opposed to just any finite set of integer points, is that the applications that use polytopes are perhaps the most naturally occurring.

Next, suppose that we want an analogue of Theorem 1.1, but we want to evaluate the integer point transform at integer lattice points, rather than evaluating it at the algebraic point $\xi^{*}:=\tfrac{1}{\pi}\left(\sqrt{2},\dots,\sqrt{p_{d}}\right)$. Then we need to evaluate at more points, and the next result, namely Theorem 6.1, part 2, gives a sufficient condition to choose such points.

In this brief section we give an equivalence for central symmetry in terms of the integer point transform, for any finite set $A\subset\mathbb{Z}^{d}$ of integer points. As a consequence, we get an equivalence for the central symmetry of any integer polytope (of arbitrary codimension) in terms of special evaluations of the integer point transform. We will use here the machinery of Section 6.

In this section we will use the box $B:=\left[-\frac{k}{2},\frac{k}{2}\right)^{d}$, for any positive integer $k$. We will suppose that $A\subset\mathbb{Z}^{d}\cap B$ is any centrally symmetric set of integer points that is contained in $B$.

It is immediate that for such a set $A$, its integer point transform is real-valued for any $\xi\in\mathbb{R}^{d}$:

In particular, for centrally symmetric polytopes, their minima and maxima may be studied without having to take norms.

Using exactly the same proof ideas of Theorem 1.1 and Theorem 4.1, we also obtain the following corollaries.

For further rumination, we mention a few threads that appeared naturally in this line of research which remain open.

1. 1.

   Perhaps the most fascinating question now is how to find the unique polytope (or set of integer points) that is guaranteed by the uniqueness property of Theorem 1.1.

   For example, suppose we seek to discover the $1$-dimensional polytope $\mathcal{P}:=[0,N]$, and suppose its integer point transform $\sigma_{\mathcal{P}}(\xi^{*})=C$ is given to us, with $\xi^{*}$ as in (4). Then we have

   $$C=\sum_{k=0}^{N}e^{2\pi ik\xi^{*}}=\frac{e^{2\pi i(N+1)\xi^{*}}-1}{e^{2\pi i\xi^{*}}-1}.$$

   Here, it is easy to solve this equation with respect to the vertex $N$ of $\mathcal{P}$, by taking complex logs, with some care being taken for picking the principal branch. However, even for an arbitrary finite set $\mathcal{P}$ of integer points in $\mathbb{Z}$, or for example an integer triangle in $\mathbb{R}^{2}$, the problem already appears to become formidable.

2. 2.

   Similarly, it would be important to reconstruct a rational polytope from the uniqueness guaranteed by the 1-point evaluation of its continuous Fourier transform, as in Theorem 4.1. This direction for future research, as well as the previous problem, involves transcendental equations. It would be very interesting to solve them over the integers, namely the coordinates of the vertices of $\mathcal{P}$.

   A similar open problem is to solve for the unique integer cone that Corollary 8.1 and Corollary 8.3 guarantee, respectively.

3. 3.

   Is it possible to strengthen Theorem 1.1, part 2, so that we can eliminate the dilation factor $k$, as follows?

   Conjecture 1.

   If $\mathcal{P},\mathcal{Q}\subset\mathbb{R}^{d}$ are any rational polytopes, then we have:

   $$\sigma_{\mathcal{P}}(\xi^{*})=\sigma_{\mathcal{Q}}(\xi^{*})\implies\mathcal{P}=\mathcal{Q},$$

   with $\xi^{*}$ as in (4). $\square$

4. 4.

   Historically, the integer point transform $\sigma_{\mathcal{P}}$ for a rational polytope $\mathcal{P}$ appeared in the work of Brion [3] in 1988, who proved the important result that we may write $\sigma_{\mathcal{P}}$ as a finite linear combination of exponential-rational functions of the vertex tangent cones $\mathcal{K}_{v}$ of $\mathcal{P}$:

   $$\sigma_{\mathcal{P}}(\xi)=\sum_{\mathrm{vertex}\,v\,{\rm of}\,\mathcal{P}}\sigma_{\mathcal{K}_{v}}(\xi),$$

   (30)

   valid for almost all $\xi\in\mathbb{R}^{d}$ (see [2], [7] for more details). We have slightly abused notation in (30) by writing $\sigma_{\mathcal{K}_{v}}(\xi)$ to mean the meromorphic continuation of these integer point transforms, which are initially defined by $\sigma_{\mathcal{K}_{v}}(\xi):=\sum_{n\in\mathcal{K}_{v}\cap\mathbb{Z}^{d}}e^{2\pi i\langle\xi,n\rangle}$. By an elementary lemma (for example [7, Theorem 10.2]), each such $\sigma_{\mathcal{K}_{v}}$ for a simplicial cone $\mathcal{K}$ also has the following finite form, as a rational-exponential function:

   $$\sigma_{\mathcal{K}_{v}}(\xi)=\frac{\sigma_{\Pi+v}(\xi)}{\prod_{k=1}^{d}\left(1-e^{\langle w_{k},\xi\rangle}\right)},$$

   where $\Pi+v:=\{\lambda_{1}w_{1}+\cdots+\lambda_{d}w_{d}+v\,\mid\text{ all }0\leq\lambda_{j}<1\}$, and where $w_{1},\dots,w_{d}$ are the edge vectors of the vertex tangent cone $\mathcal{K}_{v}$.

5. 5.

   Regarding possible extensions, it is tempting extend the integer point transform to arbitrary lattices, as follows. Suppose we are given any full-rank lattice $\mathcal{L}:=M(\mathbb{Z}^{d})$, and we define the integer point transform of a given rational polytope $\mathcal{P}\subset\mathbb{R}^{d}$, relative to $\mathcal{L}$, by

   $$\sigma_{\mathcal{P},\mathcal{L}}(\xi):=\sum_{n\in\mathcal{L}\cap\mathcal{P}}e^{2\pi i\langle n,\xi\rangle},$$

   for all $\xi\in\mathbb{R}^{d}$. Offhand, it may seem like we have a new extension, but in fact we may easily rewrite it as follows:

   $\displaystyle\sigma_{\mathcal{P},\mathcal{L}}(\xi)$

   $\displaystyle:=\sum_{n\in M(\mathbb{Z}^{d})\cap\mathcal{P}}e^{2\pi i\langle n,\xi\rangle}=\sum_{k\in\mathbb{Z}^{d}\cap\mathcal{P}}e^{2\pi i\langle Mk,\xi\rangle}=\sum_{k\in\mathbb{Z}^{d}\cap\mathcal{P}}e^{2\pi i\langle k,M^{T}\xi\rangle}:=\sigma_{\mathcal{P}}(M^{T}\xi).$

   Since $M$ is invertible ($\mathcal{L}$ has full-rank), and $\xi$ varies over all of $\mathbb{R}^{d}$, there is nothing really new in this particular extension, so we may use the usual integer point transform to sum over any lattice.