# Ehrhart quasi-polynomials and parallel translations

Akihiro Higashitani  Department of Pure and Applied Mathematics, Graduate
School of Information Science and Technology, Osaka University, Suita, Osaka
565-0871, Japan<EMAIL_ADDRESS>,  Satoshi Murai  Department of
Mathematics, Faculty of Education Waseda University, 1-6-1 NishiWaseda,
Shinjuku, Tokyo 169-8050, Japan<EMAIL_ADDRESS>and  Masahiko Yoshinaga
Masahiko Yoshinaga, Department of Mathematics, Graduate School of Science,
Osaka University, Toyonaka 560-0043, Japan<EMAIL_ADDRESS>
###### Abstract.

Given a rational polytope $P\subset\mathbb{R}^{d}$, the numerical function
counting lattice points in the integral dilations of $P$ is known to become a
quasi-polynomial, called the Ehrhart quasi-polynomial $\operatorname{ehr}_{P}$
of $P$. In this paper we study the following problem: Given a rational
$d$-polytope $P\subset\mathbb{R}^{d}$, is there a nice way to know Ehrhart
quasi-polynomials of translated polytopes $P+\bm{v}$ for all
$\bm{v}\in\mathbb{Q}^{d}$? We provide a way to compute such Ehrhart quasi-
polynomials using a certain toric arrangement and lattice point counting
functions of translated cones of $P$. This method allows us to visualize how
constituent polynomials of $\operatorname{ehr}_{P+\bm{v}}$ change in the torus
$\mathbb{R}^{d}/\mathbb{Z}^{d}$. We also prove that information of
$\operatorname{ehr}_{P+\bm{v}}$ for all $\bm{v}\in\mathbb{Q}^{d}$ determines
the rational $d$-polytope $P\subset\mathbb{R}^{d}$ up to translations by
integer vectors, and characterize all rational $d$-polytopes
$P\subset\mathbb{R}^{d}$ such that $\operatorname{ehr}_{P+\bm{v}}$ is
symmetric for all $\bm{v}\in\mathbb{Q}^{d}$.

##  1\. Introduction

Enumerations of lattice points in a convex polytope is a classical important
theme relating to algebra, combinatorics and geometry of convex polytopes. A
fundamental result on this subject is Ehrhart’s result telling that, for any
rational polytope $P\subset\mathbb{R}^{d}$, the function $\mathbb{Z}_{\geq
0}\ni t\mapsto\\#(tP\cap\mathbb{Z}^{d})$ becomes a quasi-polynomial in $t$,
where $tP$ is the $t$th dilation of $P$ and $\\#X$ denotes the cardinality of
a finite set $X$. This function is called the Ehrhart quasi-polynomial of $P$
and we denote it by $\operatorname{ehr}_{P}$. Let
$P+\bm{v}=\\{\bm{x}+\bm{v}\mid\bm{x}\in P\\}$ be the convex polytope obtained
from a convex polytope $P$ by the parallel translation by a vector
$\bm{v}\in\mathbb{R}^{d}$. The purpose of this paper is to develop a way to
understand behaviors of $\operatorname{ehr}_{P+\bm{v}}$ when $\bm{v}$ runs
over all vectors in $\mathbb{Q}^{d}$, where $P$ is a fixed rational polytope.

One motivation of studying this problem is special behaviors of
$\operatorname{ehr}_{P+\bm{v}}$ when we choose $\bm{v}\in\mathbb{Q}^{d}$
somewhat randomly. Let us give an example to explain this. Let
$T\subset\mathbb{R}^{2}$ be the trapezoid whose vertices are
$(0,0),(1,0),(2,1)$ and $(0,1)$. The Ehrhart quasi-polynomial of
$T+(\frac{17}{100},\frac{52}{100})$ becomes the following quasi-polynomial
having minimum period 100:

$\displaystyle\operatorname{ehr}_{T+(\frac{17}{100},\frac{52}{100})}(t)=\begin{cases}\frac{3}{2}t^{2}+\frac{5}{2}t+1&(\
t\equiv 0\ ),\\\ \frac{3}{2}t^{2}+\frac{3}{2}t&(\ t\equiv 25,50,75\ ),\\\
\frac{3}{2}t^{2}-\frac{1}{2}t&\left({\footnotesize\begin{array}[]{ll}t\equiv
1,3,6,7,9,12,13,15,18,19,21,23,24,26,30,\vspace{-3pt}\\\ \
32,36,38,42,44,48,49,53,55,59,61,65,66,67,\vspace{-3pt}\\\ \
69,71,72,73,78,83,84,86,89,90,92,95,96,98\end{array}}\right),\\\
\frac{3}{2}t^{2}+\frac{1}{2}t&\left({\footnotesize\begin{array}[]{ll}t\equiv
2,4,5,8,10,11,14,16,17,20,22,28,29,31,33,34,\vspace{-3pt}\\\ \
35,37,40,41,45,47,51,52,54,56,57,58,60,62,64,68,\vspace{-3pt}\\\ \
70,74,76,77,79,80,81,82,85,87,88,91,93,94,97,99\end{array}}\right),\end{cases}$

where “$t\equiv a$” means “$t\equiv a$ (mod $100$)”. This quasi-polynomial has
several special properties. For example, one can see

  * ($\alpha$)

It has a fairly large minimum period $100$, but it consists of only 4
polynomials.

  * ($\beta$)

The polynomials $\frac{3}{2}t^{2}\pm\frac{1}{2}t$ appear quite often comparing
other two polynomials.

  * ($\gamma$)

The polynomial $\frac{3}{2}t^{2}-\frac{1}{2}t$ appears when $t\equiv
1,3,6,7,\dots$, while the polynomial
$\frac{3}{2}t^{2}+\frac{1}{2}t=\frac{3}{2}(-t)^{2}-\frac{1}{2}(-t)$ appears
when $t=\dots,93,94,97,99$. There seem to be a kind of reciprocity about the
appearance of these two polynomials.

Our first goal is to explain why we get these phenomena using a certain
generalization of an Ehrhart quasi-polynomial which was considered by McMullen
[17] and is called a translated lattice point enumerators in [20].

### First result

We introduce a few notation to state our results. A function
$f:\mathbb{Z}\to\mathbb{R}$ is said to be a quasi-polynomial if there is a
natural number $q$ and polynomials $f_{0},f_{1},\dots,f_{q-1}$ such that

$f(t)=f_{i}(t)\mbox{ for all }t\in\mathbb{Z}\mbox{ with }t\equiv i\mbox{ (mod
}q).$

A number $q$ is called a period of $f$ and the polynomial $f_{k}$ is called
the $k$th constituent of $f$. For convention, we define the $k$th constituent
$f_{k}$ of $f$ for any $k\in\mathbb{Z}$ by setting $f_{k}=f_{k^{\prime}}$ with
$k^{\prime}\equiv k$ (mod $q$). For example, if $f$ has period $3$, then the
$7$th constituent equals the $1$st constituent $f_{1}$ and the $(-1)$th
constituent equals the $2$nd constituent $f_{2}$. We note that this definition
does not depend on a choice of a period. We will say that a function $L$ from
$\mathbb{Z}_{\geq 0}$ (or $\mathbb{Z}_{>0}$) to $\mathbb{R}$ is a quasi-
polynomial if there is a quasi-polynomial $f:\mathbb{Z}\to\mathbb{R}$ such
that $L(t)=f(t)$ for all $t\in\mathbb{Z}_{\geq 0}$ (or $\mathbb{Z}_{>0}$), and
in that case we regard $L$ as a function from $\mathbb{Z}$ to $\mathbb{R}$ by
identifying $L$ and $f$.

For a convex set $X\subset\mathbb{R}^{d}$ and a vector
$\bm{v}\in\mathbb{R}^{d}$, we define the function
$\mathrm{TL}_{X,\bm{v}}:\mathbb{Z}_{\geq 0}\to\mathbb{R}$ by

$\mathrm{TL}_{X,\bm{v}}(t)=\\#\big{(}(tX+\bm{v})\cap\mathbb{Z}^{d}\big{)}$

and call it the translated lattice points enumerator of $X$ w.r.t. $\bm{v}$.
Clearly $\mathrm{TL}_{P,\bm{0}}$ is nothing but the Ehrhart quasi-polynomial
of $P$. Generalizing Ehrhart’s results, McMullen [17, §4] proved that, if $P$
is a rational polytope such that $qP$ is integral then
$\mathrm{TL}_{P,\bm{v}}$ is a quasi-polynomial with period $q$, and showed
that there is a reciprocity between $\mathrm{TL}_{\mathrm{int}(P),\bm{v}}$ and
$\mathrm{TL}_{P,-\bm{v}}$, where $\mathrm{int}(P)$ is the interior of $P$. As
we will see soon in Section 2, for a rational polytope
$P\subset\mathbb{R}^{d}$ and $\bm{v}\in\mathbb{Q}^{d}$, it follows from the
above result of McMullen that

(1.1) $\displaystyle\mbox{the $k$th constituent of
$\operatorname{ehr}_{P+\bm{v}}$}=\mbox{the $k$th constituent of
$\mathrm{TL}_{P,k\bm{v}}$}$

for all $k\in\mathbb{Z}$. This equation (1.1) tells that knowing
$\operatorname{ehr}_{P+\bm{v}}$ for all $\bm{v}\in\mathbb{Q}^{d}$ is
essentially equivalent to knowing $\mathrm{TL}_{P,\bm{v}}$ for all
$\bm{v}\in\mathbb{Q}^{d}$. Our first goal is to explain that the latter
information can be described as a finite information although
$\operatorname{ehr}_{P+\bm{v}}$ could have arbitrary large minimum period.

To do this, we first discuss when $\mathrm{TL}_{P,\bm{u}}$ and
$\mathrm{TL}_{P,\bm{v}}$ equal for different $\bm{u},\bm{v}\in\mathbb{R}^{d}$
using toric arrangements. For $\bm{a}=(a_{1},\dots,a_{d})\in\mathbb{R}^{d}$
and $b\in\mathbb{R}$, let $H_{\bm{a},b}$ be the hyperplane of $\mathbb{R}^{d}$
defined by the equation $a_{1}x_{1}+\cdots+a_{d}x_{d}=b$. Let $P$ be a
rational convex $d$-polytope having $m$ facets $F_{1},\dots,F_{m}$ such that
each $F_{k}$ lies in the hyperplane $H_{\bm{a}_{k},b_{k}}$ with
$\bm{a}_{k}\in\mathbb{Z}^{d}$, $b_{k}\in\mathbb{Z}$ and
$\mathrm{gcd}(\bm{a}_{k},b_{k})=1$. We consider the arrangement of hyperplanes

$\mathcal{A}_{P}=\bigcup_{i=1}^{m}\\{H_{\bm{a}_{i},k}\mid k\in\mathbb{Z}\\}$

and let $\Delta_{P}$ be the open polyhedral decomposition of $\mathbb{R}^{d}$
determined by $\mathcal{A}_{P}$. Both $\mathcal{A}_{P}$ and $\Delta_{P}$ are
closed under translations by integer vectors, so by the natural projection
$\mathbb{R}^{d}\to\mathbb{R}^{d}/\mathbb{Z}^{d}$ they induce an arrangement of
finite hyperplanes on the torus $\mathbb{R}^{d}/\mathbb{Z}^{d}$ and a finite
open cell decomposition $\Delta_{P}/\mathbb{Z}^{d}$ of
$\mathbb{R}^{d}/\mathbb{Z}^{d}$. Let
$[\bm{v}]\in\mathbb{R}^{d}/\mathbb{Z}^{d}$ denote the natural projection of
$\bm{v}\in\mathbb{R}^{d}$ to $\mathbb{R}^{d}/\mathbb{Z}^{d}$.

######  Theorem 1.1.

With the notation as above, for $\bm{u},\bm{v}\in\mathbb{R}^{d}$, if
$[\bm{u}]$ and $[\bm{v}]$ belong to the same open cell of
$\Delta_{P}/\mathbb{Z}^{d}$ then

$\mathrm{TL}_{P,\bm{u}}(t)=\mathrm{TL}_{P,\bm{v}}(t)\ \ \mbox{ for all
}t\in\mathbb{Z}_{\geq 0}.$

For an open cell $C\in\Delta_{P}/\mathbb{Z}^{d}$, define a quasi-polynomial
$\mathrm{TL}_{P,C}$ by

$\mathrm{TL}_{P,C}=\mathrm{TL}_{P,\bm{v}}\ \ \ \mbox{ with }[\bm{v}]\in C,$

which is well-defined by Theorem 1.1. Then (1.1) tells that the $k$th
constituent of $\operatorname{ehr}_{P+\bm{v}}$ is the polynomial which appears
as the $k$th constituent of $\mathrm{TL}_{P,C}$ with $[k\bm{v}]\in C$. This
provides us a way to compute $\operatorname{ehr}_{P+\bm{v}}$ for any
$\bm{v}\in\mathbb{Q}^{d}$ from translated lattice points enumerators
$\mathrm{TL}_{P,C}$.

Let us compute $\operatorname{ehr}_{T+(\frac{17}{100},\frac{52}{100})}(t)$
using this idea, where $T$ is the trapezoid whose vertices are
$(0,0),(1,0),(2,1)$ and $(0,1)$.

Figure 1. Trapezoid $T$, arrangement $\mathcal{A}_{T}$ and the cell complex
$\Delta_{T}$.

Figure 1 shows the arrangement $\mathcal{A}_{T}$ and the cell complex
$\Delta_{T}/\mathbb{Z}^{2}$. The complex $\Delta_{T}/\mathbb{Z}^{2}$ has two
$2$-dimensional cells $F_{1},F_{2}$, three $1$-dimensional cells
$E_{1},E_{2},E_{3}$ and one $0$-dimensional cell $V_{1}$ shown in Figure 1.
Since $T$ is a lattice polygon, each $\mathrm{TL}_{P,C}$ is a polynomial by
McMullen’s result, and here are list of $\mathrm{TL}_{T,C}(t)$:

(1.6)
$\displaystyle\begin{array}[]{lll}\mathrm{TL}_{T,F_{1}}(t)=\frac{3}{2}t^{2}-\frac{1}{2}t,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\mathrm{TL}_{T,F_{2}}(t)=\mathrm{TL}_{T,E_{1}}(t)=\mathrm{TL}_{T,E_{2}}(t)=\frac{3}{2}t^{2}+\frac{1}{2}t,\vskip
3.0pt plus 1.0pt minus 1.0pt\\\
\mathrm{TL}_{T,E_{3}}(t)=\frac{3}{2}t^{2}+\frac{3}{2}t,\vskip 3.0pt plus 1.0pt
minus 1.0pt\\\
\mathrm{TL}_{T,V_{1}}(t)=\frac{3}{2}t^{2}+\frac{3}{2}t+1.\end{array}$

Also, for $k=0,1,2,\dots,99$, one has

(1.7)
$\displaystyle\left[k\left(\frac{17}{100},\frac{52}{100}\right)\right]\in\begin{cases}V_{1}&(\
k\equiv 0\ ),\\\ E_{3}&(\ k\equiv 25,50,75\ ),\\\ E_{2}&(\ k\equiv
20,40,60,80\ ),\\\ F_{1}&\left({\footnotesize\begin{array}[]{ll}k\equiv
1,3,6,7,9,12,13,15,18,19,21,23,24,26,30,\vspace{-3pt}\\\ \
32,36,38,42,44,48,49,53,55,59,61,65,66,67,\vspace{-3pt}\\\ \
69,71,72,73,78,83,84,86,89,90,92,95,96,98\end{array}}\right),\\\
F_{2}&\left({\footnotesize\begin{array}[]{ll}k\equiv
2,4,5,8,10,11,14,16,17,22,28,29,31,33,34,\vspace{-3pt}\\\ \
35,37,41,45,47,51,52,54,56,57,58,62,64,68,70,\vspace{-3pt}\\\ \
74,76,77,79,81,82,85,87,88,91,93,94,97,99\end{array}}\right).\end{cases}$

Since (1.1) tells that the $k$th constituent of
$\operatorname{ehr}_{P+\bm{v}}$ equals the $k$th constituent of
$\mathrm{TL}_{P,k\bm{v}}$, which equals $\mathrm{TL}_{P,C}$ with $[k\bm{v}]\in
C\in\Delta_{P}/\mathbb{Z}^{d}$, the equations (1.6) and (1.7) recover the
formula of $\operatorname{ehr}_{T+(\frac{17}{100},\frac{52}{100})}(t)$ given
at the beginning of this section.

As we will see, the proof of Theorem 1.1 is somewhat straightforward, and the
way of computing $\operatorname{ehr}_{P+\bm{v}}(t)$ from
$\mathrm{TL}_{P,C}(t)$ explained above may be considered as a kind of an
observation rather than a new result. But we think that this is a useful
observation. For example, this way allows us to visualize how the constituents
of $\operatorname{ehr}_{P+\bm{v}}$ change by plotting the points $[k\bm{v}]$
on $\mathbb{R}^{d}/\mathbb{Z}^{d}$. Also, we can see why properties
($\alpha$), ($\beta$) and ($\gamma$) occur from this observation. For the
property $(\alpha)$, we only see 4 polynomials in
$\operatorname{ehr}_{T+(\frac{17}{100},\frac{52}{100})}$ because we have only
4 types of translated lattice point enumerators. More generally, it can be
shown that, if we fix a rational polytope $P$, then we can only have a finite
number of polynomials as constituents of $\operatorname{ehr}_{P+\bm{v}}$
(Theorem 3.10). For the property ($\beta$), the polynomials
$\frac{3}{2}t^{2}\pm\frac{1}{2}t$ appear many times simply because they are
polynomials assigned to maximal dimensional cells of
$\Delta_{T}/\mathbb{Z}^{2}$ (indeed, if we choose $\bm{v}$ randomly, then
$[k\bm{v}]$ is likely to belong to a maximal dimensional cell). Finally, we
will see in Section 5 that the property ($\gamma$) can be figured out from the
reciprocity of $\mathrm{TL}_{P,\bm{v}}$ (see Corollary 5.4).

### Second Result

Recently real-valued extension of Ehrhart functions, namely, the function
$\operatorname{ehr}_{P}^{\mathbb{R}}:\mathbb{R}\to\mathbb{Z}_{\geq 0}$ given
by $\operatorname{ehr}_{P}^{\mathbb{R}}(t)=\\#(tP\cap\mathbb{Z}^{d})$ for all
$t\in\mathbb{R}_{\geq 0}$, catch interests [3, 4, 12, 13, 14]. One surprising
result on this topic is the following result of Royer [13, 14] proving that
$\operatorname{ehr}_{P+\bm{v}}^{\mathbb{R}}$ for all $\bm{v}\in\mathbb{Z}^{d}$
determines the polytope $P$.

######  Theorem 1.2 (Royer).

Let $P$ and $Q$ be rational polytopes in $\mathbb{R}^{d}$. If
$\operatorname{ehr}_{P+\bm{v}}^{\mathbb{R}}(t)=\operatorname{ehr}_{Q+\bm{v}}^{\mathbb{R}}(t)$
for all $\bm{v}\in\mathbb{Z}^{d}$ and $t\in\mathbb{R}_{\geq 0}$, then $P=Q$.

Our second result is somewhat analogous to this result of Royer. We prove that
$\operatorname{ehr}_{P+\bm{v}}$ for all $\bm{v}\in\mathbb{Q}^{d}$ determines
the polytope $P$ up to translations by integer vectors.

######  Theorem 1.3.

Let $P$ and $Q$ be rational $d$-polytopes in $\mathbb{R}^{d}$. If
$\operatorname{ehr}_{P+\bm{v}}(t)=\operatorname{ehr}_{Q+\bm{v}}(t)$ for all
$\bm{v}\in\mathbb{Q}^{d}$ and $t\in\mathbb{Z}_{\geq 0}$, then $P=Q+\bm{u}$ for
some $\bm{u}\in\mathbb{Z}^{d}$.

### Third result

The original motivation of this study actually comes from an attempt to
generalize results of de Vries and the third author in [20], who found a
connection between symmetries on constituents of
$\operatorname{ehr}_{P+\bm{v}}$ and geometric symmetries of $P$. Indeed, the
following result is one of the main results in [20]. We say that a quasi-
polynomial $f$ is symmetric if the $k$th constituent of $f$ equals the
$(-k)$th constituent of $f$ for all $k\in\mathbb{Z}$. Also, a convex polytope
$P\subset\mathbb{R}^{d}$ is said to be centrally symmetric if $P=-P+\bm{u}$
for some $\bm{u}\in\mathbb{R}^{d}$.

######  Theorem 1.4 (de Vries–Yoshinaga).

Let $P\subset\mathbb{R}^{d}$ be a lattice $d$-polytope. The following
conditions are equivalent.

  * (1)

$\operatorname{ehr}_{P+\bm{v}}$ is symmetric for any
$\bm{v}\in\mathbb{Q}^{d}$.

  * (2)

$P$ is centrally symmetric.

As posed in [20, Problem 6.7], it is natural to ask if there is a
generalization of this result for rational polytopes. Theorem 1.4 actually
proves that, if a rational polytope $P$ satisfies the property (1) of the
above theorem, then $P$ must be centrally symmetric (see Corollary 7.4).
Hence, to answer this question, we can assume $P=-P$. We generalize Theorem
1.4 in the following form.

######  Theorem 1.5.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope with $P=-P$. The
following conditions are equivalent.

  * (1)

$\operatorname{ehr}_{P+\bm{v}}$ is symmetric for all
$\bm{v}\in\mathbb{Q}^{d}$.

  * (2)

$2P$ is integral.

### Organization of the paper

This paper is organized as follows: We first quickly review basic known
properties of Ehrhart quasi-polynomials and translated lattice points
enumerators in Section 2. In Section 3, we study translated lattice points
enumerators using arrangement $\mathcal{A}_{P}$ and prove Theorem 1.1. Then,
after seeing two examples in Section 4, we discuss a reciprocity of translated
lattice points enumerators on maximal cells of $\Delta_{P}/\mathbb{Z}^{d}$ in
Section 5. In Section 6, we prove that translated lattice point enumerators
determine the polytope $P$ up to translations by integer vectors. In Section
7, we study translated lattice points enumerators of polytopes with some
symmetry, in particular, prove Theorem 1.5. In Section 8, we discuss a
connection to commutative algebra, more precisely, we discuss a connection
between translated lattice points enumerators and conic divisorial ideals in
Ehrhart rings. We list a few problems which we cannot solve in the last
section 9.

### Acknowledgements

We thank Katharina Jochemko for letting us know McMullen’s work in [17] and
thank Matthias Beck for letting us know the work of Royer in [13, 14]. The
first author is partially supported by KAKENHI 20K03513 and 21KK0043, the
second author is partially supported by KAKENHI 21K0319 and 21H00975, and the
third author is partially supported by KAKENHI 18H01115 and 23H00081.

##  2\. Ehrhart quasi-polynomials and translated lattice point enumerators

In this section, we recall basic results on Ehrhart quasi-polynomials and
explain a connection between Ehrhart quasi-polynomials of translated polytopes
and translated lattice point enumerators.

###  2.1. Ehrhart quasi-polynomial

We quickly recall Ehrhart’s theorems. We refer the readers to [5, 21] for
basics on convex polytopes. A convex polytope $P$ in $\mathbb{R}^{d}$ is a
convex hull of finite points in $\mathbb{R}^{d}$. The dimension of a polytope
$P$ is the dimension of its affine hull. A $k$-dimensional convex polytope
will be simply called a $k$-polytope in this paper. A convex polytope $P$ is
said to be integral (resp. rational) if all the vertices of $P$ are lattice
points (resp. rational points). The denominator of a rational polytope $P$ is
the smallest integer $k>0$ such that $kP$ is integral. The following result is
a fundamental result in Ehrhart theory. See [5, Theorems 3.23 and 4.1].

######  Theorem 2.1 (Ehrhart).

Let $P\subset\mathbb{R}^{d}$ be a rational polytope and $q$ the denominator of
$P$. Then the function $\operatorname{ehr}_{P}:\mathbb{Z}_{\geq
0}\to\mathbb{R}$ defined by

$\operatorname{ehr}_{P}(t)=\\#(tP\cap\mathbb{Z}^{d})$

is a quasi-polynomial with period $q$.

As we noted in Introduction, we regard $\operatorname{ehr}_{P}$ as a function
from $\mathbb{Z}$ to $\mathbb{R}$ by identifying it with the corresponding
quasi-polynomial $f:\mathbb{Z}\to\mathbb{R}$ that coincides with
$\operatorname{ehr}_{P}$ on $\mathbb{Z}_{\geq 0}$. Thus, if $q$ is a period of
$f$, then for a positive integer $t>0$ we set
$\operatorname{ehr}_{P}(-t)=f_{k}(-t)$, where $f_{k}$ is the $k$th constituent
of $f$ with $-t\equiv k$ (mod $q$). The quasi-polynomial
$\operatorname{ehr}_{P}$ is called the Ehrhart quasi-polynomial of $P$.

The following reciprocity result is another important result in Ehrhart
theory.

######  Theorem 2.2 (Ehrhart reciprocity).

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. Then

$\\#\big{(}\mathrm{int}(tP)\cap\mathbb{Z}^{d}\big{)}=(-1)^{d}\operatorname{ehr}_{P}(-t)\
\ \mbox{ for }\ t\in\mathbb{Z}_{>0}.$

###  2.2. Translated lattice points enumerator

Recall that, for a convex set $X\subset\mathbb{R}^{d}$ and
$\bm{v}\in\mathbb{R}^{d}$, the translated lattice points enumerator of $X$
w.r.t. $\bm{v}$ is the function $\mathrm{TL}_{X,\bm{v}}:\mathbb{Z}_{\geq
0}\to\mathbb{R}$ defined by

$\mathrm{TL}_{X,\bm{v}}(t)=\\#\big{(}(tX+\bm{v})\cap\mathbb{Z}^{d}\big{)}\ \
\mbox{ for }t\in\mathbb{Z}_{\geq 0}.$

McMullen [17, §4] proved the following generalization of Ehrhart’s results.

######  Theorem 2.3 (McMullen).

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope and $q$ the
denominator of $P$. Then

  * (1)

For any $\bm{v}\in\mathbb{R}^{d}$, the function $\mathrm{TL}_{P,\bm{v}}$ is a
quasi-polynomial with period $q$.

  * (2)

For any $\bm{v}\in\mathbb{R}^{d}$, one has

$\mathrm{TL}_{\mathrm{int}(P),\bm{v}}(t)=(-1)^{d}\mathrm{TL}_{P,-\bm{v}}(-t)\
\ \mbox{ for $t\in\mathbb{Z}_{>0}$.}$

We remark that $\bm{v}$ is not necessarily a rational point in the above
theorem and that the function $\mathrm{TL}_{P,\bm{v}}$ becomes a polynomial
when $P$ is integral.

The following connection between Ehrhart quasi-polynomials of translated
polytopes and translated lattice points enumerators, which essentially
appeared in [20, Corollary 3.4], is fundamental in the rest of this paper.

######  Lemma 2.4.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope and
$\bm{v}\in\mathbb{Q}^{d}$. For any integer $k\in\mathbb{Z}$, one has

$\mbox{\rm the $k$th constituent of $\operatorname{ehr}_{P+\bm{v}}$}=\mbox{\rm
the $k$th constituent of $\mathrm{TL}_{P,k\bm{v}}$}$

###### Proof.

We may assume $k\geq 0$. Let $\rho$ and $\rho^{\prime}$ be positive integers
such that $\rho P$ is integral and $\rho^{\prime}\bm{v}\in\mathbb{Z}^{d}$. Let
$q$ be a common multiple of $\rho$ and $\rho^{\prime}$. Then $q$ is a common
period of quasi-polynomials $\operatorname{ehr}_{P+\bm{v}}$ and
$\mathrm{TL}_{P,\bm{v}}$. For every integer $t\geq 0$ with $t\equiv k$ (mod
$q$) we have

$\displaystyle\operatorname{ehr}_{P+\bm{v}}(t)$
$\displaystyle=\\#\big{(}(tP+t\bm{v})\cap\mathbb{Z}^{d}\big{)}=\\#\big{(}(tP+k\bm{v})\cap\mathbb{Z}^{d}\big{)}=\mathrm{TL}_{P,k\bm{v}}(t),$

where the second equality follows from $(t-k)\bm{v}\in\mathbb{Z}^{d}$. Since
both $\operatorname{ehr}_{P+\bm{v}}$ and $\mathrm{TL}_{P,\bm{v}}$ are quasi-
polynomials with a period $q$, the above equation proves the desired property.
∎

######  Remark 2.5.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope and
$\bm{v}\in\mathbb{R}^{d}$. Like usual Ehrhart quasi-polynomials, each
constituent of $\mathrm{TL}_{P,\bm{v}}$ is a polynomial of degree $d$ whose
leading coefficient equals the volume of $P$. Indeed, if $f_{k}$ is the $k$th
constituent of $\mathrm{TL}_{P,\bm{v}}$ and $q$ is a period of
$\mathrm{TL}_{P,\bm{v}}$, then
$\lim_{t\to\infty}\frac{f_{k}(qt+k)}{(qt+k)^{d}}=\lim_{t\to\infty}\frac{\\#((qt+k)P\cap\mathbb{Z}^{d})}{(qt+k)^{d}}$
is the volume of $P$. Since $f_{k}$ is a polynomial, this means that $f_{k}$
has degree $d$ and the coefficient of $t^{d}$ in $f_{k}$ equals the volume of
$P$.

##  3\. Translated lattice points enumerators and toric arrangements

In this section, we study when $\mathrm{TL}_{P,\bm{v}}$ equals
$\mathrm{TL}_{P,\bm{u}}$ for different $\bm{v},\bm{u}\in\mathbb{R}^{d}$ using
toric arrangements. Our goal is to prove Theorem 1.1. To study this problem,
it is convenient to regard a translated lattice points enumerator as a
generating function of a translated cone. Let $P\subset\mathbb{R}^{d}$ be a
convex polytope and let $\mathcal{C}_{P}$ be the cone generated by
$\\{(\bm{x},1)\mid\bm{x}\in P\\}$. Then the translated lattice points
enumerator $\mathrm{TL}_{P,\bm{v}}$ can be identified with a generating
function of a translated cone $\mathcal{C}_{P}+(\bm{v},0)$ because of the
equality

$\sum_{(a_{1},\dots,a_{d+1})\in(\mathcal{C}_{P}+(\bm{v},0))\cap\mathbb{Z}^{d+1}}z^{a_{d+1}}=\sum_{t=0}^{\infty}\big{(}\mathrm{TL}_{P,\bm{v}}(t)\big{)}z^{t}.$

We refer the readers for the verification of this equation. This in particular
tells that if $\mathcal{C}_{P}+(\bm{u},0)$ and $\mathcal{C}_{P}+(\bm{v},0)$
have the same lattice points, then we have
$\mathrm{TL}_{P,\bm{u}}=\mathrm{TL}_{P,\bm{v}}$. To prove Theorem 1.1, we
mainly study when $\mathcal{C}_{P}+(\bm{u},0)$ and
$\mathcal{C}_{P}+(\bm{v},0)$ have the same lattice points.

We note that such a study is not very new. Indeed, lattice points in the
translated cone $\mathcal{C}_{P}+\bm{v}$ is closely related to conic
divisorial ideals of Ehrhart rings studied in [7, 9], and Bruns [7] explains
for which $\bm{u},\bm{v}\in\mathbb{R}^{d+1}$ the lattice points in
$\mathcal{C}_{P}+\bm{u}$ equal those in $\mathcal{C}_{P}+\bm{v}$. We will
explain this connection to commutative algebra later in Section 8.

###  3.1. Regions associated with hyperplane arrangements

We first introduce some notation on arrangements of hyperplanes. For
$\bm{x},\bm{y}\in\mathbb{R}^{d}$, we write $(\bm{x},\bm{y})$ for the standard
inner product. Also, for $\bm{a}\in\mathbb{R}^{d}\setminus\\{\bm{0}\\}$ and
$b\in\mathbb{R}$, we write

$H^{\geq}_{\bm{a},b}=\\{\bm{x}\in\mathbb{R}^{d}\mid(\bm{a},\bm{x})\geq b\\}\ \
\mbox{ and }\ \
H^{>}_{\bm{a},b}=\\{\bm{x}\in\mathbb{R}^{d}\mid(\bm{a},\bm{x})>b\\}$

for closed and open half space defined by the linear inequalities
$(\bm{a},\bm{x})\geq b$ and $(\bm{a},\bm{x})>b$, respectively, and write

$H_{\bm{a},b}=\\{\bm{x}\in\mathbb{R}^{d}\mid(\bm{a},\bm{x})=b\\}$

for the hyperplane defined by the linear equation $(\bm{a},\bm{x})=b$. In the
case where $\bm{a}$ can be chosen from $\mathbb{Z}^{d}$ and $b$ is from
$\mathbb{Z}$, we call the hyperplane $H_{\bm{a},b}$ rational. Let
$N=\\{\bm{a}_{1},\dots,\bm{a}_{m}\\}$ be a set of elements in
$\mathbb{Z}^{d}\setminus\\{\bm{0}\\}$. Define the arrangement of hyperplanes

$\mathcal{A}_{N}=\\{H_{\bm{a},k}\mid\bm{a}\in N,\ k\in\mathbb{Z}\\}.$

See Figures 2 and 3. From now on, we fix an order
$\bm{a}_{1},\dots,\bm{a}_{m}$ of elements of $N$. We define the map
$\varphi_{(\bm{a}_{1},\dots,\bm{a}_{m})}:\mathbb{R}^{d}\to\mathbb{R}^{m}$ by

$\varphi_{(\bm{a}_{1},\dots,\bm{a}_{m})}(\bm{x})=\big{(}(\bm{a}_{1},\bm{x}),(\bm{a}_{2},\bm{x}),\dots,(\bm{a}_{m},\bm{x})\big{)}.$

For $x\in\mathbb{R}$, we write $\lfloor x\rfloor=\max\\{a\in\mathbb{Z}\mid
a\leq x\\}$ and $\lceil x\rceil=\min\\{a\in\mathbb{Z}\mid a\geq x\\}$. Also,
given an integer sequence $\bm{c}=(c_{1},\dots,c_{m})\in\mathbb{Z}^{m}$, we
define

$U_{\bm{c}}^{N}=\\{\bm{x}\in\mathbb{R}^{d}\mid\lceil\varphi_{(\bm{a}_{1},\dots,\bm{a}_{m})}(\bm{x})\rceil=\bm{c}\\}=\\{\bm{x}\in\mathbb{R}^{d}\mid
c_{i}-1<(\bm{a}_{i},\bm{x})\leq c_{i}\mbox{ for }i=1,2,\dots,m\\}$

where $\lceil(x_{1},\dots,x_{m})\rceil=(\lceil x_{1}\rceil,\dots,\lceil
x_{m}\rceil)$. We call $U_{\bm{c}}^{N}$ an upper region of $N$. Note that
$U_{\bm{c}}^{N}$ could be empty. Also we have the partition

$\textstyle\mathbb{R}^{d}=\bigsqcup_{\bm{c}\in\mathbb{Z}^{d}}U_{\bm{c}}^{N}$

where $\bigsqcup$ denotes a disjoint union. We write $\Lambda_{N}$ for the set
of all upper regions of $N$. The set $\Lambda_{N}$ is stable by translations
by integer vectors, so $\mathbb{Z}^{d}$ acts on these sets. Indeed, since
$\bm{a}_{1},\dots,\bm{a}_{m}$ are integer vectors, for any
$\bm{v}\in\mathbb{Z}^{d}$, we have

$U_{\bm{c}}^{N}+\bm{v}=U_{\bm{c}+\varphi_{N}(\bm{v})}^{N}.$

We write $\Lambda_{N}/\mathbb{Z}^{d}$ for the quotient of these sets by this
$\mathbb{Z}^{d}$-action defined by translations by integer vectors. This set
can be considered as a partition of the $d$-torus
$\mathbb{R}^{d}/\mathbb{Z}^{d}$.

######  Example 3.1.

Let $N=\\{(1,0),(-1,2)\\}$. Then the set $\Lambda_{N}/\mathbb{Z}^{2}$ consists
of two elements with the following representatives:

$\displaystyle R_{1}$
$\displaystyle=U_{(1,1)}^{N}=\\{(x,y)\in\mathbb{R}^{2}\mid 0<x\leq
1,0<-x+2y\leq 1\\},$ $\displaystyle R_{2}$
$\displaystyle=U_{(1,0)}^{N}=\\{(x,y)\in\mathbb{R}^{2}\mid 0<x\leq
1,-1<-x+2y\leq 0\\}.$

See Figure 2 for the visualization of $\mathcal{A}_{N}$ and
$\Lambda_{N}/\mathbb{Z}^{2}$.

Figure 2. $\mathcal{A}_{N}$ and $\Lambda_{N}/\mathbb{Z}^{2}$ when
$N=\\{(1,-2),(0,1)\\}$.

###  3.2. Upper regions and lattice points in translated cones.

We now explain how these regions relate to lattice points in translated cones.
We first recall two basic facts on lattice points. The following lemma is an
easy consequence of Euclidian algorithm.

######  Lemma 3.2.

Let $\bm{a}\in\mathbb{Z}^{d}\setminus\\{\bm{0}\\}$, $b\in\mathbb{R}$ and
$g=\mathrm{gcd}(\bm{a})$. A linear equation $(\bm{a},\bm{x})=b$ has an
integral solution if and only if $b\in g\mathbb{Z}$.

We also need the following statement which easily follows from Lemma 3.2.

######  Lemma 3.3.

Let $H\subset\mathbb{R}^{d}$ be a hyperplane and let $\bm{v}\in\mathbb{R}^{d}$
be a point such that $H+\bm{v}\neq H$. There is an $\varepsilon>0$ such that
$H+s\bm{v}$ contains no lattice points for any $0<s\leq\varepsilon$.

######  Lemma 3.4.

Let $H\subset\mathbb{R}^{d}$ be a rational hyperplane. Any $(d-1)$-dimensional
convex cone in $H$ contains a lattice point.

###### Proof.

Let $H=H_{\bm{a},b}$ for some $\bm{a}\in\mathbb{Z}^{d}$ and $b\in\mathbb{Z}$.
Without loss of generality, we may assume $b=0$. Since any $d$-dimensional
convex cone in $\mathbb{R}^{d}$ contains a lattice point, the lemma follows
from the fact that $H\cap\mathbb{Z}^{d}\cong\mathbb{Z}^{d-1}$ as
$\mathbb{Z}$-modules. ∎

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. By the fundamental
theorem on convex cones, the cone $\mathcal{C}_{P}$ has the unique
presentation

(3.1) $\displaystyle\mathcal{C}_{P}=H_{\bm{a}_{1},0}^{\geq}\cap\cdots\cap
H_{\bm{a}_{m},0}^{\geq}$

such that

  * (1)

each $\bm{a}_{i}$ is primitive (that is, $\mathrm{gcd}(\bm{a}_{i})=1$), and

  * (2)

the presentation is irredundant, that is, each $H_{\bm{a}_{i},0}^{\geq}$
cannot be omitted from the presentation.

Note that the second condition tells that $\mathcal{C}_{P}\cap
H_{\bm{a}_{i},0}$ is a facet of $\mathcal{C}_{P}$. The vectors
$\bm{a}_{1},\dots,\bm{a}_{m}$ in (3.1) are called (inner) normal vectors of
$\mathcal{C}_{P}$ and we write
$\widetilde{N}(P)=\\{\bm{a}_{1},\dots,\bm{a}_{m}\\}$ for the set of all normal
vectors of $\mathcal{C}_{P}$. The next statement was given in [7]

######  Proposition 3.5 (Bruns).

Let $P\subset\mathbb{R}^{d}$ a convex polytope,
$\widetilde{N}(P)=\\{\bm{a}_{1},\dots,\bm{a}_{m}\\}$ and
$\bm{n}=(\bm{a}_{1},\dots,\bm{a}_{m})$. Let
$\bm{u},\bm{v}\in\mathbb{R}^{d+1}$. The following conditions are equivalent.

  * (1)

$(\mathcal{C}_{P}+\bm{u})\cap\mathbb{Z}^{d+1}=(\mathcal{C}_{P}+\bm{v})\cap\mathbb{Z}^{d+1}$.

  * (2)

$\lceil\varphi_{\bm{n}}(\bm{u})\rceil=\lceil\varphi_{\bm{n}}(\bm{v})\rceil$,
that is, $\bm{u}$ and $\bm{v}$ belong to the same upper region of
$\Lambda_{\widetilde{N}(P)}$.

###### Proof.

Let $\lceil\varphi_{\bm{n}}(\bm{u})\rceil=(c_{1},\dots,c_{m})$ and let
$\lceil\varphi_{\bm{n}}(\bm{v})\rceil=(d_{1},\dots,d_{m})$. Since

$\mathcal{C}_{P}+\bm{u}=H_{\bm{a}_{1},(\bm{a}_{1},\bm{u})}^{\geq}\cap\cdots\cap
H_{\bm{a}_{m},(\bm{a}_{m},\bm{u})}^{\geq}$

and since Lemma 3.2 tells

$H_{\bm{a},b}^{\geq}\cap\mathbb{Z}^{d+1}=H_{\bm{a},\lceil
b\rceil}^{\geq}\cap\mathbb{Z}^{d+1}\ \ \mbox{ for any
}\bm{a}\in\mathbb{Z}^{d},\ b\in\mathbb{R},$

we have

(3.2)
$\displaystyle(\mathcal{C}_{P}+\bm{u})\cap\mathbb{Z}^{d+1}=\big{(}H_{\bm{a}_{1},c_{1}}^{\geq}\cap\cdots\cap
H_{\bm{a}_{m},c_{m}}^{\geq}\big{)}\cap\mathbb{Z}^{d+1}$

and

$(\mathcal{C}_{P}+\bm{v})\cap\mathbb{Z}^{d+1}=\big{(}H_{\bm{a}_{1},d_{1}}^{\geq}\cap\cdots\cap
H_{\bm{a}_{m},d_{m}}^{\geq}\big{)}\cap\mathbb{Z}^{d+1}.$

These prove (2) $\Rightarrow$ (1).

We prove (1) $\Rightarrow$ (2). We assume $c_{1}<d_{1}$ and prove
$(\mathcal{C}_{P}+\bm{u})\cap\mathbb{Z}^{d+1}\neq(\mathcal{C}_{P}+\bm{v})\cap\mathbb{Z}^{d+1}$.
In this case $F=(\mathcal{C}_{P}+\bm{u})\cap H_{\bm{a}_{1},c_{1}}$ contains a
$(d-1)$-dimensional cone in $H_{\bm{a}_{1},c_{1}}$, so it contains a lattice
point by Lemma 3.4. On the other hand, since $c_{1}<d_{1}$ we have
$\mathbb{Z}^{d+1}\cap(\mathcal{C}_{P}+\bm{v})\cap
H_{\bm{a}_{1},c_{1}}=\varnothing$. These prove
$(\mathcal{C}_{P}+\bm{u})\cap\mathbb{Z}^{d+1}\neq(\mathcal{C}_{P}+\bm{v})\cap\mathbb{Z}^{d+1}$.
∎

######  Remark 3.6.

If $N=\widetilde{N}(P)$ for some rational $d$-polytope
$P\subset\mathbb{R}^{d}$, then the set $\Lambda_{N}$ has a special property
that every element of $\Lambda_{N}$ has dimension $d+1$. Indeed, if
$R\in\Lambda_{N}$ and $\bm{p}\in R$ then we have $\bm{p}-\bm{v}\in R$ for all
$\bm{v}\in\mathrm{int}(\mathcal{C}_{P})$ that is sufficiently close to the
origin, which tells that $R$ has dimension $d+1$. As we see in Example 3.9,
this property does not hold for a general $N$.

We have studied lattice points in translated cones $\mathcal{C}_{P}+\bm{v}$,
but we are actually interested in a special case when
$\bm{v}=(\bm{v}^{\prime},0)$ since this is the case which is related to
translated lattice points enumerators. Below we describe when
$\mathcal{C}_{P}+(\bm{u},0)$ and $\mathcal{C}_{P}+(\bm{v},0)$ have the same
lattice points. Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. By
the fundamental theorem on convex polytopes, there is the unique presentation

$P=H_{\bm{a}_{1},b_{1}}^{\geq}\cap\cdots\cap H_{\bm{a}_{m},b_{m}}^{\geq}$

such that

  * (1)

each $(\bm{a}_{i},b_{i})\in\mathbb{Z}^{d+1}$ is primitive, and

  * (2)

the presentation is irredundant.

The vectors $\bm{a}_{1},\dots,\bm{a}_{m}$ are called normal vectors of $P$. We
define

$N(P)=\\{\bm{a}_{1},\dots,\bm{a}_{m}\\}.$

We note that

$\widetilde{N}(P)=\\{(\bm{a}_{1},b_{1}),\dots,(\bm{a}_{m},b_{m})\\}$

since if $H\subset\mathbb{R}^{d}$ is a hyperplane defined by
$a_{1}x_{1}+\cdots+a_{n}x_{d}=b$ then the cone $\mathcal{C}_{H}$ is the
hyperplane defined by $a_{1}x_{1}+\cdots+a_{d}x_{d}=bx_{d+1}$. We write

$\mathcal{A}_{P}=\mathcal{A}_{N(P)}\ \ \mbox{ and }\ \
\Lambda_{P}=\Lambda_{N(P)}.$

The following statement is essentially a consequence of Proposition 3.5.

######  Corollary 3.7.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope and let
$\bm{u},\bm{v}\in\mathbb{R}^{d}$. The following conditions are equivalent.

  * (1)

$(\mathcal{C}_{P}+(\bm{u},0))\cap\mathbb{Z}^{d+1}=(\mathcal{C}_{P}+(\bm{v},0))\cap\mathbb{Z}^{d+1}$.

  * (2)

$(\bm{u},0)$ and $(\bm{v},0)$ belong to the same upper region in
$\Lambda_{\widetilde{N}(P)}$.

  * (3)

$\bm{u}$ and $\bm{v}$ belong to the same upper region in $\Lambda_{P}$

###### Proof.

The equivalence between (1) and (2) is Proposition 3.5. Let $\bm{n}$ and
$\tilde{\bm{n}}$ be the sequence of normal vectors of $P$ and
$\mathcal{C}_{P}$ respectively. The equivalence between (2) and (3) follows
from the fact that
$\varphi_{\bm{n}}(\bm{a})=\varphi_{\tilde{\bm{n}}}(\bm{a},0)$ for any
$\bm{a}\in\mathbb{R}^{d}$. ∎

We now discuss a consequence of Corollary 3.7 to translated lattice point
enumerators and Ehrhart quasi-polynomials. Recall that $[\bm{v}]$ denotes the
image of $\bm{v}\in\mathbb{R}^{d}$ by the natural projection
$\mathbb{R}^{d}\to\mathbb{R}^{d}/\mathbb{Z}^{d}$.

######  Theorem 3.8.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope and let
$\bm{u},\bm{v}\in\mathbb{R}^{d}.$

  * (1)

If $[\bm{u}]$ and $[\bm{v}]$ belong to the same region in
$\Lambda_{P}/\mathbb{Z}^{d}$, then
$\mathrm{TL}_{P,\bm{u}}(t)=\mathrm{TL}_{P,\bm{v}}(t)$ for all
$t\in\mathbb{Z}_{\geq 0}$.

  * (2)

The set $\\{\mathrm{TL}_{P,\bm{w}}\mid\bm{w}\in\mathbb{R}^{d}\\}$ is a finite
set.

###### Proof.

(1) Corollary 3.7 tells that if $[\bm{u}]$ and $[\bm{v}]$ belong to the same
region in $\Lambda_{P}/\mathbb{Z}^{d}$, then

$\big{(}\mathcal{C}_{P}+(\bm{u},0)\big{)}\cap\mathbb{Z}^{d+1}=\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}+(\bm{w},0),$

where $\bm{w}\in\mathbb{Z}^{d}$ is the vector such that $\bm{u}$ and
$\bm{v}+\bm{w}$ belong to the same region of $\Lambda_{P}$. This clearly
implies $\mathrm{TL}_{P,\bm{u}}(t)=\mathrm{TL}_{P,\bm{v}}(t)$ for all
$t\in\mathbb{Z}_{\geq 0}$.

(2) Since $\\{R\in\Lambda_{P}\mid R\cap[0,1)^{d}\neq\varnothing\\}$ is finite,
$\Lambda_{P}/\mathbb{Z}^{d}$ is a finite set. This fact and (1) prove the
desired statement. ∎

######  Example 3.9.

Consider the trapezoid $T$ in Introduction. The set of normal vectors of $T$
is $N(T)=\\{(1,0),(0,1),(0,-1),(-1,1)\\}$. Then the set
$\Lambda_{T}/\mathbb{Z}^{2}$ consists of 4 elements with the following
representatives:

$\displaystyle V$
$\displaystyle=U_{(0,0,0,0)}=\\{(x,y)\in\mathbb{R}^{2}\mid-1<x\leq 0,\
-1<y\leq 0,\ -1<-y\leq 0,\ -1<-x+y\leq 0\\},$ $\displaystyle E$
$\displaystyle=U_{(1,0,0,0)}=\\{(x,y)\in\mathbb{R}^{2}\mid 0<x\leq 1,\
-1<y\leq 0,\ -1<-y\leq 0,\ -1<-x+y\leq 0\\},$ $\displaystyle R_{1}$
$\displaystyle=U_{(1,1,0,0)}=\\{(x,y)\in\mathbb{R}^{2}\mid 0<x\leq 1,\ 0<y\leq
1,\ -1<y\leq 0,\ -1<-x+y\leq 0\\},$ $\displaystyle R_{2}$
$\displaystyle=U_{(1,1,0,1)}=\\{(x,y)\in\mathbb{R}^{2}\mid 0<x\leq 1,\ 0<y\leq
1,\ -1<-y\leq 0,\ 0<-x+y\leq 1\\}.$

See Figure 3 for the visualization of $\Lambda_{T}/\mathbb{Z}^{2}$ in the
torus $\mathbb{R}^{2}/\mathbb{Z}^{2}$. Note that $V$ is a one point set.
Theorem 3.8 tells that $\mathrm{TL}_{T,\bm{v}}$ only depends on the region in
$\Lambda_{T}/\mathbb{Z}^{2}$ where $[\bm{v}]$ belongs, and the table below is
a list of the polynomials $\mathrm{TL}_{T,C}(t)$ in each region
$C\in\Lambda_{T}/\mathbb{Z}^{2}$.

region | polynomial $\mathrm{TL}_{T,C}(t)$
---|---
$V$ | $\frac{3}{2}t^{2}+\frac{5}{2}t+1$
$E$ | $\frac{3}{2}t^{2}+\frac{3}{2}t$
$R_{1}$ | $\frac{3}{2}t^{2}+\frac{1}{2}t$
$R_{2}$ | $\frac{3}{2}t^{2}-\frac{1}{2}t$

Figure 3. $N(T)$, $\mathcal{A}_{T}$ and $\Lambda_{T}/\mathbb{Z}^{2}$.

For a quasi-polynomial $f$, let $\mathrm{Const}(f)$ be the set of constituents
of $f$. Since the $k$th constituent of $\operatorname{ehr}_{P+\bm{v}}$ is the
$k$th constituent of $\mathrm{TL}_{P,k\bm{v}}$, the second statement of the
above theorem gives the following finiteness result for constituents of
Ehrhart quasi-polynomials of translated polytopes.

######  Corollary 3.10.

If $P\subset\mathbb{R}^{d}$ is a rational $d$-polytope, then

$\textstyle\\#\big{(}\bigcup_{\bm{v}\in\mathbb{Q}^{d}}\mathrm{Const}\big{(}\operatorname{ehr}_{P+\bm{v}}\big{)}\big{)}<\infty.$

###  3.3. Polyhedral decompositions associated with hyperplane arrangements

Theorem 3.8 is slightly different to Theorem 1.1 in Introduction (indeed the
cell complex $\Lambda_{T}/\mathbb{Z}^{2}$ has $4$ cells while
$\Delta_{T}/\mathbb{Z}^{2}$ has 6 cells), but it can be considered as a
refined version of Theorem 1.1. We explain this in the rest of this section.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope and
$N(P)=\\{\bm{a}_{1},\dots,\bm{a}_{m}\\}$. The arrangement $\mathcal{A}_{P}$
gives a natural polyhedral decomposition of $\mathbb{R}^{d}$ whose maximal
open cells are connected components of
$\mathbb{R}^{d}\setminus(\bigcup_{H\in\mathcal{A}_{P}}H)$. We write
$\Delta_{P}$ for this polyhedral complex. Note that this $\Delta_{P}$ is the
same as the one defined in Introduction. Since any half line
$\bm{v}+\\{k\bm{w}\mid k\in\mathbb{R}_{\geq 0}\\}$, where
$\bm{v}\in\mathbb{R}^{d}$ and $\bm{0}\neq\bm{w}\in\mathbb{R}^{d}$, must hit
one of $H_{\bm{a}_{i},k}\in\mathcal{A}_{P}$, each connected component of
$\mathbb{R}^{d}\setminus(\bigcup_{H\in\mathcal{A}_{P}}H)$ is a bounded set, so
$\Delta_{P}$ is actually a polytopal complex. By the definition of
$\mathcal{A}_{P}$, each open cell $A$ of $\Delta_{P}$ can be written in the
form

$A=A_{1}\cap A_{2}\cap\cdots\cap A_{m}$

such that each $A_{i}$ is either $H_{\bm{a}_{i},k}$ or
$\\{\bm{x}\in\mathbb{R}^{d}\mid k<(\bm{a}_{i},\bm{x})<k+1\\}$. This means that
each region in $\Lambda_{P}$ can be written as a disjoint union of open cells
in $\Delta_{P}$, in particular, each element in $\Lambda_{P}/\mathbb{Z}^{d}$
can be written as a disjoint union of elements in $\Delta_{P}/\mathbb{Z}^{d}$.
This proves Theorem 1.1.

######  Example 3.11.

Consider the trapezoid $T$ in Introduction. As one can see from Figures 1 and
3, $\Lambda_{T}/\mathbb{Z}^{2}$ consists of 4 elements $V,E,R_{1},R_{2}$ and
$\Delta_{T}/\mathbb{Z}^{2}$ consists of $6$ elements
$V_{1},E_{1},E_{2},E_{3},F_{1},F_{2}$. We have

$V=V_{1},\ E=E_{3},\ R_{1}=E_{1}\cup E_{2}\cup F_{2},\ R_{2}=F_{1}.$

While $\Lambda_{P}$ and $\Delta_{P}$ are different in general, there is a nice
case that we have $\Lambda_{P}=\Delta_{P}$. If the set of normal vectors of
$P$ is the set of the form $\\{\pm\bm{a}_{1},\dots,\pm\bm{a}_{l}\\}$ then each
region $R\in\Lambda_{P}$ must be a region of the form

$R=\bigcap_{i=1}^{m}\big{\\{}\bm{x}\in\mathbb{R}^{d}\mid
c_{i}-1<(\bm{a}_{i},\bm{x})\leq c_{i}\ \mbox{ and
}c_{i}^{\prime}-1<(-\bm{a}_{i},\bm{x})\leq c_{i}^{\prime}\big{\\}}.$

Each non-empty content in the RHS equals either $H_{\bm{a}_{i},c_{i}}$ or
$\\{\bm{x}\in\mathbb{R}^{d}\mid c_{i}-1<(\bm{a}_{i},\bm{x})<c_{i}\\}$ so we
have $\Lambda_{P}=\Delta_{P}$ in that case. To summarize, we get the following
statement.

######  Proposition 3.12.

If $P\subset\mathbb{R}^{d}$ is a $d$-polytope with $N(P)=-N(P)$ then
$\Lambda_{P}=\Delta_{P}$.

A typical example of a polytope $P$ satisfying $N(P)=-N(P)$ is a centrally
symmetric polytope $P$ with $P=-P$ (or more generally, a polytope $P$ with
$P=-P+\bm{a}$ for some $\bm{a}\in\mathbb{Z}^{d}$).

######  Remark 3.13.

Each element of $\Delta_{P}/\mathbb{Z}^{d}$ is an open cell ball, so
$\Delta_{P}/\mathbb{Z}^{d}$ is indeed a CW complex. To see that each element
of $\Delta_{P}/\mathbb{Z}^{d}$ is a ball, it suffices to check that for each
$C\in\Delta_{P}$ the restriction of
$\mathbb{R}^{d}\to\mathbb{R}^{d}/\mathbb{Z}^{d}$ to $C$ is injective. This
injectivity follows from Corollary 3.7 since, if we have
$\bm{u}=\bm{v}+\bm{a}$ for some $\bm{u},\bm{v}\in C$ and
$\bm{0}\neq\bm{a}\in\mathbb{Z}^{d}$, then $\mathcal{C}_{P}+(\bm{v},0)$ and
$\mathcal{C}_{P}+(\bm{u},0)=\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}+(\bm{a},0)$
must have different sets of integer points.

##  4\. Some examples

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. We see in the
previous section that $\mathrm{TL}_{P,\bm{v}}$ only depends on the region $C$
in $\Delta_{P}/\mathbb{Z}^{d}$ (or $\Lambda_{P}/\mathbb{Z}^{d}$) with
$[\bm{v}]\in C$, so for $C\in\Delta_{P}/\mathbb{Z}^{d}$ (or
$C\in\Lambda_{P}/\mathbb{Z}^{d}$) we write
$\mathrm{TL}_{P,C}=\mathrm{TL}_{P,\bm{v}}$ with $[\bm{v}]\in C$. In this
section, we give a few examples of the computations of
$\operatorname{ehr}_{P+\bm{v}}$ using translated lattice points enumerators.

######  Example 4.1.

Consider the lattice parallelogram $P$ with vertices
$(0,0),(1,0),(1,3),(2,3)$. Then $N(P)=\\{(0,1),(0,-1),(3,-1),(-3,1)\\}$ and
$\Delta_{P}/\mathbb{Z}^{2}$ $(=\Lambda_{P}/\mathbb{Z}^{2})$ consists of three
vertices $V_{1},V_{2},V_{3}$, $6$ edges $E_{1},E_{2},\dots,E_{6}$ and three
$2$-dimensional open cells $R_{1},R_{2},R_{3}$ shown in Figure 4. Since $P$ is
a lattice polytope, translated lattice points enumerators of $P$ are actually
polynomials. The table below is a list of the polynomials
$\mathrm{TL}_{P,C}(t)$.

cell | polynomial $\mathrm{TL}_{P,C}(t)$
---|---
$V_{1}$ | $3t^{2}+2t+1$
$V_{2},V_{3}$ | $3t^{2}+2t$
$E_{1},\dots,E_{6}$ | $3t^{2}+t$
$R_{1},R_{2},R_{3}$ | $3t^{2}$

Figure 4. Parallelogram $P$ and the cell complex $\Delta_{P}/\mathbb{Z}^{2}$.

Now we compute $\operatorname{ehr}_{P+\bm{v}}(t)$ when
$\bm{v}=(\frac{1}{3},\frac{1}{6})$ using this information. One can compute the
constituents of $\operatorname{ehr}_{P+\bm{v}}$ visually by drawing a line of
direction $\bm{v}$ in $\mathbb{R}^{2}/\mathbb{Z}^{2}$ and plot the pints
$[t\bm{v}]$ there. First, by drawing points $[t\bm{v}]$ on
$\mathbb{R}^{2}/\mathbb{Z}^{2}$, one can see

$\displaystyle[t\bm{v}]\in\begin{cases}V_{1},&(t\equiv 0\ (\mathrm{mod}\
6)),\\\ R_{3},&(t\equiv 1\ (\mathrm{mod}\ 6)),\\\ R_{1},&(t\equiv 2,4\
(\mathrm{mod}\ 6)),\\\ E_{5},&(t\equiv 3\ (\mathrm{mod}\ 6)),\\\
R_{2},&(t\equiv 5\ (\mathrm{mod}\ 6)).\end{cases}$

See Figure 4. Lemma 2.4 tells that the $k$th constituent of
$\operatorname{ehr}_{P+\bm{v}}$ is nothing but the $k$th constituent of
$\mathrm{TL}_{P,C}$ with $[k\bm{v}]\in C$. Hence we get

$\displaystyle\operatorname{ehr}_{P+\bm{v}}(t)=\begin{cases}\mathrm{TL}_{P,V_{1}}(t)=3t^{2}+2t+1,&(t\equiv
0\ (\mathrm{mod}\ 6)),\\\ \mathrm{TL}_{P,R_{3}}(t)=3t^{2},&(t\equiv 1\
(\mathrm{mod}\ 6)),\\\ \mathrm{TL}_{P,R_{1}}(t)=3t^{2},&(t\equiv 2,4\
(\mathrm{mod}\ 6)),\\\ \mathrm{TL}_{P,E_{5}}(t)=3t^{2}+t,&(t\equiv 3\
(\mathrm{mod}\ 6)),\\\ \mathrm{TL}_{P,R_{2}}(t)=3t^{2},&(t\equiv 5\
(\mathrm{mod}\ 6)).\end{cases}$

We remark that parallelogram is a special case of a zonotope, and a nice
combinatorial formula of the Ehrhart quasi-polynomial of a translated integral
zonotope is given in [2, Proposition 3.1].

######  Example 4.2.

We give a more complicated example. Consider the rhombus
$Q\subset\mathbb{R}^{2}$ having vertices $(\pm 1,0)$ and $(0,\pm\frac{1}{2})$.
Then the cell complex $\Delta_{Q}/\mathbb{Z}^{2}$ consists of four vertices,
eight edges and four $2$-dimensional cells. See Figure 5.

Figure 5. Cell complex associated with $Q$

Since $2Q$ is integral, $\mathrm{TL}_{Q,C}$ is a quasi-polynomial having
period $2$ for each $C\in\Delta_{Q}/\mathbb{Z}^{2}$. For a quasi-polynomial
$f$ with period $2$, we write $f=(f_{0},f_{1})$, where $f_{k}$ is the $k$th
constituent of $f$. Below is the table of translated lattice points
enumerators of $Q$.

cell | quasi-polynomial $\mathrm{TL}_{Q,C}$
---|---
$V_{1}$ | $(t^{2}+t+1,t^{2}+t+1)$
$V_{2},V_{3},V_{4}$ | $(t^{2}+t,t^{2}+t)$
$E_{1},E_{2},E_{7},E_{8}$ | $(t^{2}+\frac{1}{2}t,t^{2}+\frac{1}{2}t+\frac{1}{2})$
$E_{3},E_{4},E_{5},E_{6}$ | $(t^{2}+\frac{1}{2}t,t^{2}+\frac{1}{2}t-\frac{1}{2})$
$F_{1}$ | $(t^{2},t^{2}+1)$
$F_{2},F_{3}$ | $(t^{2},t^{2})$
$F_{4}$ | $(t^{2},t^{2}-1)$

Consider $\bm{u}=(\frac{1}{8},\frac{1}{8})$ and
$\bm{w}=(\frac{1}{3},\frac{1}{3})$. Then

$\displaystyle[t\bm{u}]\in\begin{cases}V_{1},&(t\equiv 0\ (\mathrm{mod}\
8)),\\\ F_{3},&(t\equiv 1,2\ (\mathrm{mod}\ 8)),\\\ F_{4},&(t\equiv 3,4,5\
(\mathrm{mod}\ 8)),\\\ F_{2},&(t\equiv 6,7\ (\mathrm{mod}\ 8)),\end{cases}\ \
\ \mbox{ and }\ \ \ [t\bm{w}]\in\begin{cases}V_{1},&(t\equiv 0\ (\mathrm{mod}\
3)),\\\ E_{3},&(t\equiv 1\ (\mathrm{mod}\ 3)),\\\ E_{6},&(t\equiv 2\
(\mathrm{mod}\ 3)).\end{cases}$

Using that the $k$th constituent of $\operatorname{ehr}_{Q+\bm{u}}$ equals the
$k$th constituent of $\mathrm{TL}_{Q,k\bm{u}}$, it follows that

$\displaystyle\operatorname{ehr}_{Q+\bm{u}}(t)=\begin{cases}t^{2}+t+1,&(t\equiv
0\ (\mathrm{mod}\ 8)),\\\ t^{2},&(t\equiv 1,2,4,6,7\ (\mathrm{mod}\ 8)),\\\
t^{2}-1,&(t\equiv 3,5\ (\mathrm{mod}\ 8)),\end{cases}$

and

$\displaystyle\operatorname{ehr}_{Q+\bm{w}}(t)=\begin{cases}t^{2}+t+1,&(t\equiv
0,3\ (\mathrm{mod}\ 6)),\\\ t^{2}+\frac{1}{2}t-\frac{1}{2},&(t\equiv 1,5\
(\mathrm{mod}\ 6)),\\\ t^{2}+\frac{1}{2}t,&(t\equiv 2,4\ (\mathrm{mod}\
6)).\end{cases}$

One can also see from the second example that the minimum period of
$\operatorname{ehr}_{Q+\bm{w}}$ is not necessary the denominator of $\bm{w}$.

##  5\. Reciprocity in maximal regions

In this section, we explain that the quasi-polynomials $\mathrm{TL}_{P,C}$ for
maximal dimensional cells $C\in\Delta_{P}/\mathbb{Z}^{d}$ have a reciprocity
which comes from the reciprocity in Theorem 2.3.

### Reciprocity

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. The reciprocity in
Theorem 2.3 says that, for any $\bm{v}\in\mathbb{R}^{d}$, one has

(5.1)
$\displaystyle\mathrm{TL}_{\mathrm{int}(P),\bm{v}}(t)=(-1)^{d}\mathrm{TL}_{P,-\bm{v}}(-t)\
\ \ \mbox{ for }t\in\mathbb{Z}_{>0}.$

Since $\mathcal{A}_{P}$ is centrally symmetric, that is
$-\mathcal{A}_{P}=\mathcal{A}_{P}$, we have $R\in\Delta_{P}$ if and only if
$-R\in\Delta_{P}$. For each $C\in\Delta_{P}/\mathbb{Z}^{d}$ with a
representative $R\in\Delta_{P}$, we write $-C$ for the element of
$\Delta_{P}/\mathbb{Z}^{d}$ corresponding to the cell $-R$. By Theorem 1.1 and
(5.1) we have
$\mathrm{TL}_{\mathrm{int}(P),\bm{v}}(t)=\mathrm{TL}_{\mathrm{int}(P),\bm{u}}(t)$
when $[\bm{u}]$ and $[\bm{v}]$ belong to the same cell in
$\Delta_{P}/\mathbb{Z}^{d}$. Thus, for each $C\in\Delta_{P}/\mathbb{Z}^{d}$,
we write
$\mathrm{TL}_{\mathrm{int}(P),C}(t)=\mathrm{TL}_{\mathrm{int}(P),\bm{v}}(t)$
with $[\bm{v}]\in C$. Using this notation, (5.1) can be written in the
following form.

######  Proposition 5.1.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. For any
$C\in\Delta_{P}/\mathbb{Z}^{d}$, one has

$\mathrm{TL}_{\mathrm{int}(P),C}(t)=(-1)^{d}\mathrm{TL}_{P,-C}(-t)\ \ \
\mbox{for }t\in\mathbb{Z}_{>0}.$

Note that the above equation tells that $\mathrm{TL}_{\mathrm{int}(P),C}$ is a
quasi-polynomial on $\mathbb{Z}_{>0}$.

######  Example 5.2.

Consider the trapezoid $T$ in Introduction. Then we have

$F_{1}=-F_{2},\ E_{1}=-E_{1},\ E_{2}=-E_{2},E_{3}=-E_{3},V_{1}=-V_{1}.$

The following tables are lists of polynomials $\mathrm{TL}_{T,C}(t)$ and
$\mathrm{TL}_{\mathrm{int}(T),C}(t)$.

cell | polynomial $\mathrm{TL}_{T,C}(t)$
---|---
$V_{1}$ | $\frac{3}{2}t^{2}+\frac{3}{2}t+1$
$E_{1},E_{2}$ | $\frac{3}{2}t^{2}+\frac{1}{2}t$
$E_{3}$ | $\frac{3}{2}t^{2}+\frac{3}{2}t$
$F_{1}$ | $\frac{3}{2}t^{2}-\frac{1}{2}t$
$F_{2}$ | $\frac{3}{2}t^{2}+\frac{1}{2}t$
cell | polynomial $\mathrm{TL}_{\mathrm{int}(T),C}(t)$
---|---
$V_{1}$ | $\frac{3}{2}t^{2}-\frac{3}{2}t+1$
$E_{1},E_{2}$ | $\frac{3}{2}t^{2}-\frac{1}{2}t$
$E_{3}$ | $\frac{3}{2}t^{2}-\frac{3}{2}t$
$F_{2}$ | $\frac{3}{2}t^{2}+\frac{1}{2}t$
$F_{1}$ | $\frac{3}{2}t^{2}-\frac{1}{2}t$

### Maximal cells

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope with the unique
irredundant presentation

(5.2) $\displaystyle P=H_{\bm{a}_{1},b_{1}}^{\geq}\cap\cdots\cap
H_{\bm{a}_{m},b_{m}}^{\geq}.$

We write $F_{i}=P\cap H_{\bm{a}_{i},b_{i}}$ for the facet of $P$ which lies in
the hyperplane $H_{\bm{a}_{i},b_{i}}$. The next lemma follows from Lemmas 3.2
and 3.4.

######  Lemma 5.3.

With the same notation as above, for $\bm{v}\in\mathbb{R}^{d}$, the cone
$\mathcal{C}_{F_{i}}+(\bm{v},0)$ contains a lattice point if and only if
$\bm{v}\in H_{\bm{a}_{i},k}$ for some $k\in\mathbb{Z}$.

The lemma tells that the cone $\mathcal{C}_{P}+(\bm{v},0)$ has no lattice
points in its boundary if
$\bm{v}\in\mathbb{R}^{d}\setminus\bigcup_{H\in\mathcal{A}_{P}}H$,
equivalently, if $[\bm{v}]$ belongs to a $d$-dimensional cell of
$\Delta_{P}/\mathbb{Z}^{d}$. Hence we have

######  Corollary 5.4.

If $P\subset\mathbb{R}^{d}$ is a rational $d$-polytope and $C$ is a
$d$-dimensional cell of $\Delta_{P}/\mathbb{Z}^{d}$, then

$\mathrm{TL}_{P,C}(t)=(-1)^{d}\mathrm{TL}_{P,-C}(-t)\ \ \mbox{ for all
}t\in\mathbb{Z}_{\geq 0}.$

###### Proof.

Let $\bm{v}\in\mathbb{R}^{d}$ such that $[\bm{v}]\in C$. Then $\bm{v}\not\in
H$ for any $H\in\mathcal{A}_{P}$, which implies that the cone
$\mathcal{C}_{P}+(\bm{v},0)$ has no lattice points in its boundary. Thus by
Proposition 5.1 we have

$\mathrm{TL}_{P,C}(t)=\mathrm{TL}_{\mathrm{int}(P),C}(t)=(-1)^{d}\mathrm{TL}_{P,-C}(-t)\
\ \ \mbox{ for }t\in\mathbb{Z}_{>0}.$

Since $\mathrm{TL}_{P,C}$ and $\mathrm{TL}_{P,-C}$ are quasi-polynomials on
$\mathbb{Z}_{\geq 0}$, this implies the desired equality. ∎

The above reciprocity has a special meaning for centrally symmetric polytopes.
Looking quasi-polynomials $\mathrm{TL}_{Q,F_{i}}$ in Example 4.2, one may
notice that each constituent is a polynomial in $t^{2}$. In other words, the
linear term $t$ vanishes. We explain that this has a reason. We first remind
the following obvious fact.

######  Lemma 5.5.

Let $P\subset\mathbb{R}^{d}$ be a rational polytope with $-P=P+\bm{u}$ for
some $\bm{u}\in\mathbb{Z}^{d}$. Then
$\mathrm{TL}_{P,\bm{v}}(t)=\mathrm{TL}_{P,-\bm{v}}(t)$ for any
$\bm{v}\in\mathbb{R}^{d}$ and $t\in\mathbb{Z}_{\geq 0}$.

######  Theorem 5.6.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope with $-P=P+\bm{u}$ for
some $\bm{u}\in\mathbb{Z}^{d}$ and let $C\in\Delta_{P}/\mathbb{Z}^{d}$ be a
$d$-dimensional cell. Let $f(t)$ be the $k$th constituent of
$\mathrm{TL}_{P,C}$ and let $g(t)$ be the $(-k)$th constituent of
$\mathrm{TL}_{P,C}$. Then

$f(t)=(-1)^{d}g(-t).$

###### Proof.

Corollary 5.4 and Lemma 5.5 tell

$\mathrm{TL}_{P,C}(t)=(-1)^{d}\mathrm{TL}_{P,-C}(-t)=(-1)^{d}\mathrm{TL}_{P,C}(-t)\
\ \mbox{ for all }t\in\mathbb{Z}_{\geq 0}.$

By considering the $k$th constituent in the above equality, we get the desired
assertion. ∎

If a polynomial $f(t)$ of degree $d$ satisfies $f(t)=(-1)^{d}f(-t)$, then it
must be a polynomial in $t^{2}$ when $d$ is even and $t$ times a polynomial in
$t^{2}$ when $d$ is odd. Hence we get the following corollary, which explain a
reason why we get polynomials in $t^{2}$ in Example 4.2.

######  Corollary 5.7.

With the same notation as in Theorem 5.6,

  * (1)

the $0$th constituent of $\mathrm{TL}_{P,C}(t)$ is either a polynomial in
$\mathbb{Q}[t^{2}]$ or $t\mathbb{Q}[t^{2}]$;

  * (2)

if $2P$ is integral, then the $1$st constituent of $\mathrm{TL}_{P,C}(t)$ is
either a polynomial in $\mathbb{Q}[t^{2}]$ or $t\mathbb{Q}[t^{2}]$.

Note that when $2P$ is integral the quasi polynomial $\mathrm{TL}_{P,C}$ has
period 2, so its $1$st constituent equals its $(-1)$th constituent.

##  6\. Translated lattice points enumerators determine polytopes

If is clear that if $P=Q+\bm{w}$ for some integer vector $\bm{w}$, then
$\mathrm{TL}_{P,\bm{v}}=\mathrm{TL}_{Q,\bm{v}}$ for all vectors $\bm{v}$. The
goal of this section is to prove the converse of this simple fact, which is
equivalent to Theorem 1.3 in Introduction by Lemma 2.4111The condition
“$\mathrm{TL}_{P,\bm{v}}=\mathrm{TL}_{Q,\bm{v}}$ for all
$\bm{v}\in\mathbb{Q}^{d}$” is equivalent to the condition
“$\mathrm{TL}_{P,\bm{v}}=\mathrm{TL}_{Q,\bm{v}}$ for all
$\bm{v}\in\mathbb{R}^{d}$”..

######  Theorem 6.1.

Let $P$ and $Q$ be rational $d$-polytopes in $\mathbb{R}^{d}$. If
$\mathrm{TL}_{P,\bm{v}}(t)=\mathrm{TL}_{Q,\bm{v}}(t)$ for all
$\bm{v}\in\mathbb{R}^{d}$ and $t\in\mathbb{Z}_{\geq 0}$ then $P=Q+\bm{w}$ for
some $\bm{w}\in\mathbb{Z}^{d}$.

To simplify notation, we use the notation

$\Gamma_{P}=\big{\\{}\big{(}\bm{v},\mathrm{TL}_{P,\bm{v}}(t)\big{)}\in\mathbb{R}^{d}\times\mathcal{QP}\mid\bm{v}\in\mathbb{R}^{d}\big{\\}},$

where $\mathcal{QP}$ is the set of all quasi-polynomials in $t$. Thus, what we
want to prove is that $\Gamma_{P}=\Gamma_{Q}$ implies $P=Q+\bm{w}$ for some
$\bm{w}\in\mathbb{Z}^{d}$.

To prove the theorem, we first recall Minkowski’s theorem, which tells that
normal vectors and volumes of facets determine a polytope. Let
$P\subset\mathbb{R}^{d}$ be a $d$-polytope with irredundant presentation
$P=\bigcap_{i=1}^{m}H_{\bm{a}_{i},b_{i}}^{\geq}$, where $\|\bm{a}_{i}\|=1$,
and let $F_{i}=P\cap H_{\bm{a}_{i},b_{i}}$ be the facet of $P$ which lies in
the hyperplane $H_{\bm{a}_{i},b_{i}}$. We write

$\mathcal{M}(P)=\big{\\{}\big{(}\bm{a}_{1},\mathrm{vol}(F_{1})\big{)},\dots,(\bm{a}_{m},\mathrm{vol}(F_{m}))\big{\\}},$

where $\mathrm{vol}(F_{i})$ is the relative volume of $F_{i}$. The following
result is known as Minkowski’s theorem (see [1, §6.3 Theorem 1]).

######  Theorem 6.2 (Minkowski).

If $P$ and $Q$ are $d$-polytopes in $\mathbb{R}^{d}$ with
$\mathcal{M}(P)=\mathcal{M}(Q)$, then $Q=P+\bm{v}$ for some
$\bm{v}\in\mathbb{R}^{d}$.

To apply Minkowski’s theorem in our situation, we will show that we can know
volumes of facets of a polytope from translated lattice points enumerator on
codimension $1$ cells of $\Delta_{P}/\mathbb{Z}^{d}$. We say that a point
$\bm{v}\in H_{\bm{a},k}\in\mathcal{A}_{P}$ is generic in $\mathcal{A}_{P}$ if
$\bm{v}\not\in H$ for any $H\in\mathcal{A}_{P}$ with $H\neq H_{\bm{a},k}$.
Note that $\bm{v}\in H_{\bm{a},k}$ is generic if and only if it is contained
in a $(d-1)$-dimensional cell of $\Delta_{P}$.

######  Lemma 6.3.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope, $\bm{a}\in N(P)$, and
let $F$ be a facet of $P$ corresponding to the normal vector $\bm{a}$. If
$\bm{v}\in H_{\bm{a},k}$ is generic in $\mathcal{A}_{P}$, then for a
sufficiently small number $\varepsilon>0$, one has

  * (1)

$\mathrm{TL}_{P,\bm{v}}-\mathrm{TL}_{P,\bm{v}+\varepsilon\bm{a}}=\mathrm{TL}_{F,\bm{v}}\neq
0$.

  * (2)

$\mathrm{TL}_{P,\bm{v}}-\mathrm{TL}_{P,\bm{v}-\varepsilon\bm{a}}=0$ if there
are no $k\in\mathbb{R}_{>0}$ such that $-k\bm{a}\in N(P)$.

###### Proof.

The fact that $\mathrm{TL}_{F,\bm{v}}(t)\neq 0$ for some $t$ follows from
Lemma 5.3. We first assume that $F$ is the only facet of $P$ that is
orthogonal to $\bm{a}$ and prove (1) and (2). In this case, by the assumption
and Lemma 5.3, $\mathcal{C}_{F}+(\bm{v},0)$ is the only facet of
$\mathcal{C}_{P}+(\bm{v},0)$ that contains lattice points, so

$\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}=\big{(}\mathrm{int}\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cup\big{(}\mathcal{C}_{F}+(\bm{v},0)\big{)}\big{)}\cap\mathbb{Z}^{d+1}.$

Let $H_{\bm{b},0}\subset\mathbb{R}^{d+1}$ be the hyperplane that contains
$\mathcal{C}_{F}$. Then, since
$\mathcal{C}_{F}\cap(H_{\bm{b},0}^{\geq}+s(\bm{a},0))=\varnothing$ and
$\mathcal{C}_{F}\subset H_{\bm{b},0}^{\geq}-s(\bm{a},0)$ for any $s>0$, by
Lemma 3.3 there is an $\varepsilon>0$ such that

$\big{(}\mathcal{C}_{P}+(\bm{v}+\varepsilon\bm{a},0)\big{)}\cap\mathbb{Z}^{d+1}=\big{(}\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}\big{)}\setminus\big{(}\big{(}\mathcal{C}_{F}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}\big{)}$

and

$\big{(}\mathcal{C}_{P}+(\bm{v}-\varepsilon\bm{a},0)\big{)}\cap\mathbb{Z}^{d+1}=\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}.$

These prove (1) and (2).

Second, we assume that there is a facet $G\neq F$ of $P$ that is orthogonal to
$\bm{a}$. This condition is equivalent to the condition that there is
$k\in\mathbb{R}_{>0}$ such that $-k\bm{a}\in N(P)$. Also, the normal vector
corresponding to the facet $G$ must be equal to $-k\bm{a}$ and

$\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}=\big{(}\mathrm{int}\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cup\big{(}\mathcal{C}_{F}+(\bm{v},0)\big{)}\cup\big{(}\mathcal{C}_{G}+(\bm{v},0)\big{)}\big{)}\cap\mathbb{Z}^{d+1}.$

In this case
$\mathcal{C}_{F}\cap(H_{\bm{b},0}^{\geq}+s(\bm{a},0))=\varnothing$ and
$\mathcal{C}_{G}\subset H_{\bm{b},0}^{\geq}+s(\bm{a},0)$ for a sufficiently
small $s>0$, so again by Lemma 3.3 there is an $\varepsilon>0$ such that

$\big{(}\mathcal{C}_{P}+(\bm{v}+\varepsilon\bm{a},0)\big{)}\cap\mathbb{Z}^{d+1}=\big{(}\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}\big{)}\setminus\big{(}\big{(}\mathcal{C}_{F}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}\big{)},$

proving (1). ∎

######  Lemma 6.4.

If $P$ and $Q$ are rational $d$-polytopes in $\mathbb{R}^{d}$ with
$\Gamma_{P}=\Gamma_{Q}$, then $\mathcal{M}(P)=\mathcal{M}(Q)$.

###### Proof.

What we must prove is that the set $\Gamma_{P}$ determines the directions of
inner normal vectors of $P$ as well as volumes of the facets of $P$.

By Lemma 6.3(1),
$\bm{v}\in\mathbb{R}^{d}\setminus\bigcup_{H\in\mathcal{A}_{P}}H$ if and only
if there is an open ball $B\ni\bm{v}$ such that
$\mathrm{TL}_{P,\bm{v}}=\mathrm{TL}_{P,\bm{u}}$ for all $\bm{u}\in B$. This
tells that the set $\Gamma_{P}$ determines $\mathcal{A}_{P}$, and the
definition of $\mathcal{A}_{P}$ tells that $\mathcal{A}_{P}$ determines the
set $\overline{N}=\\{\pm(\bm{a}/\|\bm{a}\|)\mid\bm{a}\in N(P)\\}$. For each
$\bm{a}\in\overline{N}$, Lemma 6.3 also tells $k\bm{a}\in N(P)$ for some $k>0$
if and only if, for a generic $\bm{v}\in H_{\bm{a},0}\in\mathcal{A}_{P}$, we
have $\mathrm{TL}_{P,\bm{v}}\neq\mathrm{TL}_{P,\bm{v}+\varepsilon\bm{a}}$ for
a sufficiently small number $\varepsilon>0$. Hence the set $\Gamma_{P}$
determines $\\{(\bm{a}/\|\bm{a}\|)\mid\bm{a}\in N(P)\\}$.

It remains to prove that $\Gamma_{P}$ determines the volumes of facets of $P$.
Let $F$ be a facet of $P$ and let $\bm{a}\in N(P)$ be the normal vector
associated with the facet $F$. For any $\bm{v}\in\mathbb{R}^{d}$, let
$\mathrm{TL}_{P,\bm{v}}^{0}(t)$ denote the $0$th constituent of
$\mathrm{TL}_{P,\bm{v}}$, which must be a degree $d$ polynomial whose leading
coefficient it the normalized volume of $P$. If we take a generic point
$\bm{v}\in H_{\bm{a},0}$ in $\mathcal{A}_{P}$, then by Lemma 6.3 we have

$\textstyle\mathrm{lim}_{t\to\infty}\frac{1}{t^{d-1}}\mathrm{TL}^{0}_{F,\bm{v}}(t)=\mathrm{lim}_{t\to\infty}\frac{1}{t^{d-1}}\big{(}\mathrm{TL}^{0}_{P,\bm{v}}(t)-\mathrm{TL}^{0}_{P,\bm{v}+\varepsilon\bm{a}}(t)\big{)},$

where $\varepsilon$ is a sufficiently small integer. Since
$\mathrm{TL}_{P,\bm{v}}^{0}(t)-\mathrm{TL}^{0}_{P,\bm{v}+\varepsilon\bm{a}}(t)$
is a polynomial of degree $\leq d-1$, this limit exists and must be equal to
the relative volume of $F$ since $\mathrm{TL}_{F,\bm{v}}$ can be considered as
a translated lattice points enumerator in the Euclidian space
$H_{\bm{a},0}\cong\mathbb{R}^{d-1}$ with the lattice
$H_{\bm{a},0}\cap\mathbb{Z}^{d}\cong\mathbb{Z}^{d-1}$. Thus volumes of facets
of $P$ are determined by $\Gamma_{P}$. ∎

Let $\pi_{i}:\mathbb{R}^{d}\to\mathbb{R}^{d-1}$ be the projection given by

$\pi_{i}(x_{1},\dots,x_{d})=(x_{1},\dots,x_{i-1},x_{i+1},\dots,x_{d}).$

We next show that translated lattice points enumerators of $\pi_{i}(P)$ can be
determined from those of $P$. Let $P\subset\mathbb{R}^{d}$ be a $d$-polytope.
We define

$\partial_{i}^{-}P=\\{\bm{x}\in P\mid\bm{x}\not\in
P+\varepsilon\mathbf{e}_{i}\mbox{ for any }\varepsilon>0\\},$

where $\mathbf{e}_{1},\dots,\mathbf{e}_{d}$ are the standard vectors of
$\mathbb{R}^{d}$. Intuitively, $\partial_{i}^{-}P$ is the set of points in $P$
which is visible from $-\infty\mathbf{e}_{i}$. Indeed, $\partial_{i}^{-}P$ has
the following description: Let $\mathrm{Facets}(P)$ be the set of facets of
$P$, and, for each $F\in\mathrm{Facets}(P)$, let $\bm{a}_{F}\in N(P)$ be the
inner normal vector associated with $F$. An alternative description of
$\partial_{i}^{-}P$ is

(6.1) $\displaystyle\partial_{i}^{-}P=\bigcup_{F\in\mathrm{Facets}(P),\
(\bm{a}_{F},\mathbf{e}_{i})>0}F.$

See Figure 6.

Figure 6. $\partial_{1}^{-}P$ and $\partial_{2}^{-}P$.

######  Lemma 6.5.

With the same notation as above, for any $\bm{v}\in\mathbb{R}^{d}$, there is
an $\varepsilon_{i,\bm{v}}>0$ such that

$(tP+\bm{v})\cap\mathbb{Z}^{d}=\left(\big{(}t(\partial_{i}^{-}P)+\bm{v}\big{)}\bigsqcup\big{(}tP+\bm{v}+\varepsilon_{i,\bm{v}}\mathbf{e}_{i}\big{)}\right)\cap\mathbb{Z}^{d}\
\ \mbox{ for all $t\in\mathbb{Z}_{\geq 0}.$}$

###### Proof.

By Lemma 3.3 there is an $\varepsilon>0$ such that

$\displaystyle\big{(}\mathcal{C}_{P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}$
$\displaystyle=\left(\big{(}\mathcal{C}_{P}\\!+\\!(\bm{v}\\!+\\!\varepsilon\mathbf{e}_{i},0)\big{)}\bigsqcup\big{\\{}\bm{x}\\!+\\!(\bm{v},0)\in\mathcal{C}_{P}\\!+\\!(\bm{v},0)\mid\bm{x}\not\in\mathcal{C}_{P}\\!+\\!s(\mathbf{e}_{i},0)\
\mbox{ for any }s>0\big{\\}}\right)\cap\mathbb{Z}^{d+1}.$

Cutting the above equation by the hyperplane $x_{d+1}=t$, we get the desired
equality. ∎

We define $\mathrm{TL}_{P,\bm{v}}^{(-i)}(t)$ by

$\mathrm{TL}_{P,\bm{v}}^{(-i)}(t)=\\#\big{(}\big{(}t(\partial_{i}^{-}P)+\bm{v}\big{)}\cap\mathbb{Z}^{d}\big{)}.$

Lemma 6.5 tells that

$\mathrm{TL}_{P,\bm{v}}^{(-i)}(t)=\mathrm{TL}_{P,\bm{v}}(t)-\mathrm{TL}_{P,\bm{v}+\varepsilon_{i,\bm{v}}\mathbf{e}_{i}}(t),$

where $\varepsilon_{i,\bm{v}}$ is a number given in Lemma 6.5. We note that
the function $\mathrm{TL}_{P,\bm{v}}^{(-i)}$ is zero for almost all
$\bm{v}\in\mathbb{R}^{d}$. Indeed, we have the following statement.

######  Lemma 6.6.

With the same notation as above, $\mathrm{TL}_{P,\bm{v}}^{(-i)}$ is not a zero
function only when there is $\bm{a}\in N(P)$ and $k\in\mathbb{Z}$ such that
$\bm{v}\in H_{\bm{a},k}$ and $(\bm{a},\mathbf{e}_{i})>0$.

###### Proof.

We have $\mathrm{TL}_{P,\bm{v}}^{(-i)}(t)\neq 0$ for all $t\in\mathbb{Z}_{\geq
0}$ only when

$\big{(}\mathcal{C}_{\partial_{i}^{-}P}+(\bm{v},0)\big{)}\cap\mathbb{Z}^{d+1}\neq\varnothing.$

By (6.1) and Lemma 5.3 this condition is equivalent to $\bm{v}\not\in
H_{\bm{a},k}$ for some $\bm{a}\in N(P)$ and $k\in\mathbb{Z}$ with
$(\bm{a},\mathbf{e}_{i})>0$. ∎

The next proposition shows that translated lattice points enumerators of
$\pi_{i}(P)$ can be determined from those of $P$.

######  Proposition 6.7.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. For any
$\bm{v}\in\mathbb{R}^{d}$ and $t\in\mathbb{Z}_{\geq 0}$, one has

$\mathrm{TL}_{\pi_{i}(P),\pi_{i}(\bm{v})}(t)=\sum_{0\leq s<1,\
\mathrm{TL}_{P,\bm{v}+s\mathbf{e}_{i}}^{(-i)}(t)\neq
0}\mathrm{TL}_{P,\bm{v}+s\mathbf{e}_{i}}^{(-i)}(t).$

We note that the RHS in the proposition is a finite sum by Lemma 6.6 since the
segment $\\{\bm{v}+s\mathbf{e}_{i}\mid 0\leq s<1\\}$ meets only finitely many
hyperplanes in $\mathcal{A}_{P}$. See Figure 7 for a visualization of the
proposition.

Figure 7. Lattice points in the projection.

###### Proof.

We may assume $i=d.$ Fix $t\in\mathbb{Z}_{\geq 0}$ and a lattice point
$\bm{u}=(u_{1},\dots,u_{d-1})\in\pi_{d}(tP+\bm{v})$. It suffices to prove that
there is a unique integer $r\in\mathbb{Z}$ such that
$(\bm{u},r)\in\bigcup_{0\leq
s<1}(t(\partial_{d}^{-}P)+\bm{v}+s\mathbf{e}_{d})$.

(Existence) By the assumption, there is an $\alpha\in\mathbb{R}$ such that

$(\bm{u},\alpha)\in t(\partial_{d}^{-}P)+\bm{v}.$

Then, $r=\lceil\alpha\rceil$ satisfies the desired condition since
$(\bm{u},\lceil\alpha\rceil)$ is contained in
$t(\partial_{d}^{-}P)+\bm{v}+(\lceil\alpha\rceil-\alpha)\mathbf{e}_{d}$.

(Uniqueness) The uniqueness of $r$ follows from the fact that, for any
$(\bm{u},\alpha),(\bm{u},\alpha^{\prime})$ which are contained in
$\bigcup_{0\leq s<1}(t(\partial_{n}^{-}P)+\bm{v}+s\mathbf{e}_{d})$, we have
$|\alpha-\alpha^{\prime}|<1$. ∎

We will also use the following variation of Proposition 6.7. For a
$d$-polytope $P$, let

$\partial_{i}^{+}P=\\{\bm{x}\in P\mid\bm{x}\not\in
P-\varepsilon\mathbf{e}_{i}\mbox{ for any }\varepsilon>0\\}$

and

$\mathrm{TL}_{P,\bm{v}}^{(+i)}(t)=\\#\big{(}\big{(}t(\partial_{i}^{+}P)+\bm{v}\big{)}\cap\mathbb{Z}^{d}\big{)}\
\ \mbox{ for }t\in\mathbb{Z}_{\geq 0}.$

The next statement can be proved by the same argument given in the proof of
Proposition 6.7.

######  Proposition 6.8.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. For any
$\bm{v}\in\mathbb{R}^{d}$ and $t\in\mathbb{Z}_{\geq 0}$, one has

$\mathrm{TL}_{\pi_{i}(P),\pi_{i}(\bm{v})}(t)=\sum_{0\leq s<1,\
\mathrm{TL}_{P,\bm{v}-s\mathbf{e}_{i}}^{(+i)}(t)\neq
0}\mathrm{TL}_{P,\bm{v}-s\mathbf{e}_{i}}^{(-i)}(t).$

We now prove Theorem 6.1.

###### Proof of Theorem 6.1.

We use induction on $d$. Suppose $d=1$ and $\Gamma_{P}=\Gamma_{Q}$. Then $P$
and $Q$ are line segments in $\mathbb{R}$ such that $P+a$ and $Q+a$ contains
the same number of integers for any $a\in\mathbb{R}$. It is easy to see that
this condition implies $P=Q+v$ for some $v\in\mathbb{Z}$.

Assume $d>1$ and $\Gamma_{P}=\Gamma_{Q}$. By Lemma 6.4, we already know
$Q=P+\bm{v}$ for some $\bm{v}\in\mathbb{R}^{d}$. By Proposition 6.7 and the
assumption $\Gamma_{P}=\Gamma_{Q}$, we have
$\Gamma_{\pi_{1}(P)}=\Gamma_{\pi_{1}(Q)}$ and
$\Gamma_{\pi_{2}(P)}=\Gamma_{\pi_{2}(Q)}$. Since $Q=P+\bm{v}$, the induction
hypothesis tells that $\pi_{1}(\bm{v}),\pi_{2}(\bm{v})\in\mathbb{Z}^{d-1}$
which guarantees $\bm{v}\in\mathbb{Z}^{d}$. ∎

##  7\. Group symmetry

In the previous section, we see that translated lattice points enumerators
determine polytopes up to translations by integer vectors. In this last
section, we study translated lattice points enumerators of polytopes having
some symmetries, in particular, we prove Theorem 1.5 in Introduction.

Let $\mathrm{GL}_{d}(\mathbb{Z})$ be the subgroup of the general linear group
$\mathrm{GL}_{d}(\mathbb{R})$ consisting of all elements
$g\in\mathrm{GL}_{d}(\mathbb{R})$ with $g(\mathbb{Z}^{d})=\mathbb{Z}^{d}$. If
we identify each element of $\mathrm{GL}_{d}(\mathbb{R})$ with $d\times d$
non-singular matrix in a standard way, then $\mathrm{GL}_{d}(\mathbb{Z})$ may
be considered as the set of unimodular matrices. For a rational $d$-polytope
$P\subset\mathbb{R}^{d}$, we define

$\mathrm{Aut}_{\mathbb{Z}}(P)=\\{g\in\mathrm{GL}_{d}(\mathbb{Z})\mid
g(P)=P+\bm{v}\mbox{ for some }\bm{v}\in\mathbb{Z}^{d}\\}$

and

$\mathrm{Aut}_{\mathbb{Z}}(\Gamma_{P})=\\{g\in\mathrm{GL}_{d}(\mathbb{Z})\mid\mathrm{TL}_{P,g(\bm{v})}=\mathrm{TL}_{P,\bm{v}}\mbox{
for all }\bm{v}\in\mathbb{R}^{d}\\}.$

######  Proposition 7.1.

For a rational $d$-polytope $P\subset\mathbb{R}^{d}$, one has
$\mathrm{Aut}_{\mathbb{Z}}(\Gamma_{P})=\mathrm{Aut}_{\mathbb{Z}}(P)$.

###### Proof.

We first prove “$\subset$”. Let $g\in\mathrm{Aut}_{\mathbb{Z}}(\Gamma_{P})$.
Then, for any $\bm{v}\in\mathbb{R}^{d}$, we have

$\mathrm{TL}_{P,\bm{v}}(t)=\mathrm{TL}_{P,g(\bm{v})}(t)=\\#\big{(}\big{(}tP+g(\bm{v})\big{)}\cap\mathbb{Z}^{d}\big{)}=\\#\big{(}(tg^{-1}(P)+\bm{v})\cap\mathbb{Z}^{d}\big{)}=\mathrm{TL}_{g^{-1}(P),\bm{v}}(t)$

for all $t\in\mathbb{Z}_{\geq 0}$. Thus we have
$\Gamma_{P}=\Gamma_{g^{-1}(P)}$ so $P=g^{-1}(P)+\bm{v}$ for some
$\bm{v}\in\mathbb{Z}^{d}$ by Theorem 6.1. Then
$g\in\mathrm{Aut}_{\mathbb{Z}}(P)$ since $P=g(P)-g(\bm{v})$ and
$g(\bm{v})\in\mathbb{Z}^{d}$.

We next prove “$\supset$”. Let $g\in\mathrm{Aut}_{\mathbb{Z}}(P)$. Then for
any $\bm{v}\in\mathbb{R}^{d}$, we have

$\\#\big{(}\big{(}tP+g(\bm{v})\big{)}\cap\mathbb{Z}^{d}\big{)}=\\#\big{(}\big{(}tg(P)+g(\bm{v})\big{)}\cap\mathbb{Z}^{d}\big{)}=\\#\big{(}(tP+\bm{v})\cap\mathbb{Z}^{d}\big{)}$

for any $t\in\mathbb{Z}_{\geq 0}$, where the last equality follows from the
fact that $g\in\mathrm{GL}_{d}(\mathbb{Z})$. This implies
$\mathrm{TL}_{P,g(\bm{v})}=\mathrm{TL}_{P,\bm{v}}$ for all
$\bm{v}\in\mathbb{R}^{d}$. ∎

######  Example 7.2.

Consider the rhombus $Q$ in Example 4.2. From the list of translated lattice
points enumerators in the example, one can see that they are equal on
$E_{1},E_{2},E_{7}$ and $E_{8}$. This can be explained using the symmetry. Let
$\rho_{1},\rho_{2}\in\mathrm{GL}_{2}(\mathbb{Z})$ be a reflection by the
$x$-axis and the $y$-axis, respectively. Then $\rho_{1},\rho_{2}$ do not
change $Q$ so they are elements of $\mathrm{Aut}_{\mathbb{Z}}(Q)$. We have

$\rho_{1}(E_{1})=E_{7},\ \rho_{1}(E_{2})=E_{8},\mbox{ and
}\rho_{2}(E_{1})=E_{2},$

which tell that translated lattice points enumerators are equal on
$E_{1},E_{2},E_{7}$ and $E_{8}$.

We now focus on centrally symmetric polytopes. Recall that a quasi-polynomial
$f$ is said to be symmetric if its $k$th constituent equals its $(-k)$th
constituent for all $k\in\mathbb{Z}$.

We first prove the following criterion for the symmetry of Ehrhart quasi-
polynomials of $P+\bm{v}$.

######  Lemma 7.3.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. The following
conditions are equivalent.

  * (i)

$\operatorname{ehr}_{P+\bm{v}}$ is symmetric for any
$\bm{v}\in\mathbb{Q}^{d}$.

  * (ii)

For any $\bm{v}\in\mathbb{Q}^{d}$ and any $k\in\mathbb{Z}_{\geq 0}$, one has

$\mbox{the $k$th constituent of $\mathrm{TL}_{P,\bm{v}}$}=\mbox{the $(-k)$th
constituent of $\mathrm{TL}_{P,-\bm{v}}$}.$

###### Proof.

We first prove “(i) $\Rightarrow$ (ii)”. Fix $\bm{v}\in\mathbb{Q}^{d}$ and
$k\in\mathbb{Z}_{\geq 0}$. Then

$k$th constituent of $\mathrm{TL}_{P,\bm{v}}$ $\displaystyle=\mbox{$k$th
constituent of $\operatorname{ehr}_{P+\frac{1}{k}\bm{v}}$}$ (by Lemma 2.4)
$\displaystyle=\mbox{$(-k)$th constituent of
$\operatorname{ehr}_{P+\frac{1}{k}\bm{v}}$}$ (by (i))
$\displaystyle=\mbox{$(-k)$th constituent of $\mathrm{TL}_{P,-\bm{v}}$}$
$\displaystyle\mbox{(by Lemma \ref{ehrToTrans})},$

as desired.

The proof for “(ii) $\Rightarrow$ (i)” is similar. Indeed, we have

$k$th constituent of $\operatorname{ehr}_{P+\bm{v}}$
$\displaystyle=\mbox{$k$th constituent of $\mathrm{TL}_{P,k\bm{v}}$}$ (by
Lemma 2.4) $\displaystyle=\mbox{$(-k)$th constituent of
$\mathrm{TL}_{P,-k\bm{v}}$}$ (by (ii)) $\displaystyle=\mbox{$(-k)$th
constituent of $\operatorname{ehr}_{P+\bm{v}}$}$ $\displaystyle\mbox{(by Lemma
\ref{ehrToTrans})},$

as desired. ∎

Recall that a polytope $P\subset\mathbb{R}^{d}$ is said to be centrally
symmetric if $-P=P+\bm{v}$ for some $\bm{v}\in\mathbb{R}^{d}$.

######  Corollary 7.4.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. If
$\operatorname{ehr}_{P+\bm{v}}$ is symmetric for any
$\bm{v}\in\mathbb{Q}^{d}$, then

  * (i)

$P$ is centrally symmetric.

  * (ii)

$\operatorname{ehr}_{\pi_{i}(P)+\bm{u}}$ is symmetric for any
$\bm{u}\in\mathbb{Q}^{d-1}$ and $i\in\\{1,2,\dots,d\\}$.

###### Proof.

(i) Let $q$ be a positive integer such that $qP$ is integral. It suffices to
prove that $qP$ is centrally symmetric. Since
$\operatorname{ehr}_{qP+q\bm{v}}(s)=\operatorname{ehr}_{P+\bm{v}}(qs)$ for any
$s\in\mathbb{Z}_{\geq 0}$, the $k$th constituent of
$\operatorname{ehr}_{qP+q\bm{v}}$ is obtained from the $qk$th constituent of
$\operatorname{ehr}_{P+\bm{v}}$ by substituting $t$ with $\frac{t}{q}$ (as
polynomials in $t$). This fact and the assumption tell that
$\operatorname{ehr}_{qP+q\bm{v}}$ is symmetric for any
$\bm{v}\in\mathbb{Q}^{d}$. Since $qP$ is a lattice polytope, it follows from
Theorem 1.4 that $qP$ is centrally symmetric.

(ii) For any $\bm{v}\in\mathbb{Q}^{d}$, we have

$k$th constituent of $\mathrm{TL}_{\pi_{i}(P),\pi_{i}(\bm{v})}$
$\displaystyle=\mbox{$k$th constituent of $\sum_{0\leq
s<1}\mathrm{TL}_{P,\bm{v}+s\mathbf{e}_{i}}^{(-i)}$}$ $\displaystyle(\mbox{by
Proposition \ref{7.6}})$ $\displaystyle=\mbox{$(-k)$th constituent of
$\sum_{0\leq s<1}\mathrm{TL}_{P,-\bm{v}-s\mathbf{e}_{i}}^{(+i)}$}$ (by Lemma
7.3) $\displaystyle=\mbox{$(-k)$th constituent of
$\mathrm{TL}_{\pi_{i}(P),-\pi_{i}(\bm{v})}$}$ $\displaystyle(\mbox{by
Proposition \ref{7.6*}}).$

This tells that $\pi_{i}(P)$ satisfies the condition (ii) of Lemma 7.3. ∎

We now come to the goal of this section. Let $P$ be a centrally symmetric
polytope with $-P=P+\bm{u}$. Then $\frac{1}{2}\bm{u}$ is a center of $P$, and
for any vertex $\bm{v}=\frac{1}{2}\bm{u}+\bm{v}^{\prime}$ of $P$, the point
$\frac{1}{2}\bm{u}-\bm{v}^{\prime}$ is also a vertex of $P$ by the central
symmetry. We write this vertex $\frac{1}{2}\bm{u}-\bm{v}^{\prime}$ as
$\bm{v}^{*}$.

######  Theorem 7.5.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. The following
conditions are equivalent.

  * (i)

$\operatorname{ehr}_{P+\bm{v}}$ is symmetric for any
$\bm{v}\in\mathbb{Q}^{d}$.

  * (ii)

$P$ is centrally symmetric and $\bm{v}-\bm{v}^{*}\in\mathbb{Z}^{d}$ for any
vertex $\bm{v}$ of $P$.

To prove the theorem, we recall the following basic fact on
$\mathbb{Z}$-modules.

######  Lemma 7.6.

For any $d$-dimensional cone $X\subset\mathbb{R}^{d}$ with apex $\bm{0}$,
there is a $\mathbb{Z}$-basis of $\mathbb{Z}^{d}$ which are contained in
$\mathrm{int}(X)$.

###### Proof.

Take any integer vector $\bm{a}\in\mathrm{int}(X)$ with $\gcd(\bm{a})=1$.

First, we claim that there are
$\bm{a}_{1},\dots,\bm{a}_{d-1}\in\mathbb{Z}^{d}$ such that
$\bm{a},\bm{a}_{1},\dots,\bm{a}_{d-1}$ is a $\mathbb{Z}$-basis of
$\mathbb{Z}^{d}$. In fact, by the assumption, the $\mathbb{Z}$-module
$\mathbb{Z}^{d}/(\mathbb{Z}\bm{a})$ is a free $\mathbb{Z}$-module of rank
$d-1$ since it is torsionfree. If we choose
$\bm{a}_{1},\dots,\bm{a}_{d-1}\in\mathbb{Z}^{d}$ so that they form a
$\mathbb{Z}$-basis for $\mathbb{Z}^{d}/(\mathbb{Z}\bm{a})$, the sequence
$\bm{a},\bm{a}_{1},\dots,\bm{a}_{d-1}$ becomes a $\mathbb{Z}$-basis of
$\mathbb{Z}^{d}$.

Now, we may check that we can choose $\bm{a}_{1},\ldots,\bm{a}_{d-1}$ from
$\mathrm{int}(X)$. For each $\bm{a}_{i}$, since $\bm{a}$ is in the interior of
$X$, by taking a sufficiently large integer $k_{i}$, the point
$\bm{a}_{i}+k_{i}\bm{a}$ is contained in $\mathrm{int}(X)$. Then
$\bm{a},\bm{a}_{1}+k_{1}\bm{a},\dots,\bm{a}_{d-1}+k_{d-1}\bm{a}$ is a desired
$\mathbb{Z}$-basis. ∎

###### Proof of Theorem 7.5.

((ii) $\Rightarrow$ (i)) By taking an appropriate translation, we may assume
$P=-P$. Then $\bm{v}^{*}=-\bm{v}$ for any vertex $\bm{v}$ of $P$, so the
condition (ii) tells that $2P$ is integral. In particular, every quasi-
polynomial $\mathrm{TL}_{P,\bm{v}}$ has period 2. We prove that $P$ satisfies
the condition (ii) of Lemma 7.3.

Let $\bm{v}\in\mathbb{Q}^{d}$ and $k\in\\{0,1\\}$. Since $P=-P$ we have

the $k$th constituent of $\mathrm{TL}_{P,\bm{v}}$ $\displaystyle=\mbox{the
$k$th constituent of $\mathrm{TL}_{P,-\bm{v}}$}.$

However, since $\mathrm{TL}_{P,-\bm{v}}$ has period $2$, the RHS in the above
equation equals the $(-k)$th constituent of $\mathrm{TL}_{P,-\bm{v}}$.

((i) $\Rightarrow$ (ii)) We have already seen that (i) implies that $P$ is
centrally symmetric in Corollary 7.4. We prove the second condition of (ii) by
induction on $d$. Suppose $d=1$. Then we may assume

$\textstyle P=[0,x+\frac{p}{q}]$

for some $x,p,q\in\mathbb{Z}_{\geq 0}$ with $0\leq p<q$. Then we have

$\textstyle\mbox{ the first constituent of
$\operatorname{ehr}_{P}$}=\mathrm{vol}(P)t-\frac{p}{q}+1$

and

$\textstyle\mbox{ the $(q-1)$th constituent of
$\operatorname{ehr}_{P}$}=\mathrm{vol}(P)t-(q-1)\frac{p}{q}+\lfloor(q-1)\frac{p}{q}\rfloor+1.$

Then the condition (i) tells $p-2p/q=\lfloor p(q-1)/q\rfloor$, but it implies
$2p/q\in\mathbb{Z}$. Hence $2P$ is integral which guarantees the condition
(ii).

Suppose $d>1$. Let $\bm{v}\in\mathbb{Q}^{d}$ be a vertex of $P$. Consider the
normal cone at the vertex $\bm{v}$

$X=\\{\bm{a}\in\mathbb{R}^{d}\mid\max\\{(\bm{a},\bm{x})\mid\bm{x}\in
P\\}=(\bm{a},\bm{v})\\}.$

This is a $d$-dimensional cone with apex $\bm{0}$. By Lemma 7.6, there is a
$\mathbb{Z}^{d}$-basis $\mathbf{e}^{\prime}_{1},\dots,\mathbf{e}^{\prime}_{d}$
which are contained in $\mathrm{int}(X)$. Consider the linear transformation
$g\in\mathrm{GL}_{n}(\mathbb{Z})$ which changes the hyperplane
$\\{\bm{x}\in\mathbb{R}^{d}\mid(\bm{x},\mathbf{e}^{\prime}_{i})=0\\}$ to
$\\{\bm{x}=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}\mid x_{i}=0\\}$. Since
$g(\mathbb{Z}^{d})=\mathbb{Z}^{d}$, we have
$\operatorname{ehr}_{g(P)+\bm{u}}(t)=\operatorname{ehr}_{P+g^{-1}(\bm{u})}(t)$
for any $\bm{u}\in\mathbb{Q}^{d}$, so $g(P)$ also satisfies the condition (i).
Let $g(\bm{v})=(y_{1},\dots,y_{d})$. By the choice of
$\mathbf{e}_{1}^{\prime},\dots,\mathbf{e}_{d}^{\prime}$, we have

$g(P)\cap\\{(x_{1},\dots,x_{d})\in\mathbb{R}^{d}\mid
x_{i}=y_{i}\\}=\\{g(\bm{v})\\}\ \ \mbox{ for any $1\leq i\leq d$}.$

This tells that $\pi_{j}(g(\bm{v}))$ is a vertex of $\pi_{j}(g(P))$ for
$j=1,2,\dots,d$ and the same hold for $\pi_{j}(g(\bm{v}^{*}))$ by the central
symmetry. For each $j=1,2,\dots,d,$ Lemma 7.4 tells that $\pi_{j}(g(P))$
satisfies the condition (i), so we have that
$\pi_{j}(g(\bm{v})-g(\bm{v}^{*}))\in\mathbb{Z}^{d-1}$ by the induction
hypothesis. But then we must have $g(\bm{v}-\bm{v}^{*})\in\mathbb{Z}^{d}$ and
therefore $\bm{v}-\bm{v}^{*}\in\mathbb{Z}^{d}$. ∎

If $P=-P$, then the condition (ii) of Theorem 7.5 is equivalent to saying that
$2P$ is integral. So, Theorem 7.5 is essentially equivalent to Theorem 1.5 in
Introduction.

##  8\. A connection to commutative algebra

In this section, we briefly explain a connection between translated lattice
points enumerators and conic divisorial ideals of Ehrhart rings in commutative
algebra. In particular, we explain that Theorem 2.3 can be proved
algebraically using the duality of Cohen–Macaulay modules.

###  8.1. Conic divisorial ideals

Let $S=\mathbb{F}[x_{1}^{\pm},\dots,x_{d+1}^{\pm}]$ be the Laurent polynomial
ring over a field $\mathbb{F}.$ We will consider the grading of $S$ defined by
$\deg(x_{1})=\cdots=\deg(x_{d})=0$ and $\deg(x_{d+1})=1$. For
$\bm{a}=(a_{1},\dots,a_{d+1})\in\mathbb{Z}^{d+1}$, we write

$x^{\bm{a}}=x_{1}^{a_{1}}\cdots x_{d+1}^{a_{d+1}}.$

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. The Ehrhart ring
$\mathbb{F}[P]$ of $P$ (over $\mathbb{F}$) is the monoid algebra generated by
the monomials $x^{\bm{a}}$ such that $\bm{a}$ is in the monoid
$\mathcal{C}_{P}\cap\mathbb{Z}^{d+1}$. As vector spaces, we can write

(8.1)
$\displaystyle\mathbb{F}[P]=\mathrm{span}_{\mathbb{F}}\\{x^{\bm{a}}\mid\bm{a}\in\mathcal{C}_{P}\cap\mathbb{Z}^{d+1}\\}.$

For a finitely generated graded $\mathbb{F}[P]$-module $M$, its Hilbert
function is the function defined by
$\mathrm{hilb}(M,k)=\dim_{\mathbb{F}}M_{k}$ for $k\in\mathbb{Z}$, where
$M_{k}$ is the degree $k$ component of $M$, and the Hilbert series of $M$ is
the formal power series
$\operatorname{Hilb}(M,z)=\sum_{k\in\mathbb{Z}}\mathrm{hilb}(M,k)z^{k}$.
Ehrhart rings are closely related to Ehrhart quasi-polynomials. Indeed, from
(8.1), we can see that the Hilbert function of $\mathbb{F}[P]$ is nothing but
the Ehrhart quasi-polynomial of $P$.

For any $\bm{v}\in\mathbb{R}^{d+1}$, the vector space

$I_{\bm{v}}=\mathrm{span}_{\mathbb{F}}\\{x^{\bm{a}}\mid\bm{a}\in(\mathcal{C}_{P}+\bm{v})\cap\mathbb{Z}^{d+1}\\}\subset
S$

becomes a finitely generated graded $\mathbb{F}[P]$-module. The modules
$I_{\bm{v}}$ are called conic divisorial ideals of $\mathbb{F}[P]$. We note
that different vectors in $\mathbb{R}^{d+1}$ could give the same conic
divisorial ideal, more precisely, we have $I_{\bm{v}}=I_{\bm{u}}$ if and only
if the cones $\mathcal{C}_{P}+\bm{v}$ and $\mathcal{C}_{P}+\bm{u}$ have the
same lattice points.

Let us call a conic divisorial ideal $I$ standard if $I=I_{(\bm{v},0)}$ for
some $\bm{v}\in\mathbb{R}^{d}$. Hilbert functions of standard conic divisorial
ideals are nothing but translated lattice points enumerators. Indeed, for any
$\bm{v}\in\mathbb{R}^{d}$, we have

(8.2)
$\displaystyle\dim_{\mathbb{F}}(I_{(\bm{v},0)})_{t}=\\#\\{\bm{a}\\!=\\!(a_{1},\dots,a_{d+1})\in(C_{P}+(\bm{v},0))\cap\mathbb{Z}^{d+1}\\!\mid
a_{d+1}\\!=\\!t\\}=\\#\big{(}(tP+\bm{v})\\!\cap\mathbb{Z}^{d}\big{)}.$

We will not explain algebraic backgrounds on (conic) divisorial ideals of
Ehrhart rings since it is not relevant to the theme of this paper. But in the
rest of this section we briefly explain how algebraic properties of conic
divisorial ideals can be used to consider properties of translated lattice
points enumerators. For more detailed information on conic divisorial ideals,
see [9] and [10, §4.7].

###  8.2. Hilbert series of conic divisorial ideals and an algebraic proof of
Theorem 2.3

We need some basic tools on commutative algebra such as the Cohen–Macaulay
property and canonical modules. We refer the readers to [8, §3 and §4] for
basics on commutative algebra.

We introduce one more notation. For $\bm{v}\in\mathbb{R}^{d+1}$, we define

(8.3) $\displaystyle
I^{\circ}_{\bm{v}}=\mathrm{span}_{\mathbb{F}}\big{\\{}x^{\bm{a}}\mid\bm{a}\in\big{(}\mathrm{int}(\mathcal{C}_{P})+\bm{v}\big{)}\cap\mathbb{Z}^{d+1}\big{\\}}.$

The space $I^{\circ}_{\bm{v}}$ is also a conic divisorial ideal. Indeed, if
$\bm{w}\in\mathrm{int}(\mathcal{C}_{P})$ is a vector which is sufficiently
close to the origin, then we have

$\big{(}\mathrm{int}(\mathcal{C}_{P})+\bm{v}\big{)}\cap\mathbb{Z}^{d+1}=(\mathcal{C}_{P}+\bm{v}+\bm{w})\cap\mathbb{Z}^{d+1},$

which tells $I_{\bm{v}}^{\circ}=I_{\bm{v}+\bm{w}}$. The following facts are
known. See [10, Corollary 3.3 and Remark 4.4(b)].

  * •

$I_{\bm{v}}$ is a $(d+1)$-dimensional Cohen–Macaulay module.

  * •

$I_{\bm{v}}^{\circ}$ is the canonical module of $I_{-\bm{v}}$, more precisely,
we have

$\operatorname{Hom}_{\mathbb{F}[P]}(I_{\bm{v}},\omega)\cong\mathrm{span}_{\mathbb{F}}\\{x^{\bm{a}}\mid\bm{a}\in(\mathrm{int}(\mathcal{C}_{P})-\bm{v})\cap\mathbb{Z}^{d+1}\\}=I_{-\bm{v}}^{\circ},$

where
$\omega=\mathrm{span}_{\mathbb{F}}\\{x^{\bm{a}}\mid\bm{a}\in\mathrm{int}(\mathcal{C}_{P})\cap\mathbb{Z}^{d+1}\\}$
is the graded canonical module of $\mathbb{F}[P]$.

These properties give the following consequences on Hilbert series of conic
divisorial ideals.

######  Proposition 8.1.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope and $q$ the
denominator of $P$. Let $\bm{v}=(v_{1},\dots,v_{d+1})\in\mathbb{R}^{d+1}$ and
$\alpha=\lceil v_{d+1}\rceil$.

  * (1)

$\operatorname{Hilb}(I_{\bm{v}}^{\circ},z)=(-1)^{d+1}\operatorname{Hilb}(I_{-\bm{v}},z^{-1})$.

  * (2)

$\operatorname{Hilb}(I_{\bm{v}},z)=\frac{z^{\alpha}}{(1-z^{q})^{d+1}}Q(z)$ for
some polynomial $Q(z)\in\mathbb{Z}_{\geq 0}[z]$ of degree $<q(d+1)$.

###### Proof.

The equality (1) is the well-known formula of the Hilbert series of a
canonical module. See [8, Theorem 4.45]. We prove (2). Consider the subring

$A=\mathrm{span}_{\mathbb{F}}\big{\\{}x^{\bm{a}}\mid
x^{\bm{a}}\in\mathcal{C}_{P}\mbox{ and $\deg(x^{\bm{a}})\in
q\mathbb{Z}$}\big{\\}}\subset\mathbb{F}[P].$

Since $qP$ is integral, $\mathbb{F}[qP]$ is a semi-standard graded
$\mathbb{F}$-algebra, that is, $\mathbb{F}[qP]$ is a finitely generated as a
module over a standard graded $\mathbb{F}$-algebra
$\mathbb{F}[x^{\bm{a}}x_{d+1}\mid x^{\bm{a}}\in P\cap\mathbb{Z}^{d}]$ (see
[19, Theorem 9.3.6](d)). Then, since $A\cong\mathbb{F}[qP]$, where the degree
$k$ part of $\mathbb{F}[qP]$ corresponds to the degree $qk$ part of $A$, any
finitely generated $A$-module $M$ of Krull dimension $m$ has the Hilbert
series of the form $Q(z)/(1-z^{q})^{m}$ for some polynomial $Q(z)$, and if $M$
is Cohen–Macaulay then $Q(z)\in\mathbb{Z}_{\geq 0}[z]$ ([8, Corollaries 4.8
and 4.10]).

Since $\mathbb{F}[P]$ is a finitely generated $A$-module, $I_{\bm{v}}$ is a
finitely generated Cohen–Macaulay $A$-module of Krull dimension $d+1$. Thus
there is a polynomial $Q(z)\in\mathbb{Z}_{\geq 0}[z]$ such that

$\operatorname{Hilb}(I_{\bm{v}},z)=\frac{1}{(1-z^{q})^{d+1}}Q(z).$

Since $(I_{\bm{v}})_{k}=0$ for $k<\alpha=\lceil v_{d+1}\rceil$ by the
definition of $I_{\bm{v}}$, the polynomial $Q(z)$ must be of the form

$Q(z)=c_{0}t^{\alpha}+c_{1}t^{\alpha+1}+\cdots+c_{m}t^{\alpha+m}$

for some $m\geq 0$, where $c_{0},\dots,c_{m}\in\mathbb{Z}_{\geq 0}$ and
$c_{m}\neq 0$, so it follows that

$\operatorname{Hilb}(I_{\bm{v}},z)=\frac{z^{\alpha}}{(1-z^{q})^{d+1}}(c_{0}+c_{1}z+\cdots+c_{m}z^{m}).$

Now it remains to prove $m<q(d+1)$. By statement (1), we have

$\displaystyle\operatorname{Hilb}(I_{-\bm{v}}^{\circ},z)$
$\displaystyle=(-1)^{d+1}\frac{z^{-\alpha}}{(1-z^{-q})^{d+1}}(c_{0}+c_{1}z^{-1}+\cdots+c_{m}z^{-m})$
$\displaystyle=\frac{z^{q(d+1)-\alpha-m}}{(1-z^{q})^{d+1}}(c_{m}+c_{m}z+\cdots+c_{0}z^{m}).$

This tells

$-\alpha<\min\\{k\in\mathbb{Z}\mid(I_{-\bm{v}}^{\circ})_{k}\neq
0\\}=q(d+1)-\alpha-m$

proving the desired inequality $m<q(d+1)$. ∎

The statements in Proposition 8.1 are known to imply the quasi-polynomiality
and reciprocity of translated lattice points enumerators in Theorem 2.3.
Recall that $\mathrm{TL}_{P,\bm{v}}$ coincides with the Hilbert function of
$I_{(\bm{v},0)}$. Proposition 8.1(2) tells that the Hilbert series of
$I_{(\bm{v},0)}$ can be written in the form $\frac{1}{(1-z^{q})^{d+1}}Q(z)$
for some polynomial $Q(z)$ of degree $<q(d+1)$, which is known to imply that
$\mathrm{hilb}(I_{(\bm{v},0)},t)$ is a quasi-polynomial with period $q$. See
e.g., [5, §3.8] or [18, §4]. Also, Proposition 8.1(1) is essentially
equivalent to the reciprocity in Theorem 2.3(2). See [5, §4.3].

Finally, we note that the proposition gives some restriction to the possible
values of $\mathrm{TL}_{P,\bm{v}}$. If $P\subset\mathbb{R}^{d}$ is a lattice
$d$-polytope and $\bm{v}\in\mathbb{R}^{d}$, then the proposition tells

$\operatorname{Hilb}(I_{(\bm{v},0)},z)=\frac{1}{(1-z)^{d+1}}(h_{0}+h_{1}z+\cdots+h_{d}z^{d})$

for some $h_{0},h_{1},\dots,h_{d}\in\mathbb{Z}_{\geq 0}$. These $h$-numbers
must satisfy the following conditions

  * •

$h_{0}=1$ if $\bm{v}\in\mathbb{Z}^{d}$ and $h_{0}=0$ if
$\bm{v}\not\in\mathbb{Z}^{d}$;

  * •

$h_{0}+\cdots+h_{d}=d!\mathrm{vol}(P)$.

The first condition follows from
$h_{0}=\dim_{\mathbb{F}}(I_{(\bm{v},0)})_{0}$, and the second condition
follows since $\frac{1}{d!}(h_{0}+\cdots+h_{d})$ is the top degree coefficient
of the polynomial $\mathrm{hilb}(I_{(\bm{v},0)},t)$. Below we give a simple
application of this. Consider a lattice polygon $P\subset\mathbb{R}^{2}$ whose
volume is $\frac{3}{2}$. Then the possible values of $h_{0}+h_{1}z+h_{2}z^{2}$
are

$1+z+z^{2},\ 1+2z,\ 1+2z^{2},\ 3z,\ 2z+z^{2},\ z+2z^{2},\ 3z^{2}.$

If $f(t)$ is a polynomial
$\sum_{t=0}^{\infty}f(t)z^{t}=\frac{1}{(1-z)^{3}}(h_{0}+h_{1}z+h_{2}z^{2})$,
then $f(t)=h_{0}{t+2\choose 2}+h_{1}{t+1\choose 2}+h_{2}{t\choose 2}$. So a
translated lattice points enumerator of an integral polygon with volume
$\frac{3}{2}$ must be one of the following polynomials

$\textstyle\frac{3}{2}t^{2}+\frac{3}{2}t+1,\ \frac{3}{2}t^{2}+\frac{5}{2}t+1,\
\frac{3}{2}t^{2}+\frac{1}{2}t+1,\ \frac{3}{2}t^{2}+\frac{3}{2}t,\
\frac{3}{2}t^{2}+\frac{1}{2}t,\ \frac{3}{2}t^{2}-\frac{1}{2}t,\
\frac{3}{2}t^{2}-\frac{3}{2}t.$

Four of them appear as translated lattice points enumerators of the trapezoid
in Introduction. See (1.1).

##  9\. Problems

In this last section, we list a few problems which we cannot answer.

### Gcd property and zonotopes

A quasi-polynomial $f$ with period $q$ is said to have the gcd property if its
$k$th constituent only depends on the gcd of $k$ and $q$ for all
$k\in\mathbb{Z}$. We note that if $f$ has the gcd property then $f$ must be
symmetric. It was proved in [20] that, for a lattice $d$-polytope
$P\subset\mathbb{R}^{d}$, $\operatorname{ehr}_{P+\bm{v}}$ has the gcd property
for all $\bm{v}\in\mathbb{Q}^{d}$ if and only if $P$ is a zonotope.
Considering the statement in Theorem 1.5, one may ask if a similar statement
holds for zonotopes $P$ such that $2P$ is integral, but this is not the case.
Indeed, the rhombus $Q$ in Example 4.2 is a zonotope and $2Q$ is integral but
the computation given in the example tells that
$\operatorname{ehr}_{Q+(\frac{1}{8},\frac{1}{8})}$ does not satisfy the gcd
property. We repeat the following question asked in [20, Problem 6.7(2)].

######  Problem 9.1.

Let $P\subset\mathbb{R}^{d}$ be a rational $d$-polytope. Is it true that, if
$\operatorname{ehr}_{P+\bm{v}}$ has the gcd property for all
$\bm{v}\in\mathbb{Q}^{d}$, then $P=Q+\bm{u}$ for some integral zonotope $Q$
and some $\bm{u}\in\mathbb{Q}^{d}$?

To consider this problem we can assume that $P$ is a zonotope by the argument
similar to the proof of Corollary 7.4(i) and $2P$ is integral by Theorem 1.5.

### Period collapse

Recall that the denominator of a rational polytope $P$ is always a period of
$\operatorname{ehr}_{P}$. If the minimum period of $\operatorname{ehr}_{P}$ is
not equal to the denominator of $P$, we say that period collapse occurs to
$P$. A period collapse is a major subject in the study of Ehrhart quasi-
polynomials (see e.g., [6, 11, 15, 16]). We ask the following vague question:
Is a way of computing $\operatorname{ehr}_{P+\bm{v}}$ from $\mathrm{TL}_{P,C}$
applicable to produce polytopes giving period collapse? For translations of a
lattice polytope, a period collapse cannot occur. Indeed, if $P$ is a lattice
polytope, then the minimum period of $\operatorname{ehr}_{P+\bm{v}}$ must be
the smallest integer $k$ such that $k\bm{v}$ is integral since
$\mathrm{TL}_{P,\bm{v}}(0)\neq 0$ only when $\bm{v}$ is integral.

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