# Higher Dimensional Fourier Quasicrystals from Lee–Yang Varieties

Lior Alon  Massachusetts Institute of Technology, USA<EMAIL_ADDRESS>,
Mario Kummer  Technische Universität, Dresden, Germany  mario.kummer@tu-
dresden.de ,  Pavel Kurasov  Stockholm University, Sweden<EMAIL_ADDRESS>and  Cynthia Vinzant  University of Washington, Seattle, USA<EMAIL_ADDRESS>
###### Abstract.

In this paper, we construct Fourier quasicrystals with unit masses in
arbitrary dimensions. This generalizes a one-dimensional construction of
Kurasov and Sarnak. To do this, we employ a class of complex algebraic
varieties avoiding certain regions in ${\mathbb{C}}^{n}$, which generalize
hypersurfaces defined by Lee–Yang polynomials. We show that these are Delone
almost periodic sets that have at most finite intersection with every discrete
periodic set.

###### 2020 Mathematics Subject Classification:

52C23, 42B10, 42A75, 42A38, 14P05

LA was supported by the Simons Foundation Grant 601948, DJ. MK has been
partially supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), Grant number 502861109. PK was partially supported by
the Swedish Research Council Grant 2020-03780. CV was partially supported by
NSF grant No. DMS-2153746 and a Sloan Research Fellowship.

##  1\. Introduction

In this paper, we describe a general construction of Delone sets
$\Lambda\subseteq{\mathbb{R}}^{d}$, in every dimension $d$, whose counting
measures $\mu_{\Lambda}=\sum_{x\in\Lambda}\delta_{x}$ are Fourier
quasicrystals with unit masses. Our construction is based on the interplay
between real algebraic geometry and discrete geometry, using a new class of
complex algebraic varieties that avoid certain regions in $\mathbb{C}^{n}$.
This directly generalizes previous work by Kurasov and Sarnak [kurasovsarnak],
who constructed one-dimensional Fourier quasicrystals with unit masses using
Lee–Yang polynomials. The notion of a Fourier quasicrystal was introduced and
used in the context of measures, see e.g. [lev2015quasicrystals,
favorov2016fourier, kurasovsarnak, lagarias2000mathematical, lev2017fourier].

A _crystalline measure_ is a tempered measure of the form
$\mu=\sum_{x\in\Lambda}a_{x}\delta_{x}$, with discrete support $\Lambda$, such
that the distributional Fourier transform is also a (complex) measure with
discrete support $\hat{\mu}=\sum_{\xi\in\Lambda^{\prime}}c_{\xi}\delta_{\xi},\
c_{\xi}\in{\mathbb{C}}$, see [meyerquestion]. A measure $\mu$ is a _Fourier
quasicrystal_ if it is crystalline and the absolute value measures $|\mu|$ and
$|\hat{\mu}|$ are tempered.111Not all crystalline measures are Fourier
quasicrystals, see e.g. [favorov2024crystalline].  We say $\mu$ has unit
masses if $a_{x}=1$ for all $x\in\Lambda$, in which case $\mu$ is the counting
measure of $\Lambda$. To ease the reading, we say that a set
$\Lambda\subseteq\mathbb{R}^{d}$ is a Fourier quasicrystal if its counting
measure $\mu_{\Lambda}=\sum_{x\in\Lambda}\delta_{x}$ is a Fourier
quasicrystal. Explicitly, we have the following.

######  Definition 1.1 (FQ).

A set $\Lambda\subseteq{\mathbb{R}}^{d}$ is a _Fourier quasicrystal_ (FQ) if
there exists a set $\Lambda^{\prime}\subseteq{\mathbb{R}}^{d}$ and non-zero
complex coefficients $(c_{\xi})_{\xi\in\Lambda^{\prime}}$, called the
_spectrum_ and the _Fourier coefficients_ of $\Lambda$, respectively, such
that

  1. (1)

for every Schwartz function $f\in\mathcal{S}({\mathbb{R}}^{d})$,

$\sum_{x\in\Lambda}\hat{f}(x)=\sum_{\xi\in\Lambda^{\prime}}c_{\xi}f(\xi),$

  2. (2)

both $\Lambda$ and $\Lambda^{\prime}$ are discrete (locally finite) with
polynomial growth bound: there exist $C>0$ and $m\in{\mathbb{N}}$ such that

$\left|\Lambda\cap B_{R}(0)\right|+\sum_{\xi\in\Lambda^{\prime}\cap
B_{R}(0)}|c_{\xi}|\leq CR^{m}\quad\text{for all }\ R>1,$

where $B_{R}(0)$ is the ball of radius $R$ around the origin.

Figure 1. From 1.3, the curve $\\{\theta\mid\exp(2\pi i\theta)\in X\\}$, the
Fourier quasicrystal $\Lambda$ (blue dots), and a discrete set which contains
the spectrum of $\Lambda$ (red dots).

Throughout the paper, the term _Fourier quasicrystal_ (FQ) is always
understood in the set-theoretic sense of 1.1, except for the introductory
section. In the rest of the introduction, we use the terms _FQ-sets_ and _FQ-
measures_ to make a clear distinction between sets in ${\mathbb{R}}^{d}$ that
are Fourier quasicrystals in the sense of 1.1 and measures on
${\mathbb{R}}^{d}$ that are Fourier quasicrystals.

######  Definition 1.2 (Delone).

A set $\Lambda\subseteq{\mathbb{R}}^{d}$ is _Delone_ if it is uniformly
discrete and relatively dense. Recall that $\Lambda$ is uniformly discrete if
there exists $r>0$ such that $|x-x^{\prime}|\geq r$ for every pair of distinct
points $x,x^{\prime}\in\Lambda$, and relatively dense if there exists $R>0$
such that the balls of radius $R$ around points of $\Lambda$ cover
${\mathbb{R}}^{d}$:

$\bigcup_{x\in\Lambda}B_{R}(x)={\mathbb{R}}^{d}.$

Lagarias states in [lagarias2000mathematical]*Corollary 3.3 (following
[cordoba1989dirac]) that if $\Lambda\subseteq{\mathbb{R}}^{d}$ is a Delone FQ-
set whose Fourier transform is translation bounded, then $\Lambda$ is
periodic. Lagarias conjectured that if an FQ-measure has Delone support and
Delone spectrum, then it is periodic. The conjecture was proven by Lev and
Olevskii in [lev2015quasicrystals] for $d=1$, and also for $d>1$ under an
additional positivity assumption. Favorov showed that the positivity
assumption is needed by constructing a non-positive two-dimensional
counterexample [favorov2016fourier]. Since counting measures are positive, the
conjecture is true for FQ-sets in arbitrary dimension. This leads naturally to
the following question:

Are there non-periodic Delone FQ-sets?

We provide a positive answer in every dimension $d$, which also answers
questions asked by Meyer in [meyerquestion]*p. 3158. The first construction of
a non-periodic Delone FQ-set was given by Kurasov and Sarnak [kurasovsarnak]
for $d=1$. They provided a general construction of FQ-sets in ${\mathbb{R}}$
using multivariate stable (Lee–Yang) polynomials. We call
$p\in{\mathbb{C}}[z_{1},\ldots,z_{n}]$ a Lee–Yang polynomial if $p(z)\neq 0$
whenever $z=(z_{1},\ldots,z_{n})\in{\mathbb{C}}^{n}$ has $|z_{i}|<1$ for all
$i$ or $|z_{i}|>1$ for all $i$. Lee–Yang polynomials first appeared in the
work of Lee and Yang [lee1952statistical] in which they showed that certain
univariate polynomials222The grand partition functions of two dimensional
Ising models have all their roots on the circle, by showing that each of those
is the restriction $s\mapsto p(s,s,\ldots,s)$ of a Lee–Yang polynomial $p$.
More generally, for every Lee–Yang polynomial $p$ and every positive vector
$\ell\in{\mathbb{R}}_{+}^{n}$ the trigonometric polynomial

$f(x)=p(\exp(2\pi i\ell x))\quad\text{ with }\quad x\in{\mathbb{C}}$

has all of its roots in ${\mathbb{R}}$, where
$\exp\colon{\mathbb{C}}^{n}\to({\mathbb{C}}^{*})^{n}$ is taken entry-wise.
That is, $\exp(2\pi i\ell x)=(e^{2\pi i\ell_{1}x},e^{2\pi
i\ell_{2}x},\ldots,e^{2\pi i\ell_{n}x})$. They show that when $p(z)$ and
$\nabla p(z)$ have no common zeros in the torus
$\operatorname{T}^{n}=\\{z\in{\mathbb{C}}^{n}\mid|z_{i}|=1\text{ for all
}i\\}$, the set $\Lambda=\\{x\in{\mathbb{R}}\mid f(x)=0\\}$ is a Delone FQ-
set. Moreover, under mild conditions on $\ell$ and $p$, the set $\Lambda$
intersects every arithmetic progression in at most finitely many points.
Properties of this construction are further investigated in [AlonVinzant].

It turns out that every FQ-set in ${\mathbb{R}}$ arises from this
construction. Namely, Olevskii and Ulanovskii showed that every FQ-set
$\Lambda\subseteq{\mathbb{R}}$ is the zero set of a real-rooted trigonometric
polynomial [olevskii2020fourier] and Alon, Cohen and Vinzant showed that every
real-rooted trigonometric polynomial is the restriction of a Lee–Yang
polynomial [AlonCohenVinzant]. Arbitrary one-dimensional FQ-measures are less
well-understood.

The Kurasov–Sarnak construction can be seen geometrically as follows. The
vector $\ell\in{\mathbb{R}}_{+}^{n}$ defines a map from ${\mathbb{C}}$ to
${\mathbb{C}}^{n}$ given by $x\mapsto\exp(2\pi i\ell x)$ that maps
${\mathbb{R}}$ into $\operatorname{T}^{n}$. When the entries of $\ell$ are
linearly independent over ${\mathbb{Q}}$, the image of ${\mathbb{R}}$ under
this map is dense in $\operatorname{T}^{n}$. The FQ-set $\Lambda$ is the set
of $x\in{\mathbb{R}}$ whose image under this map belongs to the algebraic
variety $\\{z\in\operatorname{T}^{n}\mid p(z)=0\\}$.

In this paper we generalize this geometric construction by considering
algebraic varieties of arbitrary codimension $d$ and maps
${\mathbb{C}}^{d}\to{\mathbb{C}}^{n}$ given by $x\mapsto\exp(2\pi iLx)$, where
$L$ is a $n\times d$ matrix with real entries and positive $d\times d$ minors.
The higher codimensional analogue of the zero set of a Lee–Yang polynomial
will be a Lee–Yang variety, defined in 4.1. The intimate relation between
real-rootedness and Lee–Yang polynomials extends to Lee–Yang varieties.
Namely, for every Lee–Yang variety $X$ of codimension $d$, $\exp(2\pi iLx)\in
X$ implies that $x\in{\mathbb{R}}^{d}$ and analogously to the Kurasov–Sarnak
construction, we consider the set
$\Lambda=\\{x\in{\mathbb{R}}^{d}\mid\exp(2\pi iLx)\in X\\}$. Similar notions
of “real-rootedness” for algebraic varieties have been studied in [SV18, KV19,
KS20a, KS20b, RVY21]. This existing theory enables us to provide a large
collection of Lee–Yang varieties and resulting Delone FQ-sets with various
properties in ${\mathbb{R}}^{d}$ for arbitrary $d\in{\mathbb{N}}$. Our proofs
are mainly based on the interplay between real algebraic geometry and discrete
geometry, rather than on complex analysis.

If $\Lambda\subseteq{\mathbb{R}}$ is an FQ-set, then by [lev2015quasicrystals]
it is the zero set of a of a trigonometric polynomial in one complex variable,
all of whose zeroes are real. Similarly, the FQ-sets we construct in this
paper are solution sets to systems of multivariate trigonometric equations in
several complex variables, all of whose solutions are real. More precisely, if
our Lee–Yang variety is defined by equations $p_{1}(z)=0,\ldots,p_{m}(z)=0$
where $p_{j}(z)\in{\mathbb{C}}[z_{1},\ldots,z_{n}]$, then we obtain the system
of trigonometric equations

$p_{1}(\exp(2\pi iLx))=0,\ldots,p_{m}(\exp(2\pi iLx))=0$

all of whose solutions in ${\mathbb{C}}^{d}$ lie in ${\mathbb{R}}^{d}$. The
FQ-set $\Lambda$ is the set of solutions.

######  Example 1.3.

Consider the algebraic curve $X\subseteq{\mathbb{C}}^{3}$ defined by equations

$\displaystyle p_{1}(z)=(1-2i)-z_{1}-z_{2}+(1+2i)z_{1}z_{2}$
$\displaystyle=0,$ $\displaystyle p_{2}(z)=1-(1+i)z_{1}-(1-i)z_{3}+z_{1}z_{3}$
$\displaystyle=0.$

This curve can be parametrized (up to closure) by the map

$t\mapsto\left(\frac{-1+i+t}{-1-i+t},\frac{-i+t}{i+t},\frac{1+i+t}{1-i+t}\right).$

Take the linear map $x=(x_{1},x_{2})\mapsto
Lx=(x_{1},x_{2},-\sqrt{2}x_{1}+\sqrt{3}x_{2})$, given by a $3\times 2$ matrix
$L$ whose $2\times 2$ minors are $1$, $\sqrt{2}$, and $\sqrt{3}$. We will see
that a compactification of $X$ is a Lee–Yang variety and that
$\Lambda=\\{x\in{\mathbb{R}}^{2}\mid\exp(2\pi iLx)\in X\\}$ is a Delone FQ-
set. This set $\Lambda$ consists of the solutions to the two trigonometric
equations

$\displaystyle-4\sin(\pi\theta_{1})\cos(\pi\theta_{2})-4\sin(\pi\theta_{2})(\sin(\pi\theta_{1})+\cos(\pi\theta_{1}))$
$\displaystyle=0,$ $\displaystyle
2\sin(\pi\theta_{1})\cos(\pi\theta_{3})-2\sin(\pi\theta_{3})(2\sin(\pi\theta_{1})+\cos(\pi\theta_{1}))$
$\displaystyle=0.$

where
$(\theta_{1},\theta_{2},\theta_{3})=(x_{1},x_{2},-\sqrt{2}x_{1}+\sqrt{3}x_{2})$.
The left hand side of these equations are $e^{-\pi
i(\theta_{1}+\theta_{2})}p_{1}(\exp({2\pi i\theta}))$ and $e^{-\pi
i(\theta_{1}+\theta_{3})}p_{2}(\exp(2\pi i\theta))$. The density of $\Lambda$
is

$\lim_{R\to\infty}\frac{\left|\Lambda\cap
B_{R}\right|}{\operatorname{Vol}(B_{R})}=1+\sqrt{2}+\sqrt{3}.$

The spectrum of $\Lambda$, as defined in 1.1, is contained in the discrete set

$\Lambda^{\prime}=\left\\{(k_{1}-\sqrt{2}k_{3},k_{2}+\sqrt{3}k_{3})\mid
k\in{\mathbb{Z}}^{3},{\rm sign}(k)\not\in\\{(+,-,+),(-,+,-)\\}\right\\},$

where ${\rm sign}(k)$ records the signs in $\\{-,0,+\\}$ of the coordinates of
$k$. As shown in 3.7, the number of points in $\Lambda^{\prime}\cap[-R,R]^{2}$
is $\frac{1}{9}(10\sqrt{2}+3\sqrt{3})R^{3}+O(R^{2})$ as $R\to\infty$. We will
also see that there is no affine line $W\subsetneq{\mathbb{R}}^{2}$ such that
$W\cap\Lambda$ contains a set isometric to an FQ-set in ${\mathbb{R}}$. The
torus points of the curve $X$, the FQ-set $\Lambda$, and set
$\Lambda^{\prime}$ are shown in Figure 1.

Meyer in [Meyer2023-2, Meyer2023] and de Courcy-Ireland and Kurasov [KudCI]
have used real-rooted systems of trigonometric polynomials to construct FQ-
measures. They obtain weights that are in general not integer valued. In
particular, their FQ-measures are not counting measures of subsets of
${\mathbb{R}}^{d}$. In a recent preprint [lawton2024fourier], Lawton and Tsikh
construct FQ-sets $\Lambda\subseteq{\mathbb{R}}^{d}$ using an approach that is
similar to ours in that $\Lambda$ is also the solution set to a system of
multivariate trigonometric equations with only real solutions. We prove that
every Delone FQ-set arising from their method can also be obtained from our
construction (see Section 8). On the other hand, their method only applies to
systems of trigonometric equations where the number of equations equals the
number of variables while our construction is not subject to this restriction.

###  1.1. Notations and conventions

Throughout the paper, we use $d,n\in{\mathbb{N}}$ and $c=n-d\geq 0$ as
dimensions. The Fourier quasicrystals in this paper will be subsets of
$d$-dimensional Euclidean space, constructed using algebraic varieties of
dimension $c=n-d\geq 0$ in an ambient space of dimension $n\geq d$.

We consider ${\mathbb{C}}^{*}={\mathbb{C}}\setminus\\{0\\}$ as a subset of the
projective line $\mathbb{P}^{1}={\mathbb{C}}\cup\\{\infty\\}$. For a subset
$X\subseteq(\mathbb{P}^{1})^{n}$ we denote
$X(\operatorname{T})=X\cap\operatorname{T}^{n}$, where

$\operatorname{T}^{n}=\\{z\in({\mathbb{C}}^{*})^{n}\mid|z_{1}|=\cdots=|z_{n}|=1\\}.$

We use entrywise functions, for $z=(z_{1},\ldots,z_{n})$

$\exp:{\mathbb{C}}^{n}\to({\mathbb{C}}^{*})^{n}\quad\exp(z)=(e^{z_{1}},\ldots,e^{z_{n}}),$

and similarly, $\log|z|=(\log|z_{1}|,\ldots,\log|z_{n}|)$, with the convention
$\log|0|:=-\infty$ and $\log|\infty|:=\infty$, so that $\log|\cdot|$ is a map
from $(\mathbb{P}^{1})^{n}\to({\mathbb{R}}\cup\\{\pm\infty\\})^{n}$.

We denote $[n]:=\\{1,2,\ldots,n\\}$ and $\binom{[n]}{d}=\\{I\subseteq[n]\mid\
|I|=d\\}$ the collection of $d$-tuples of distinct indices in $[n]$. Given
$I\subseteq[n]$, a set $S$ and the product $S^{n}$, the canonical projection
$\pi_{I}\colon S^{n}\to S^{I}$ is given by
$\pi_{I}(s_{1},\ldots,s_{n})=(s_{i})_{i\in I}$, where $S^{I}$ denotes the set
of tuples indexed by $I$.

We use $R^{n\times d}$ to denote the set of $n\times d$ matrices with entries
in a ring $R$. Given a matrix $M\in R^{n\times d}$ (or $R^{d\times n}$) and
$I\in\binom{[n]}{d}$, we use $M_{I}$ to denote the determinant of the $d\times
d$ submatrix of $M$ obtained by restricting to the rows (or columns) indexed
by $I$. We use $M^{t}$ to denote the transpose of $M$. By convention, all
vectors are assumed to be column vectors.

Finally, for a function $f$, we use the convention that its Fourier transform
is

$\hat{f}(y)=\int_{\mathbb{R}^{n}}f(x)e^{-2\pi i\langle x,y\rangle}\,dx.$

##  2\. The construction and main results

Here we present a rather general construction of multidimensional Fourier
quasicrystals in detail and formulate main theorems describing their
properties.

###  2.1. General construction and main theorem

Our Fourier quasicrystals will result from the following construction. The
input to our construction is:

  * •

an algebraic variety $X\subseteq(\mathbb{P}^{1})^{n}$ of dimension $c=n-d$,
and

  * •

a real matrix $L\in{\mathbb{R}}^{n\times d}$.

We consider the points whose image under the map $x\mapsto\exp(2\pi iLx)$
belongs to $X$:

(1) $\Lambda=\Lambda(X,L):=\\{x\in{\mathbb{C}}^{d}\mid\exp(2\pi iLx)\in X\\}.$

If all $d\times d$ minors of $L$ are positive and $X$ is a strict Lee–Yang
variety in the sense of 2.2 below, then $\Lambda\subseteq{\mathbb{R}}^{d}$,
and our main result states that $\Lambda$ is a $d$-dimensional Fourier
quasicrystal. In Section 4.2 we provide a method for explicit construction of
one-dimensional strict Lee–Yang varieties in $(\mathbb{P})^{n}$, hence $d=n-1$
dimensional Fourier quasicrystals, for every $n\geq 2$. In Section 8, we will
see that many algebraic varieties can be transformed to a strict Lee–Yang
variety by a change of coordinates. Strict Lee–Yang varieties are defined as
follows.

######  Definition 2.1.

For a vector
$b=(b_{1},\ldots,b_{n})\in\left({\mathbb{R}}\cup\\{\pm\infty\\}\right)^{n}$ we
let $\operatorname{var}(b)$ be the number of sign changes in the sequence
$b_{1},\ldots,b_{n}$ after discarding the zeroes.

######  Definition 2.2 (Strict Lee–Yang variety).

An equidimensional333An algebraic variety is equidimensional if it is a finite
union of irreducible algebraic varieties of the same dimension. closed
algebraic subvariety $X$ in $(\mathbb{P}^{1})^{n}$, of dimension $c=n-d\geq
0$, is called a _strict Lee–Yang variety_ if it satisfies the following three
conditions:

  1. (1)

$X$ is invariant under the coordinate-wise involution $z\mapsto
1/\overline{z}$;

  2. (2)

for every $z\in X$ we either have $z\in
X(\operatorname{T})=X\cap\operatorname{T}^{n}$ or
$\operatorname{var}(\log|z|)\geq d$;

  3. (3)

$X(\operatorname{T})$ is contained in the smooth part of $X$.

Now we can formulate our main result:

######  Theorem 2.3 (Construction of Delone FQ).

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a strict Lee–Yang variety of dimension
$c=n-d$ and let $L\in{\mathbb{R}}^{n\times d}$ be a real matrix all of whose
$d\times d$ minors are positive. Then the set $\Lambda=\Lambda(X,L)$ is real,
$\Lambda\subseteq{\mathbb{R}}^{d}$, it is a Delone set, and it is a Fourier
quasicrystal. Its spectrum is contained in the discrete set

(2) $\Lambda^{\prime}=\Lambda^{\prime}(L)=\\{L^{t}k\mid k\in{\mathbb{Z}}^{n},\
\operatorname{var}(k)<d\\}.$

The proof of 2.3 is given at the end of Section 7. Note that neither 2.2 nor
the positivity of the minors of $L$ are invariant under arbitrary permutations
of the coordinates.

######  Remark 2.4 (Real-rooted system of trigonometric equations).

The statement that $\Lambda(X,L)$ is real is equivalent to the system of
trigonometric equations

${\mathbb{C}}^{d}\ni x\mapsto P(\exp(2\pi iLx))$

having only real solutions, where ${\mathbb{C}}^{n}\ni z\mapsto P(z)$ is a
system of (possibly more than $d$) polynomial equations that determine $X$. An
independent recent work [lawton2024fourier] provided a construction of
$d$-dimensional FQs from sufficiently generic real-rooted systems of $d$
trigonometric equations. In Section 8 we show that every such system is
obtained via our construction if the FQ is Delone.

###  2.2. Properties of the constructed Fourier quasicrystals

Similarly to the one-dimensional case [kurasovsarnak, AlonVinzant], under mild
conditions on $X$ and $L$ the set $\Lambda(X,L)$ is far from being periodic.
Let $\dim_{{\mathbb{Q}}}(A)$ of a set $A\subseteq{\mathbb{R}}^{d}$ denote the
dimension, as a rational vector space, of the linear span of its elements over
${\mathbb{Q}}$. For example, the set $\\{1,\pi,2+\pi\\}\subseteq{\mathbb{R}}$
has $\dim_{{\mathbb{Q}}}=2$. A lattice in ${\mathbb{R}}^{d}$ has
$\dim_{{\mathbb{Q}}}\leq d$, and a set contained in the projection of a
lattice in ${\mathbb{R}}^{d}\times{\mathbb{R}}^{N}$ to ${\mathbb{R}}^{d}$ has
$\dim_{{\mathbb{Q}}}\leq d+N$.

######  Theorem 2.5 (Quantitative non-periodicity).

Let $\Lambda=\Lambda(X,L)$ be defined as in 2.3, and further assume that

  1. (1)

the $d\times d$ minors of $L$ are ${\mathbb{Q}}$-linearly independent, and

  2. (2)

$X$ is irreducible and $X\cap({\mathbb{C}}^{*})^{n}$ is not a coset of an
algebraic subtorus444A coset of a $c$-dimensional algebraic subtorus of
$({\mathbb{C}}^{*})^{n}$ is a set $\\{\exp(x+My)\mid y\in{\mathbb{C}}^{c}\\}$
with $x\in{\mathbb{C}}^{n}$ and $M\in{\mathbb{Z}}^{n\times c}$ of full rank.
of $({\mathbb{C}}^{*})^{n}$.

Then $\dim_{{\mathbb{Q}}}(\Lambda)=\infty$, and there is a constant
$r=r(X)>0$, that only depends on $n$ and the degree of $X$, such that the
following uniform bounds hold

$|\Lambda\cap A|<r^{m+1},\ \text{ for every set }\ A\subseteq{\mathbb{R}}^{d}\
\text{with}\ \dim_{{\mathbb{Q}}}(A)=m.$

In particular, for every $a,b\in{\mathbb{R}}^{d}$ there are at most $r^{3}$
points $a+jb\in\Lambda$ with $j\in{\mathbb{Z}}$.

This statement, including an explicit expression for $r$, is a special case of
6.7.

######  Theorem 2.6 (Properties of our construction).

Let $\Lambda=\Lambda(X,L)\subseteq{\mathbb{R}}^{d}$ and $\Lambda^{\prime}$ as
in 2.3. Let $(c_{\xi})_{\xi\in\Lambda^{\prime}}$ be the coefficients such that
for every Schwartz function $f\in\mathcal{S}({\mathbb{R}}^{d})$, we have
$\sum_{x\in\Lambda}\hat{f}(x)=\sum_{\xi\in\Lambda^{\prime}}c_{\xi}f(\xi)$.
Denote the $d\times d$ minors of $L$ by $(L_{I})_{I\in\binom{[n]}{d}}$. Then:

  1. (1)

The set $\Lambda$ is Bohr almost periodic with density equal to

$\lim_{R\to\infty}\frac{\left|\Lambda\cap
B_{R}\right|}{\operatorname{Vol}(B_{R})}=c_{0}=\sum_{I\in\binom{[n]}{d}}L_{I}\cdot
d_{I}$

where $d({X})=(d_{J})_{J\in\binom{[n]}{d}}$ is the multidegree of the strict
Lee–Yang variety ${X}$, and $B_{R}$ is the ball of radius $R$ around the
origin.

  2. (2)

The set $\Lambda^{\prime}=\Lambda^{\prime}(L)$ as in (2) has rational
dimension $\dim_{{\mathbb{Q}}}(\Lambda^{\prime})\leq n$ and growth rate
$\left|\Lambda^{\prime}\cap B_{R}\right|=CR^{n}+O(R^{n-1})$, for
$R\rightarrow\infty$ and some $C\geq 0$. If the rows of $L$ are linearly
independent over ${\mathbb{Q}}$, then $C>0$.

  3. (3)

There are coefficients $c_{L,k}\in{\mathbb{C}}$, for $k\in{\mathbb{Z}}^{n}$,
such that for all $\xi\in\Lambda^{\prime}$

$c_{\xi}=\sum_{k\in{\mathbb{Z}}^{n},\ L^{t}k=\xi}c_{L,k}$

where almost all summands, and at least those with $\operatorname{var}(k)\geq
d$, in the sum on the right-hand side are zero. The coefficients $c_{L,k}$ are
bounded, $|c_{L,k}|\leq c_{0}$, and depend linearly on
$(L_{I})_{I\in\binom{[n]}{d}}$. Specifically, there is a collection of
measures $m_{J}$ on $X(\operatorname{T})$, indexed by $J\in\binom{[n]}{n-d}$,
such that

$c_{L,k}=\sum_{I\in\binom{[n]}{d}}L_{I}\widehat{m}_{[n]\setminus I}(k).$

The measures $m_{J}$ and their Fourier transform are defined in Section 7
where we also prove 2.6, except for the statement that the density of
$\Lambda$ is given by $c_{0}$, which is proven independently in 2.9. One
possibility to construct a multidimensional FQ is by taking the product of
one-dimensional FQs, or a rotation and translation thereof. In Section 9 we
give sufficient criteria on $X$ and $L$ that guarantee that $\Lambda(X,L)$ is
not of this form. More precisely, we prove that, under mild assumptions, our
Fourier quasicrystals are genuinely high-dimensional in the sense that they do
not intersect any affine line in a set that is isometric to a one dimensional
Fourier quasicrystal.

######  Theorem 2.7.

Let $\Lambda=\Lambda(X,L)\subseteq{\mathbb{R}}^{d}$ be as in 2.3 and assume in
addition that:

  1. (1)

The variety $X$ is a curve, i.e. $\dim(X)=1$, and so $d=n-1$.

  2. (2)

$X$ is irreducible and $X(\operatorname{T})=X\cap\operatorname{T}^{n}$ is not
contained in a coset of a proper subtorus.

  3. (3)

The $d\times d$ minors of $L$ are ${\mathbb{Q}}$-linearly independent.

Then, there is no affine line $W\subsetneq{\mathbb{R}}^{d}$ such that
$W\cap\Lambda$ contains a set isometric to a Fourier quasicrystal in
${\mathbb{R}}$.

In Section 11 we construct explicit examples of Fourier quasicrystals that
have all the properties described in the above theorems.

###  2.3. Diffraction and hyperuniformity

Next, we provide a rather straight forward application, showing that every
discrete set $\Lambda\subseteq{\mathbb{R}}^{d}$ whose counting measure is a
Fourier quasicrystal is _hyperuniform_ , and in fact _stealthy hyperuniform_.
See [torquato2018hyperuniform] for a thorough review of hyperuniform materials
and the physical implications of a hyperuniform state of matter. See also
[bjorklund2023hyperuniformity, ouguz2017hyperuniformity] where it is shown
that crystals (i.e. lattices) and certain quasicrystals555not to be confused
with Fourier quasicrystals are stealthy hyperuniform and hyperuniform
respectively. What we show next suggests that the physical properties of
Fourier quasicrystals should be closer to crystals than to quasicrystals. In
the physics literature there are two equivalent definitions of hyperuniform
sets, and in [bjorklund2023hyperuniformity]*Theorem 1.1 this equivalence is
proven rigorously.

######  Definition 2.8.

A discrete set $\Lambda\subseteq{\mathbb{R}}^{d}$ is called _hyperuniform_ if
one of the following equivalent conditions hold.

  1. (1)

Hyperuniformity via physical space: Suppose $\Lambda$ has density $C>0$, and
let $N_{R}(x)=|\Lambda\cap B_{R}(x)|$ for $x\in{\mathbb{R}}^{d}$. Then
$\Lambda$ is called _hyperuniform_ if the variance of
$\frac{N_{R}(x)}{\operatorname{Vol}(B_{R})}$ over $x\in B_{R}(0)$ goes to zero
as $R\to\infty$. That is,

$\lim_{R\to\infty}\frac{1}{\operatorname{Vol}(B_{R})}\int_{x\in
B_{R}}\left|\frac{N_{R}(x)}{\operatorname{Vol}(B_{R})}-C\right|^{2}dx=0.$

  2. (2)

Hyperuniformity via Fourier space: Suppose that the following limit of
tempered measures exists (in the vague topology) and is unique,

$\gamma=\lim_{R\to\infty}\frac{1}{\operatorname{Vol}(B_{R})}\sum_{x,y\in\Lambda\cap
B_{R}}\delta_{x-y}.$

It is called the _auto-correlation_ of $\Lambda$. Its distributional Fourier
transform $\widehat{\gamma}$ is called the _diffraction measure_ of $\Lambda$.
In such case $\Lambda$ is hyperuniform if

$\lim_{\epsilon\to
0}\frac{\widehat{\gamma}(B_{\epsilon}\setminus\\{0\\})}{\epsilon^{d}}=0.$

Another related quantity in the physics literature is the _structure factor_
$S(\xi)$, which is the density of the diffraction measure $\widehat{\gamma}$,
namely $d\widehat{\gamma}=S(\xi)d\xi$, and is well defined at $\xi$ only if
$\widehat{\gamma}$ is absolutely continuous in a neighborhood of $\xi$.
Hyperuniformity is sometimes defined by $S(\xi)\to 0$ when $0\neq\xi\to 0$.

A hyperuniform set $\Lambda$ is called _stealthy hyperuniform_ if $\\{0\\}$ is
an isolated point in the support of $\widehat{\gamma}$.

######  Theorem 2.9.

If a discrete set $\Lambda\subseteq{\mathbb{R}}^{d}$ is a Fourier
quasicrystal, then it is stealthy hyperuniform. In more details, suppose that
$\Lambda$ is a Fourier quasicrystal with spectrum $\Lambda^{\prime}$ and
Fourier coefficients $(c_{\xi})_{\xi\in\Lambda^{\prime}}$. Then:

  1. (1)

The auto-correlation $\gamma$ of $\Lambda$ exists and the diffraction measure
is equal to
$\widehat{\gamma}=\sum_{\xi\in\Lambda^{\prime}}|c_{\xi}|^{2}\delta_{\xi}$. In
particular the support of the diffraction measure is the discrete set
$\Lambda^{\prime}$.

  2. (2)

The Fourier coefficient $c_{0}$ is real, positive, and is the density of
$\Lambda$, with

$\sup_{x\in{\mathbb{R}}^{d}}\left|N_{R}(x)-c_{0}\operatorname{Vol}(B_{R})\right|=O(R^{d-1}).$

The proof is in Section 10.

###  2.4. Open questions

We conclude this section with the following questions:

  1. (1)

_Can all Delone FQ-sets in every dimension be obtained via our construction,
namely as $\Lambda(X,L)$ with strict Lee–Yang $X$ and $L$ with positive
minors?_

  2. (2)

_If $X$ is Lee–Yang but $X(\operatorname{T})$ is not smooth, is it still true
that $\Lambda(X,L)$ is an FQ-set, possibly with multiplicities? If so, can
every FQ-set be obtained in this way?_

  3. (3)

_If there is a negative answer to question (1) or (2), can every (Delone) FQ-
set be obtained from some suitable modification of our construction, namely as
$\Lambda(X,L)$ with an algebraic variety $X$ and a matrix $L$?_

  4. (4)

_Can all FQ-sets in every dimension be obtained as the common zeros of a real-
rooted system of trigonometric equations?_

  5. (5)

_Can the zero set of every real-rooted system of trigonometric equations be
obtained as $\Lambda(X,L)$ with $X$ Lee–Yang and $L$ with positive minors?_

Positive answers for these questions are known in some special cases. All
these questions have positive answers for the one dimensional case.
Furthermore, in Section 8 we show that the answer to the first question is
positive for Delone FQ-sets that arise from the construction in
[lawton2024fourier].

##  3\. The positive Grassmannian

We denote by ${\rm Gr}(d,n)$ the Grassmannian of $d$-dimensional linear
subspaces of ${\mathbb{R}}^{n}$. Any subspace $V\in{\rm Gr}(d,n)$ can be
expressed as the column span of an $n\times d$ matrix $L$, whose columns
$v_{1},\ldots,v_{d}$ form a basis of $V$. For $I\subseteq[n]$ with $|I|=d$,
let $L_{I}$ denote the determinant of the $d\times d$ submatrix of $L$
obtained by restricting to rows indexed by $I$. We use these minors to specify
$V$ as follows.

Let $1\leq d\leq n$ and take $e_{1},\ldots,e_{n}$ to be the standard basis for
${\mathbb{R}}^{n}$. For each subset $I=\\{i_{1},\ldots,i_{d}\\}$ of $[n]$ with
$i_{1}<\cdots<i_{d}$, we denote $e_{I}=e_{i_{1}}\wedge\cdots\wedge e_{i_{d}}$.
The collection of wedge products $\\{e_{I}\mid I\in\binom{[n]}{d}\\}$ forms a
basis of $\bigwedge^{d}{\mathbb{R}}^{n}$. For a linear space $V={\rm
span}\\{v_{1},\ldots,v_{d}\\}$ in $\operatorname{Gr}(d,n)$, we can express
$v_{1}\wedge\ldots\wedge v_{d}$ as $\sum_{I\in\binom{[n]}{d}}L_{I}e_{I}$ in
$\bigwedge^{d}{\mathbb{R}}^{n}$ where $L$ is the $n\times d$ matrix with
columns $v_{1},\ldots,v_{d}$. The coefficients, $(L_{I})_{I\in\binom{n}{d}}$,
are known as the _Plücker coordinates_ of $V$ and are independent of the basis
$\\{v_{1},\ldots,v_{d}\\}$ of $V$ up to global scaling. Thus the map sending
${\rm span}\\{v_{1},\ldots,v_{d}\\}$ to $[v_{1}\wedge\ldots\wedge
v_{d}]\in\mathbb{P}(\bigwedge^{d}{\mathbb{R}}^{n})$ is well defined. This is
the _Plücker embedding_ of the Grassmannian $\operatorname{Gr}(d,n)$ into
$\mathbb{P}(\bigwedge^{d}{\mathbb{R}}^{n})\cong\mathbb{P}^{\binom{n}{d}-1}({\mathbb{R}})$.
Finally, we will define $s(I)\in\\{\pm 1\\}$ such that $e_{I}\wedge
e_{[n]\setminus I}=s(I)\cdot e_{[n]}$.

Let ${\rm Gr}_{+}(d,n)$ resp. ${\rm Gr}_{\geq}(d,n)$ denote the positive and
nonnegative Grassmannian, respectively. This is the collection of
$d$-dimensional subspaces of ${\mathbb{R}}^{n}$ that can be written as the
column span of an $n\times d$ matrix all of whose $d\times d$ minors are
strictly positive or nonnegative, respectively. In order to formulate a
characterization of points lying on an element of the positive Grassmannian we
recall and extend 2.1.

######  Definition 3.1.

For a vector
$b=(b_{1},\ldots,b_{n})\in\left({\mathbb{R}}\cup\\{\pm\infty\\}\right)^{n}$ we
let $\operatorname{var}(b)$ be the number of sign changes in the sequence
$b_{1},\ldots,b_{n}$ after discarding the zeroes. We further let
$\operatorname{\overline{var}}(b)$ be the number of sign changes in the
sequence $b_{1},\ldots,b_{n}$ where the zeroes are assigned signs that
maximize the number of sign changes.

######  Lemma 3.2 ([Karp]*Lemma 4.1).

Let $v\in{\mathbb{R}}^{n}\setminus\\{0\\}$.

  1. (1)

There exists a linear subspace in ${\rm Gr}_{+}(d,n)$ containing $v$ iff
$\overline{\operatorname{var}}(v)<d$.

  2. (2)

There exists a linear subspace in ${\rm Gr}_{\geq}(d,n)$ containing $v$ iff
$\operatorname{var}(v)<d$.

######  Theorem 3.3 ([GK, KWZ]).

For a $d$-dimensional linear subspace $V\subseteq{\mathbb{R}}^{n}$, we have

  1. (1)

$V\in\operatorname{Gr}_{\geq}(d,n)\Leftrightarrow\operatorname{var}(v)<d\text{
for all }v\in
V\backslash\\{0\\}\Leftrightarrow\operatorname{\overline{var}}(w)\geq d\text{
for all }w\in V^{\perp}\backslash\\{0\\}.$

  2. (2)

$V\in\operatorname{Gr}_{+}(d,n)\
\Leftrightarrow\operatorname{\overline{var}}(v)<d\text{ for all }v\in
V\backslash\\{0\\}\Leftrightarrow\operatorname{var}(w)\geq d\text{ for all
}w\in V^{\perp}\backslash\\{0\\}.$

######  Corollary 3.4.

If $V\in\operatorname{Gr}_{\geq}(d,n)$ and $W\in\operatorname{Gr}_{+}(d,n)$,
then $V\cap W^{\perp}=\\{0\\}$ and $V^{\perp}\cap W=\\{0\\}$.

We conclude this section with a statement that will enable us to prove that
the support of the Fourier transform of our Fourier quasicrystals is discrete.
For positive integers $d\leq n$, define

(3)
${\mathbb{R}}^{n}_{\operatorname{var}<d}=\\{y\in{\mathbb{R}}^{n}\mid\operatorname{var}(y)<d\\}\
\ \ \text{ and }\ \ \
{\mathbb{Z}}^{n}_{\operatorname{var}<d}={\mathbb{R}}^{n}_{\operatorname{var}<d}\cap{\mathbb{Z}}^{n}.$

This is a union of orthants in ${\mathbb{R}}^{n}$ and ${\mathbb{Z}}^{n}$,
respectively.

######  Lemma 3.5.

Let $L\in{\mathbb{R}}^{n\times d}$ be a $n\times d$ matrix all of whose
$d\times d$ minors are positive. Then
$\\{y\in{\mathbb{R}}^{n}_{\operatorname{var}<d}\mid L^{t}y\in[-1,1]^{d}\\}$ is
closed, bounded, and has nonempty interior in the Euclidean topology on
${\mathbb{R}}^{n}$.

###### Proof.

First, note that ${\mathbb{R}}^{n}_{\operatorname{var}<d}$ and $\\{y\mid
L^{t}y\in[-1,1]^{d}\\}$ are closed subsets of ${\mathbb{R}}^{n}$, so their
intersection is also closed. To see that its interior is nonempty, consider a
point $y=\lambda(1,\ldots,1)^{t}$. For sufficiently small $\lambda>0$,
$L^{t}y=\lambda L^{t}(1,\ldots,1)^{t}$ belongs to $(-1,1)^{d}$. Then for every
$z$ in a sufficiently small neighborhood of $y$ in ${\mathbb{R}}^{n}$,
$z\in{\mathbb{R}}_{>0}^{n}$ and $L^{t}z\in(-1,1)^{d}$. Therefore $y$ belongs
to the interior of this set.

To see that it is bounded, let $W$ the column span of $L$. By assumption we
have $W\in{\rm Gr}_{+}(d,n)$. Suppose, for the sake of contradiction, that
$\\{y\in{\mathbb{R}}^{n}_{\operatorname{var}<d}\mid L^{t}y\in[-1,1]^{d}\\}$ is
unbounded. Note that ${\mathbb{R}}^{n}_{\operatorname{var}<d}$ is a union of
orthants in ${\mathbb{R}}^{n}$. It follows that the set
$\\{y\in{\mathbb{R}}^{n}_{\operatorname{var}<d}\mid L^{t}y\in[-1,1]^{d}\\}$ is
a union of finitely many polyhedra. Since it is unbounded, it therefore must
contain a ray $\\{y+\lambda v\mid\lambda\in{\mathbb{R}}_{\geq 0}\\}$ for some
$y,v\in{\mathbb{R}}^{n}$ with $v\neq 0$. By assumption, $L^{t}(y+\lambda
v)=L^{t}y+\lambda L^{t}v$ belongs to $[-1,1]^{d}$ for all $\lambda\geq 0$. As
$[-1,1]^{d}$ is bounded, we must have $L^{t}v=0$ and thus $v\in
W^{\perp}\setminus\\{0\\}$. By 3.3, it follows that $\operatorname{var}(v)\geq
d$. Then for sufficiently large $\lambda\geq 0$, $\operatorname{var}(y+\lambda
v)\geq d$. Indeed, since $\operatorname{var}$ does not count zero coordinates
towards sign changes, the entries of $y$ can only increase
$\operatorname{var}(y+\lambda v)$ for sufficiently large $\lambda$. This
contradicts $y+\lambda v\in{\mathbb{R}}^{n}_{\operatorname{var}<d}$ for all
$\lambda\geq 0$. ∎

######  Corollary 3.6.

Let $L\in{\mathbb{R}}^{n\times d}$ be an $n\times d$ matrix all of whose
$d\times d$ minors are positive. The image of
${\mathbb{Z}}^{n}_{\operatorname{var}<d}$ under the map $k\mapsto L^{t}k$ is
discrete. More precisely, for $R>0$, the number points whose image lies in
$[-R,R]^{d}$ is

$|\\{k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d}\mid
L^{t}k\in[-R,R]^{d}\\}|={\rm vol}\cdot R^{n}+O(R^{n-1})$

where ${\rm vol}>0$ is the volume of
$\\{y\in{\mathbb{R}}^{n}_{\operatorname{var}<d}\mid L^{t}y\in[-1,1]^{d}\\}$.
Moreover, if the rows of $L$ are linearly independent over ${\mathbb{Q}}$,
then

$|\\{L^{t}k\mid
k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d}\\}\cap[-R,R]^{d}|={\rm vol}\cdot
R^{n}+O(R^{n-1}).$

###### Proof.

Let $P=\\{y\in{\mathbb{R}}^{n}_{\operatorname{var}<d}\mid
L^{t}y\in[-1,1]^{d}\\}$. By 3.5, $P$ is closed, bounded, and has nonempty
interior. Note that $L^{t}y$ belongs to $[-R,R]^{d}$ if and only if
$L^{t}((1/R)y)$ belongs to $[-1,1]^{d}$ and
${\mathbb{R}}^{n}_{\operatorname{var}<d}$ is invariant under positive scaling.
Thus for every $R>0$,

$\\{y\in{\mathbb{R}}^{n}_{\operatorname{var}<d}\mid
L^{t}y\in[-R,R]^{d}\\}=R\cdot P=\\{R\cdot y\mid y\in P\\}.$

Since $R\cdot P$ is bounded, the number of integer points it contains is
finite, showing that the image of ${\mathbb{Z}}^{n}_{\operatorname{var}<d}$
under the map $k\mapsto L^{t}k$ is discrete.

To get the statement on the asymptotic growth, we write $P$ as a union of
polyhedra $P_{\sigma}=\\{y\in{\mathbb{R}}^{n}_{\sigma}\mid
L^{t}y\in[-1,1]^{d}\\}$ where $\sigma$ runs over all vectors in $\\{\pm
1\\}^{n}$ with $\operatorname{var}(\sigma)<d$ and
${\mathbb{R}}^{n}_{\sigma}=\\{y\in{\mathbb{R}}^{n}:\sigma_{j}y_{j}\geq 0\text{
for }j=1,\ldots,n\\}$. Since $P_{\sigma}$ is bounded, it is a polytope. By
[BordaThesis]*Corollary 1.2, for every polytope $Q\subseteq{\mathbb{R}}^{n}$,
the number of integer points in $R\cdot Q$ is

$|(R\cdot Q)\cap{\mathbb{Z}}^{n}|={\rm vol}(Q)R^{n}+O(R^{n-1}).$

For every $\sigma\neq\sigma^{\prime}$, the intersection of $P_{\sigma}$ and
$P_{\sigma^{\prime}}$ is a polytope of dimension $\leq n-1$ and so the number
of integer points in $R\cdot(P_{\sigma}\cap P_{\sigma^{\prime}})$ is
$O(R^{n-1})$. It follows that

$\displaystyle|(R\cdot P)\cap{\mathbb{Z}}^{n}|$
$\displaystyle=\sum_{\sigma}|(R\cdot
P_{\sigma})\cap{\mathbb{Z}}^{n}|+O(R^{n-1})$ $\displaystyle=\sum_{\sigma}{\rm
vol}(P_{\sigma})R^{n}+O(R^{n-1})={\rm vol}(P)R^{n}+O(R^{n-1}).$

Since $P$ has nonempty interior, ${\rm vol}={\rm vol}(P)>0$.

Finally, if the rows of $L$ are linearly independent over ${\mathbb{Q}}$, then
the map $k\mapsto L^{t}k$ is injective on ${\mathbb{Z}}^{n}$. The second
equality then follows from the first. ∎

######  Example 3.7 (1.3 continued).

The $2\times 2$ minors of the matrix
$L^{t}={\small\begin{pmatrix}1&0&-\sqrt{2}\\\ 0&1&\sqrt{3}\end{pmatrix}}$ are
all positive. 3.5 then implies that the set

$P=\\{y\in{\mathbb{R}}^{3}\mid\operatorname{var}(y)<2,(y_{1}-\sqrt{2}y_{3},y_{2}+\sqrt{3}y_{3})\in[-1,1]^{2}\\}$

is closed and bounded. We can write $P$ as a union of its intersections,
$P_{\sigma}$, with the orthants ${\mathbb{R}}^{3}_{\sigma}$, where $\sigma$
ranges over the six elements of $\\{\pm
1\\}^{3}\backslash\\{(1,-1,1),(-1,1,-1)\\}$. See Figure 2. The volume of $P$
is $\frac{1}{9}(10\sqrt{2}+3\sqrt{3})\approx 2.1487$, so, by 3.6, the number
of integer points in $R\cdot P$ is given by

$|(R\cdot
P)\cap{\mathbb{Z}}^{3}|=\frac{1}{9}(10\sqrt{2}+3\sqrt{3})R^{3}+O(R^{2}).$

Since the rows of $L$ are linearly independent over ${\mathbb{Q}}$, the map
${\mathbb{Z}}^{3}\to{\mathbb{R}}^{2}$ given by $k\mapsto L^{t}k$ is injective.
Therefore for $\Lambda^{\prime}=\\{L^{t}k\mid
k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d}\\}$, the number of points in
$\Lambda^{\prime}\cap[-R,R]^{2}$ is also given by
$\frac{1}{9}(10\sqrt{2}+3\sqrt{3})R^{3}+O(R^{2})$. The set $\Lambda^{\prime}$
appears on the right in Figure 1.

Figure 2. From 3.7, the set of $y\in{\mathbb{R}}^{3}$ with
$\operatorname{var}(y)<2$ and
$(y_{1}-\sqrt{2}y_{3},y_{2}+\sqrt{3}y_{3})\in[-1,1]^{2}$. Points with
$\operatorname{var}=0$ and $1$ are in blue and red, respectively.

##  4\. Lee–Yang varieties

In this section we introduce a class of algebraic varieties which we call
Lee–Yang varieties. These arise from relaxing condition (2) in 2.2 of strict
Lee–Yang varieties. This section is technical and is partitioned into five
subsections. First, we define Lee–Yang varieties and study their connection to
real-rootedness. Second, we provide examples of Lee–Yang varieties and
explicit construction methods. Third, we discuss the transversality of
$x\mapsto(\exp(2\pi iLx))$ and a Lee–Yang variety, when $L$ has positive
minors. Then we focus on strict Lee–Yang varieties and the topology of their
intersection with the torus. In the final subsection, we prove that certain
integrals over the torus part of a strict Lee–Yang variety vanish. This is
crucial for proving that the spectrum of the constructed Fourier quasicrystal
$\Lambda(X,L)$ is discrete.

###  4.1. Definitions and real-rootedness

We recall and add to 2.2.

######  Definition 4.1.

An equidimensional closed subvariety $X$ in $(\mathbb{P}^{1})^{n}$ of
dimension $c=n-d$ with $0\leq d\leq n$ is called a _strict Lee–Yang variety_
if it satisfies the following three conditions:

  1. (1)

$X$ is invariant under the coordinate-wise involution $z\mapsto
1/\overline{z}$,

  2. (2)

for every $z\in X$ we either have $z\in
X(\operatorname{T})=X\cap\operatorname{T}^{n}$ or
$\operatorname{var}(\log|z|)\geq d$, and

  3. (3)

$X(\operatorname{T})$ is contained in the smooth part of $X$.

We call $X$ a _Lee–Yang variety_ if it satisfies the following two conditions:

  1. (1)

$X$ is invariant under the coordinate-wise involution $z\mapsto
1/\overline{z}$, and

  2. (2′)

for every $z\in X$ we either have $z\in X(\operatorname{T})$ or
$\operatorname{\overline{var}}(\log|z|)\geq d$.

######  Remark 4.2.

  1. (1)

Every strict Lee–Yang variety is a Lee–Yang variety and one can check that not
every Lee–Yang variety is a strict Lee–Yang variety.

  2. (2)

Lee–Yang varieties of codimension one are exactly the zeros sets of Lee–Yang
polynomials, as considered in [AlonVinzant]*§3, or of stable pairs of
polynomials, considered in [kurasovsarnak]*§II.

  3. (3)

The smoothness assumption (3) in 4.1 will ensure that the support of the
Fourier quasicrystals, which we will construct, is uniformly discrete, see
Section 5. Property (2) in 4.1 makes the definition robust with respect to
small perturbations. Indeed, if $z\in(\mathbb{P}^{1})^{n}$ satisfies
$\operatorname{var}(\log|z|)\geq d$, then this is also the case for every
$z^{\prime}$ in a sufficiently small neighbourhood of $z$.

  4. (4)

The conclusion of 2.3 is in general false for varieties that satisfy (1), (2)
and (3) from 4.1 but are not equidimensional. Indeed, let
$X\subseteq(\mathbb{P}^{1})^{n}$ be a strict Lee–Yang variety of dimension
$c=n-d<n$ and $L\in{\mathbb{R}}^{n\times d}$ a real matrix all of whose
$d\times d$ minors are positive. For every $y\not\in\Lambda(X,L)$ the closed
subset $X\cup\\{\exp(2\pi iLy)\\}\subseteq(\mathbb{P}^{1})^{n}$ satisfies (1),
(2) and (3) of 4.1 but the Fourier transform of
$\delta_{y}+\sum_{x\in\Lambda}\delta_{x}$ is not discrete.

We now observe some first properties of Lee–Yang varieties.

######  Lemma 4.3.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a closed subvariety. For
$a\in\operatorname{T}^{n}$ we consider the automorphism

$m_{a}\colon(\mathbb{P}^{1})^{n}\to(\mathbb{P}^{1})^{n}$

which multiplies the $i$-th coordinate by $a_{i}\in\operatorname{T}$. Then $X$
is a (strict) Lee–Yang variety if and only if $m_{a}(X)$ is a (strict)
Lee–Yang variety.

###### Proof.

This follows directly from the definitions because for every
$z\in(\mathbb{P}^{1})^{n}$ one has $\log|z|=\log|m_{a}(z)|$ and
${1}/{\overline{m_{a}(z)}}=m_{a}({1}/{\overline{z}})$. ∎

######  Proposition 4.4.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be an equidimensional closed subvariety
of codimension $d$ and let $L\in{\mathbb{R}}^{n\times d}$ be a matrix of full
rank. Further let $x\in{\mathbb{C}}^{d}$ such that the point $\exp(2\pi iLx)$
belongs to $X$.

  1. (1)

If $X$ is a Lee–Yang variety and if all $d\times d$ minors of $L$ are
positive, then $x\in{\mathbb{R}}^{d}$.

  2. (2)

If $X$ is a strict Lee–Yang variety and if all $d\times d$ minors of $L$ are
nonnegative, then $x\in{\mathbb{R}}^{d}$.

###### Proof.

We prove part (1). Because $X$ is a Lee–Yang variety, we either have that
$\exp(2\pi iLx)\in\operatorname{T}^{n}$, which means
$L\operatorname{Im}(x)=0$, or we have

$d\leq\operatorname{\overline{var}}(\log|\exp(2\pi
iLx)|)=\operatorname{\overline{var}}(-2\pi
L\operatorname{Im}(x))=\operatorname{\overline{var}}(L\operatorname{Im}(x)).$

On the other hand, because the column span of $L$ is in ${\rm Gr}_{+}(d,n)$,
3.2 then also implies that $L\operatorname{Im}(x)=0$. Since $L$ is full rank,
we find that $\operatorname{Im}(x)=0$. The proof for part (2) is analogous. ∎

######  Notation 4.5.

For $M=(m_{ij})_{i,j}\in{\mathbb{Z}}^{m\times n}$ we consider the map

$\chi_{M}\colon({\mathbb{C}}^{*})^{n}\to({\mathbb{C}}^{*})^{m},\,(t_{1},\ldots,t_{n})\mapsto(t_{1}^{m_{i1}}\cdots
t_{n}^{m_{in}})_{i=1,\ldots,m}.$

######  Lemma 4.6.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a Lee–Yang variety of dimension
$c=n-d$ and $M\in{\mathbb{Z}}^{c\times n}$ be a matrix of rank $c$ whose
(right) kernel belongs to ${\rm Gr}_{+}(d,n)$. For
$X^{*}=X\cap({\mathbb{C}}^{*})^{n}$ we have
$\chi_{M}^{-1}(\operatorname{T}^{c})\cap X=X^{*}(\operatorname{T})$.

###### Proof.

Since $\chi_{M}$ takes $\operatorname{T}^{n}$ to $\operatorname{T}^{c}$, it is
clear that
$X^{*}(\operatorname{T})\subseteq\chi_{M}^{-1}(\operatorname{T}^{c})$. We now
show the other inclusion. Let $z=\exp(x+iy)\in X$ with
$x,y\in{\mathbb{R}}^{n}$. Since $X$ is Lee–Yang, either $x=0$, in which case
we are done, or $\operatorname{\overline{var}}(x)\geq d$. If
$\chi_{M}(\exp(x+iy))=\exp(Mx+iMy)\in\operatorname{T}^{c}$ then $x\in\ker(M)$.
3.2 implies that $\operatorname{\overline{var}}(v)<d$ for every non-zero
$v\in\ker(M)$, so we conclude that $x=0$ and $z\in\operatorname{T}^{n}$. ∎

######  Lemma 4.7.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a strict Lee–Yang variety of dimension
$c=n-d$. For every $I\in\binom{[n]}{c}$ the projection $\pi_{I}\colon
X\to(\mathbb{P}^{1})^{I}$ satisfies
$\pi_{I}^{-1}(\operatorname{T}^{I})=X(\operatorname{T})$.

###### Proof.

Since $\pi_{I}$ takes $\operatorname{T}^{n}$ to $\operatorname{T}^{I}$, it is
clear that $X(\operatorname{T})\subseteq\pi_{I}^{-1}(\operatorname{T}^{I})$.
We now show the other inclusion. Let $z\in X$ such that
$\pi_{I}(z)\in\operatorname{T}^{I}$ . This means that $\log|z_{i}|=0$ for all
$i\in I$. Because $|I|=n-d$, this implies that
$\operatorname{var}(\log|z|)<d$. Property (2) in 4.1 then implies that $z\in
X(\operatorname{T})$. ∎

######  Proposition 4.8.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a Lee–Yang variety of dimension
$c=n-d$ and denote by $X_{\rm sm}$ the smooth part of $X$. The closure of
$X_{\rm sm}(\operatorname{T})$ is $X(\operatorname{T})$.

###### Proof.

Let $M\in\operatorname{{\mathbf{G}L}}_{n}({\mathbb{Z}})$ such that the right
kernel of the matrix consisting of the first $c$ rows of $M$ is in
$\operatorname{Gr}_{+}(d,n)$. Let $X_{0}^{\prime}$ denote the image of
$X\cap({\mathbb{C}}^{*})^{n}$ under the map $\chi_{M}$. Because $M$ is
invertible over ${\mathbb{Z}}$, the map $\chi_{M}$ is an isomorphism.
Therefore, it suffices to show that the closure of $(X_{0}^{\prime})_{\rm
sm}(\operatorname{T})$ is $X_{0}^{\prime}(\operatorname{T})$. Letting
$X_{1}^{\prime}$ be the closure of $X_{0}^{\prime}$ in $(\mathbb{P}^{1})^{n}$,
we have that
$X_{1}^{\prime}(\operatorname{T})=X_{0}^{\prime}(\operatorname{T})$. Thus, it
suffices to show that the closure of $(X_{1}^{\prime})_{\rm
sm}(\operatorname{T})$ is $X_{1}^{\prime}(\operatorname{T})$. By 4.6 and our
choice of $M$ the projection

$\pi\colon X_{1}^{\prime}\to(\mathbb{P}^{1})^{c}$

onto the first $c$ coordinates satisfies
$\pi^{-1}(\operatorname{T}^{c})=X_{1}^{\prime}(\operatorname{T})$. Let
$Z\subsetneq(\mathbb{P}^{1})^{c}$ be the branch locus of $\pi$. Note that $Z$
contains in particular the image of the singular locus of $X_{1}^{\prime}$
under $\pi$. Let $x\in X_{1}^{\prime}(\operatorname{T})$ and $X_{2}^{\prime}$
an irreducible component of $X_{1}^{\prime}$ containing $x$. It suffices to
show that $x$ is in the closure of
$X_{2}^{\prime}(\operatorname{T})\setminus\pi^{-1}(Z)$. Let $f$ be the
restriction of $\pi$ to $X_{2}^{\prime}$. We have that
$f^{-1}(\operatorname{T}^{c})=X_{2}^{\prime}(\operatorname{T})$. This implies
that elements of $\operatorname{T}^{c}$ have finite fiber under $f$.
Therefore, the map $f$ is generically finite. Because, by equidimensionality,
$\dim(X_{2}^{\prime})=c$ and because $f$ is proper, this implies that $f$ is
surjective. By semicontinuity of fiber dimensions, there exists an open subset
$U\subseteq(\mathbb{P}^{1})^{c}$ which contains $\operatorname{T}^{c}$ such
that every element of $U$ has finite fiber under $f$. Letting $V=f^{-1}(U)$ it
suffices to show that $x$ is in the closure of
$V(\operatorname{T})\setminus\pi^{-1}(Z)$. The restriction of $f$ to $V$ is
finite because it is proper with finite fibers. This implies that every
semialgebraic continuous path $\alpha\colon(0,1)\to U(\operatorname{T})$ with
$\lim_{t\to 0}\alpha(t)=f(x)$ lifts to $\deg(f)$ distinct semialgebraic paths
in $V(\operatorname{T})$. Because there are, counted with multiplicity, at
most $\deg(f)$ many preimages of $f(x)$ under $f$, at least one of these lifts
must converge to $x$. This shows the claim. ∎

###  4.2. Examples

We first consider zero dimensional (strict) Lee–Yang varieties.

######  Lemma 4.9.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a finite subset. The following are
equivalent:

  1. (1)

$X\subseteq\operatorname{T}^{n}$,

  2. (2)

$X$ is a Lee–Yang variety,

  3. (3)

$X$ is a strict Lee–Yang variety.

###### Proof.

This immediately follows from the fact that
$\operatorname{var}(a),\operatorname{\overline{var}}(a)<n$ for every $a$ of
length $n$. ∎

Next we provide an explicit way of constructing strict Lee–Yang varieties of
dimension one and arbitrary high codimension.

######  Definition 4.10.

A _separating curve_ is a smooth irreducible real projective curve $X$ such
that $X({\mathbb{C}})\setminus X({\mathbb{R}})$ has two connected components.

######  Example 4.11.

For example $\mathbb{P}^{1}$ is a separating curve, the two connected
components of
$\mathbb{P}^{1}({\mathbb{C}})\setminus\mathbb{P}^{1}({\mathbb{R}})$ being the
(open) upper and lower half-plane $H_{+}$ and $H_{-}$. More generally, if the
smooth irreducible real projective curve $X$ has genus $g$ and
$X({\mathbb{R}})$ has $g+1$ connected components, then $X$ is separating.

Consider a separating curve $X$ and denote the two connected components of
$X({\mathbb{C}})\setminus X({\mathbb{R}})$ by $X_{+}$ and $X_{-}$. We will
explain how to embed $X$ to $(\mathbb{P}^{1})^{n}$ as a strict Lee–Yang
variety. We say that a non-constant real rational function $f\colon
X\to\mathbb{P}^{1}$ is _separating_ if $f^{-1}(H_{+})=X_{+}$. In this case one
also has $f^{-1}(H_{-})=X_{-}$ and
$f^{-1}({\mathbb{R}}\cup\\{\infty\\})=X({\mathbb{R}})$. It was shown by
Ahlfors [ahlfors]*§4.2 that every separating curve admits a separating
rational function. Several methods for constructing separating functions are
provided in [gabard, KS20b].

######  Proposition 4.12.

Let $f_{1},\ldots,f_{n}$ be separating real rational functions on the
separating curve $X$. Let $\psi\colon X\to(\mathbb{P}^{1})^{n}$ be the
morphism

$f=(f_{1},-f_{2},f_{3},\ldots,(-1)^{n-1}f_{n})\colon X\to(\mathbb{P}^{1})^{n}$

composed with the coordinate-wise Möbius transformation
$z\mapsto\frac{z+i}{z-i}$. If the restriction of $f$ to $X({\mathbb{R}})$ is
injective, then the image $\tilde{X}\subseteq(\mathbb{P}^{1})^{n}$ of $\psi$
is a one-dimensional strict Lee–Yang variety.

###### Proof.

The Möbius transformation $z\mapsto\frac{z+i}{z-i}$ maps the lower open half-
plane to the open unit disc. Because $X$ is separating, so in particular a
real curve, and the $f_{i}$ are real rational functions, this implies that
$\tilde{X}$ is is invariant under the coordinate-wise involution $z\mapsto
1/\overline{z}$. Next we show that for every
$z\in\tilde{X}\setminus\tilde{X}(\operatorname{T})$ we have
$\operatorname{var}(\log|z|)\geq n-1$. Let $z=\psi(x)$ for $x\in X$. Without
loss of generality, assume that $x\in X_{+}$. Then
$\operatorname{Im}(f_{i}(x))>0$ for all $i=1,\ldots,n$ because $f_{i}$ is
separating. This shows that

$\operatorname{var}(\operatorname{Im}(f_{1}(x),-f_{2}(x),f_{3}(x),\ldots,(-1)^{n-1}f_{n}(x)))=n-1$

which implies $\operatorname{var}(\log|z|)=n-1$. It remains to show that
$\tilde{X}(\operatorname{T})$ is contained in the smooth part of $\tilde{X}$.
Letting $X^{\prime}$ be the image of $f$, this is equivalent to
$X^{\prime}({\mathbb{R}})$ being contained in the smooth part of $X^{\prime}$.
We first observe that, because $f_{1}$ is separating, the preimage of
$X^{\prime}({\mathbb{R}})$ under $f$ is $X({\mathbb{R}})$. Therefore, because
$X$ is smooth and $f$ is injective on $X({\mathbb{R}})$, it suffices to show
that the differential of $f$ at every point $x\in X({\mathbb{R}})$ is
injective. This follows because the differential of $f_{1}=\pi_{1}\circ f$ is
injective at every point $x\in X({\mathbb{R}})$ by [KS20a]*Theorem 2.19. ∎

######  Example 4.13.

Let $X=\mathbb{P}^{1}$ and $f_{1},\ldots,f_{n}$ be some real univariate
polynomials of degree one with positive leading coefficients. These are
clearly separating. Then the map $\psi$ from 4.12 is an embedding and
therefore $\tilde{X}$ is a strict Lee–Yang variety in $(\mathbb{P}^{1})^{n}$
of codimension $n-1$. The (closure in $(\mathbb{P}^{1})^{3}$ of the) curve
considered in 1.3 is of this form and thus a strict Lee–Yang variety.

In order to construct separating functions on curves other than
$\mathbb{P}^{1}$, the following criterion turns out to be convenient.

######  Lemma 4.14 ([KS20b]*Lemma 2.10).

Let $f$ be a real non-constant rational function on the separating curve $X$.
Then $f$ or $-f$ is separating if and only if every connected component of

$X({\mathbb{R}})\setminus\\{P\mid f(P)=0\\}$

contains exactly one pole of $f$. In other words, on each connected component
of $X({\mathbb{R}})$ zeros and poles of $f$ _interlace_.

Finally, we note that products of strict Lee–Yang varieties are Lee–Yang
varieties.

######  Example 4.15.

If for each $j=1,\ldots,r$, $X_{j}\subseteq(\mathbb{P}^{1})^{n_{j}}$ is a
strict Lee–Yang variety of codimension $1\leq d_{j}<n_{j}$, then the product
$X=X_{1}\times\cdots\times
X_{r}\subseteq(\mathbb{P}^{1})^{n_{1}+\ldots+n_{r}}$ is a Lee–Yang variety of
codimension $d=\sum_{i=1}^{r}d_{j}$. To see this, note that for any point
$z=(z_{j})_{j}\in X$ where $z_{j}\in X_{j}$,
$\operatorname{\overline{var}}(\log|z_{j}|)\geq d_{j}$. Therefore
$\operatorname{\overline{var}}(\log|z|)\geq\sum_{j=1}^{r}d_{j}=d$. The torus
part $X(\operatorname{T})$ is contained in the smooth part of $X$ but in
general $X$ will not satisfy part (2) of 4.1.

We will see more examples in Section 11.

###  4.3. Transversality

We discuss some transversality properties of Lee–Yang varieties. Throughout
this subsection we fix a Lee–Yang variety $X\subseteq(\mathbb{P}^{1})^{n}$ of
dimension $c=n-d$ and a matrix $L\in{\mathbb{R}}^{n\times d}$ with positive
$d\times d$ minors. Further let

$Y=\\{z\in{\mathbb{C}}^{n}\mid\exp(2\pi iz)\in X\\}$

and $Y({\mathbb{R}})=Y\cap{\mathbb{R}}^{n}$. We further let $X_{\rm sm}$ be
the smooth part of $X$ and $Y_{\rm sm}$ the set of points $z\in Y$ such that
$\exp(2\pi iz)\in X_{\rm sm}$. The following Lemma is a straightforward
consequence of 4.4.

######  Lemma 4.16.

If $x\in{\mathbb{C}}^{d}$ such that $Lx\in Y$, then $x\in{\mathbb{R}}^{d}$.

###### Proof.

The assumption $Lx\in Y$ implies that $\exp(2\pi iLx)\in X$. Now the claim
follows from 4.4. ∎

######  Lemma 4.17.

Let $x\in Y_{\rm sm}({\mathbb{R}})$. The following holds true:

  1. (1)

The tangent space $T_{x}Y\subseteq{\mathbb{C}}^{n}$ is the ${\mathbb{C}}$-span
of $T_{x}Y({\mathbb{R}})$.

  2. (2)

The orthogonal complement of $T_{x}Y({\mathbb{R}})$ is in ${\rm
Gr}_{\geq}(d,n)$.

###### Proof.

Because $X$ is invariant under the coordinate-wise involution $z\mapsto
1/\overline{z}$, it follows that $Y$ is invariant under complex conjugation.
This implies part (1). Now we prove part (2). Consider a non-zero tangent
vector $w\in T_{x}Y({\mathbb{R}})$. By 3.3 we have to show that
$\operatorname{\overline{var}}(w)\geq d$. By part (1) we have $\xi=i\cdot w\in
T_{x}Y$. Let $\gamma\colon[0,1]\to Y$ be a smooth path with $\gamma(0)=x$ and
$\gamma^{\prime}(0)=\xi$. For small $t\in(0,1]$ we have by the definition of
Lee–Yang varieties that

$\operatorname{\overline{var}}(\operatorname{Im}[\frac{1}{t}\left(\gamma(t)-\gamma(0)\right)])=\operatorname{\overline{var}}(\operatorname{Im}[\gamma(t)])\geq
d$

since $\operatorname{Im}[\gamma(0)]=0$ and $\exp(2\pi i\gamma(t))\in X$. Thus

$\operatorname{\overline{var}}(w)=\operatorname{\overline{var}}(\operatorname{Im}[\xi])\geq
d$

since $\gamma^{\prime}(0)=\xi$ and $\operatorname{\overline{var}}$ is upper-
semicontinuous. ∎

######  Lemma 4.18.

Let $X$ be a strict Lee–Yang variety, let $x\in X(\operatorname{T})$ and
$I\in\binom{[n]}{c}$. Then the projection map $\pi_{I}\colon
X\to(\mathbb{P}^{1})^{I}$ is unramified at $x$.

###### Proof.

Let $\psi_{1}\colon(\mathbb{P}^{1})^{n}\to(\mathbb{P}^{1})^{n}$ and
$\psi_{2}\colon(\mathbb{P}^{1})^{I}\to(\mathbb{P}^{1})^{I}$ be the coordinate-
wise Möbius transformations $z\mapsto\frac{z+i}{z-i}$. Let
$\tilde{X}=\psi_{1}^{-1}(X)$ and $\tilde{x}=\psi_{1}^{-1}(x)$. We consider the
morphism

$f=\psi_{2}^{-1}\circ\pi_{I}\circ\psi_{1}\colon\tilde{X}\to(\mathbb{P}^{1})^{I}.$

Note that $f$ is just the projection from $\tilde{X}$ onto the coordinates
from $I$. Because $\psi_{1}$ and $\psi_{2}$ are isomorphisms, it suffices to
show that $f$ is unramified at $\tilde{x}$. Part (1) of 4.1 implies that
$\tilde{X}$ is invariant under $z\mapsto\overline{z}$, namely a real variety
with $\tilde{X}({\mathbb{R}})=\psi_{1}^{-1}(X(\operatorname{T}))$. Here
$\tilde{X}({\mathbb{R}})$ denotes the set of fixed points under
$z\mapsto\overline{z}$, i.e.
$\tilde{X}({\mathbb{R}})=\tilde{X}\cap({\mathbb{R}}\cup\\{\infty\\})^{n}$.
Furthermore, the point $\tilde{x}\in\tilde{X}({\mathbb{R}})$ is a smooth point
of $\tilde{X}$. Now let $y\in\tilde{X}$ such that $f(y)$ is real. Then
$\pi_{I}(\psi_{1}(y))\in\operatorname{T}^{I}$ and 4.7 implies that
$\psi_{1}(y)\in X(\operatorname{T})$ which in turn shows that
$y\in\tilde{X}({\mathbb{R}})$. Thus $f$ is _real-fibered_ in the sense of
[KS20a]*Definition 2.1 and [KS20a]*Theorem 2.19 implies that $f$ is unramified
at $\tilde{x}$. ∎

######  Corollary 4.19.

Let $X$ be a strict Lee–Yang variety and $I\in\binom{[n]}{c}$. Then the
projection map $\pi_{I}\colon Y({\mathbb{R}})\to{\mathbb{R}}^{I}$ is
everywhere unramified.

######  Remark 4.20.

In the case that $X$ is a strict Lee–Yang variety of dimension $c$, we can use
4.19 to define an orientation on the smooth manifold $Y({\mathbb{R}})$. To
this end, for $I\in\binom{[n]}{c}$ let $\pi_{I}\colon
Y({\mathbb{R}})\to{\mathbb{R}}^{I}\cong{\mathbb{R}}^{c}$ be the projection
onto the coordinates from $I$. By 4.19 each $\pi_{I}$ is unramified. This
implies in particular that for every subset $I=\\{i_{1},\ldots,i_{c}\\}$ with
$1\leq i_{1}<\cdots<i_{c}\leq n$ the top-dimensional differential form

$s(I)\cdot dx_{I}=s(I)\cdot dx_{i_{1}}\wedge\cdots\wedge dx_{i_{c}}$

is nowhere vanishing on $Y({\mathbb{R}})$ and thus defines an orientation on
$Y({\mathbb{R}})$. We claim that all these orientations agree. To see this,
consider the tangent space of $Y({\mathbb{R}})$ at some point. We can write
this tangent space as the range of an $n\times c$ matrix $M$ such the
determinant of the first $c$ rows is positive. The differential form $dx_{I}$
evaluates at this basis to the determinant $M_{I}$ of the submatrix of $M$
whose rows are indexed by $I$. Because the determinant of the first $c$ rows
is positive, the columns of $M$ form a reference frame for the orientation
induced by $dx_{[c]}$. Because the orthogonal complement of the range of $M$
is in $\operatorname{Gr}_{\geq}(n-c,n)$ by 4.17, it follows that the sign of
$M_{I}$ is $s(I)$ which implies the claim. In the same way we can define an
orientation on $X(\operatorname{T})$. The measures on $Y({\mathbb{R}})$ and
$X(\operatorname{T})$ defined by these orientations and the differential forms
$dx_{I}$ will play an important role in Section 7.

Let $z\cdot z^{\prime}:=(z_{1}z_{1}^{\prime},\ldots,z_{n}z_{n}^{\prime})$
denote point-wise multiplication in $({\mathbb{C}}^{*})^{n}$.

######  Theorem 4.21.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a Lee–Yang variety of dimension
$c=n-d$ and let $L\in{\mathbb{R}}^{n\times d}$ be a matrix with positive
$d\times d$ minors. We denote by $X_{\rm sm}$ the smooth part of $X$. For
every $z_{0}\in\operatorname{T}^{n}$ the map $x\mapsto z_{0}\cdot\exp(2\pi
iLx)$ is transverse to $X_{\rm sm}$ as a map from ${\mathbb{C}}^{d}$ to
${\mathbb{C}}^{n}$, and is transverse to $X_{\rm sm}(\operatorname{T})$ as a
map from ${\mathbb{R}}^{d}$ to $\operatorname{T}^{n}$.

###### Proof.

Since $z_{0}^{-1}\cdot X=\\{z_{0}^{-1}\cdot z\mid z\in X\\}$ is a Lee–Yang
variety of the same codimension as $X$ by 4.3, it is enough to only consider
$z_{0}=(1,1,\ldots,1)$. For the first claim we need to show that if $x=Ly\in
Y$ for some $y\in{\mathbb{C}}^{d}$ such that $\exp(2\pi ix)$ is a smooth point
of $X$, then the tangent spaces $T_{x}Y$ and
$T_{x}L{\mathbb{C}}^{d}=L{\mathbb{C}}^{d}$ add up to ${\mathbb{C}}^{n}$. 4.16
implies that $y\in{\mathbb{R}}^{d}$ and thus $x\in Y({\mathbb{R}})$. Thus by
part (1) of 4.17 it suffices to show that
$T_{x}Y({\mathbb{R}})+L{\mathbb{R}}^{d}={\mathbb{R}}^{n}$ which then also
implies the second claim. For dimension reasons, this is equivalent to showing
that $T_{x}Y({\mathbb{R}})\cap L{\mathbb{R}}^{d}=\\{0\\}$. But this follows
now directly from 3.4 and part (2) of 4.17. ∎

For the rest of this subsection we assume that $X(\operatorname{T})$ is
contained in the smooth part of $X$. Then 4.21 implies in particular that for
all $y\in{\mathbb{R}}^{n}$ the set

$\Lambda_{y}=\\{x\in{\mathbb{R}}^{d}\mid\exp(2\pi i(Lx+y))\in X\\}$

is discrete. The last goal of this subsection is to bound the Hausdorff
distance of $\Lambda_{y}$ to $\Lambda_{0}$ in terms of $y$. To this end let
$V=L{\mathbb{R}}^{d}\subseteq{\mathbb{R}}^{n}$ and
$V^{\perp}\subseteq{\mathbb{R}}^{n}$ its orthogonal complement. We denote by
$\pi_{1}\colon{\mathbb{R}}^{n}\to V$ and $\pi_{2}\colon{\mathbb{R}}^{n}\to
V^{\perp}$ the orthogonal projections.

######  Lemma 4.22.

Let $K\subseteq V^{\perp}$ be simply connected and let
$Y_{K}=\pi_{2}^{-1}(K)\cap Y({\mathbb{R}})$.

  1. (1)

The set $Y_{K}$ has countably many connected components
$(Y_{i})_{i\in{\mathbb{N}}}$.

  2. (2)

The restriction $\pi_{2}|_{Y_{i}}\colon Y_{i}\to K$ is a homeomorphism for
every $i\in{\mathbb{N}}$.

  3. (3)

The inverse $\psi_{i}\colon K\to Y_{i}$ of $\pi_{2}|_{Y_{i}}\colon Y_{i}\to K$
is continuously differentiable for every $i\in{\mathbb{N}}$.

###### Proof.

By transversality, see 3.4 and part (2) of 4.17, the projection $\pi_{2}\colon
Y({\mathbb{R}})\to V^{\perp}$ is a covering map and using that
$Y({\mathbb{R}})$ is an analytic variety, each fiber, $\pi_{2}^{-1}(w)$ with
$w\in V^{\perp}$, is a zero dimensional semianalytic set, hence discrete and
countable. Therefore, $Y_{K}$ has countably many connected components
$(Y_{j})_{j\in{\mathbb{N}}}$, each homeomorphic to $K$, proving (1) and (2).
The inverse function theorem further implies part (3). ∎

######  Theorem 4.23.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a Lee–Yang variety of dimension
$c=n-d$ such that $X(\operatorname{T})$ is contained in the smooth part of
$X$. Let $L\in{\mathbb{R}}^{n\times d}$ be a matrix with positive $d\times d$
minors. There are differentiable functions

$\varphi_{i}\colon\left[-\frac{1}{2},\frac{1}{2}\right]^{n}\to{\mathbb{R}}^{d}$

for $i\in{\mathbb{N}}$ such that $\Lambda_{y}=\\{\varphi_{i}(y)\mid
i\in{\mathbb{N}}\\}$ for all $y\in\left[-\frac{1}{2},\frac{1}{2}\right]^{n}$.
Moreover, there exists a positive constant $C>0$ such that each $\varphi_{i}$
is $C$-Lipschitz continuous.

###### Proof.

Let $f\colon V\to{\mathbb{R}}^{d}$ be the inverse of ${\mathbb{R}}^{d}\to
V,\,x\mapsto Lx$. Let $K=\pi_{2}([-N,N]^{n})$ for some large enough
$N\in{\mathbb{N}}$ and use the notation from 4.22. By construction we have
that $\psi_{i}(\pi_{2}(y))-y\in V$ for all $y\in[-N,N]^{n}$. Thus the map

$\varphi_{i}\colon[-N,N]^{n}\to{\mathbb{R}}^{d},\,y\mapsto
f(\psi_{i}(\pi_{2}(y))-y)$

is well-defined. By construction these maps are continuously differentiable
and satisfy $\Lambda_{y}=\\{\varphi_{i}(y)\mid i\in{\mathbb{N}}\\}$ for all
$y\in\left[-\frac{1}{2},\frac{1}{2}\right]^{n}$. Since $[-N,N]^{n}$ is
compact, this implies that each $\varphi_{i}$ is $c_{i}$-Lipschitz continuous
for some $c_{i}\geq 0$. In order to find a uniform Lipschitz constant, we let
$m\in{\mathbb{N}}$ such that every $i\in{\mathbb{N}}$ with
$Y_{i}\cap\left[-\frac{1}{2},\frac{1}{2}\right]^{n}\neq\emptyset$ satisfies
$i\leq m$. Then for every $j\in{\mathbb{N}}$ there exists
$k_{j}\in{\mathbb{Z}}^{n}$ and $i\leq m$ such that

$Y_{j}\cap\pi_{2}^{-1}(\pi_{2}(\left[-\frac{1}{2},\frac{1}{2}\right]^{n}))\subseteq
k_{j}+Y_{i}.$

This implies that the Lipschitz constant of
$\varphi_{j}|_{\left[-\frac{1}{2},\frac{1}{2}\right]^{n}}$ can be bounded by
the Lipschitz constant $c_{i}$ of $\varphi_{i}$. In particular, we can choose
$C=\max_{i=1}^{m}(c_{i})$. ∎

######  Corollary 4.24.

There exists a positive constant $C>0$ such that for every
$y\in\left[-\frac{1}{2},\frac{1}{2}\right]^{n}$ the Hausdorff distance of
$\Lambda_{0}$ and $\Lambda_{y}$ can be bounded as follows

$\mathrm{dist}(\Lambda_{0},\Lambda_{y})\leq C\|y\|.$

###### Proof.

This follows directly from 4.23. ∎

###  4.4. Topology of strict Lee–Yang varieties

In this short subsection we determine the topology of strict Lee–Yang
varieties. We will not make use of this later on. For $I\subseteq[n]$ we
denote by

$\pi_{I}\colon(\mathbb{P}^{1})^{n}\to(\mathbb{P}^{1})^{I}\cong(\mathbb{P}^{1})^{|I|}$

the projection onto the coordinates from $I$.

######  Definition 4.25.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a closed subvariety of pure dimension
$c$. The _multidegree_ $d(X)$ of $X$ is the tuple $(d_{[n]\setminus
I})_{I\in\binom{[n]}{c}}$ of nonnegative integers such that $d_{[n]\setminus
I}$ is the cardinality of the fiber of a generic point in $\mathbb{P}^{c}$
under the map $\pi_{I}\colon X\to\mathbb{P}^{c}$.

######  Example 4.26.

If $X\subseteq(\mathbb{P}^{1})^{n}$ is a hypersurface defined by an
$n$-variate polynomial having degree $d_{i}$ in the $i$-th variable, then the
multidegree of $X$ is $(d_{1},\ldots,d_{n})$.

For the rest of the subsection we fix a strict Lee–Yang variety
$X\subseteq(\mathbb{P}^{1})^{n}$ of dimension $c$ with multidegree
$(d_{[n]\setminus I})_{I\in\binom{[n]}{c}}$.

######  Lemma 4.27.

For every $I\in\binom{[n]}{c}$ the projection $\pi_{I}\colon
X(\operatorname{T})\to\operatorname{T}^{c}$ is a covering map with
$d_{[n]\setminus I}$ sheets. In particular, we have $d_{[n]\setminus I}>0$.

###### Proof.

It was shown in 4.18 that $\pi_{I}$ is everywhere unramified and thus it is a
covering map. This implies in particular that $d_{I}>0$. Moreover, by 4.7 the
number of sheets equals $d_{I}$. ∎

By 4.20 and 4.27 we can choose an orientation on $X(\operatorname{T})$ with

(4) $d_{[n]\setminus I}=\frac{s(I)}{(2\pi
i)^{c}}\int_{X(\operatorname{T})}\frac{dz_{i_{1}}\wedge\cdots\wedge
dz_{i_{c}}}{z_{i_{1}}\cdots z_{i_{c}}}$

for all $I=\\{i_{1},\ldots,i_{c}\\}\subseteq[n]$ with $i_{1}<\cdots<i_{c}$.
Using this we can determine the topology of $X(\operatorname{T})$ as a subset
of $\operatorname{T}^{n}$. Namely, 4.27 implies that each connected component
of $X(\operatorname{T})$ is a compact connected covering space of
$\operatorname{T}^{c}$ and thus itself homeomorphic to $\operatorname{T}^{c}$.
In order to the determine the homology class of $X(\operatorname{T})$ in
$H_{c}(\operatorname{T}^{n})$ note that the differential $c$-forms

$\frac{s(I)}{(2\pi i)^{c}}\cdot\frac{dz_{i_{1}}\wedge\cdots\wedge
dz_{i_{c}}}{z_{i_{1}}\cdots z_{i_{c}}}$

are a basis of
$H^{c}(\operatorname{T}^{n})=\operatorname{Hom}(H_{c}(\operatorname{T}^{n}),{\mathbb{Z}})$.
Thus letting $(\beta_{I})_{I\in\binom{[n]}{c}}\subseteq
H_{c}(\operatorname{T}^{n})$ be the dual basis, we obtain

$[X(\operatorname{T})]=\sum_{I\in\binom{[n]}{c}}d_{[n]\setminus
I}\cdot\beta_{I}.$

###  4.5. The complex geometry of strict Lee–Yang varieties

The goal of this section is to prove that certain integrals on strict Lee–Yang
varieties vanish, that will play a role when computing Fourier
transformations. Namely, we will show that for every strict Lee–Yang variety
$X\subseteq(\mathbb{P}^{1})^{n}$ of dimension $c=n-d$

(5) $\int_{X(\operatorname{T})}z^{-k}\cdot\frac{dz_{i_{1}}\wedge\cdots\wedge
dz_{i_{c}}}{z_{i_{1}}\cdots z_{i_{c}}}=0,$

whenever $k\in{\mathbb{Z}}^{n}$ with $\operatorname{var}(k)\geq d$ and
$i_{1},\ldots,i_{c}\in[n]$. We start with the instructive example when the
strict Lee–Yang is isomorphic to $\mathbb{P}^{1}$ and embedded to
$(\mathbb{P}^{1})^{n}$ as in 4.12.

######  Example 4.28.

Let $f_{1},\ldots,f_{n}\colon\mathbb{P}^{1}\to\mathbb{P}^{1}$ be some real
rational functions such that $f_{j}^{-1}(H_{+})=H_{+}$ for $j=1,\ldots,n$
where $H_{+}\subseteq{\mathbb{C}}$ is the upper half-plane. In this case one
also has $f_{j}^{-1}(H_{-})=H_{-}$ and
$f_{j}^{-1}({\mathbb{R}}\cup\\{\infty\\})={\mathbb{R}}\cup\\{\infty\\}$. We
further let $g_{j}$ be the rational function obtained by post-composing
$(-1)^{j+1}\cdot f_{j}$ by the Möbius transformation $z\mapsto\frac{z+i}{z-i}$
which maps the lower half-plane to the open unit disc. This implies for even
$j$ that $g_{j}$ has all its poles in the upper half-plane and all zeros in
the lower half-plane, and vice versa for odd $j$. Let $X$ be the image of the
map

$\mathbb{P}^{1}\to(\mathbb{P}^{1})^{n},\,z\mapsto(g_{1}(z),\ldots,g_{n}(z)).$

In this situation Equation 5 is of the form

(6) $\int_{X(\operatorname{T})}z_{1}^{k_{1}}\cdots
z_{n}^{k_{n}}\cdot\frac{dz_{j}}{z_{j}}=0$

for all $k\in{\mathbb{Z}}^{n}$ with $\operatorname{var}(k)=n-1$ and all
$j=1,\ldots,n$. Here $z_{1},\ldots,z_{n}$ are the coordinates on
$(\mathbb{P}^{1})^{n}$. If we replace $k$ by $-k$, then the resulting integral
will be the complex conjugate of the original integral. Therefore, without
loss of generality, we can assume that $k_{1}\leq 0$. The integral in Equation
6 can be expressed as

$\int_{{\mathbb{R}}}g_{1}^{k_{1}}\cdots
g_{n}^{k_{n}}\cdot\frac{dg_{j}}{g_{j}}.$

The condition $\operatorname{var}(k)=n-1$ implies that $(-1)^{i}k_{i}>0$ for
$i=1,\ldots,n$. Therefore, by our assumption on the location of the zeros and
poles of $g_{i}$, the expression

$g_{1}^{k_{1}}\cdots g_{n}^{k_{n}}\cdot\frac{dg_{j}}{g_{j}}$

has no pole in the closed lower half-plane which shows that the integral in
question vanishes by Cauchy’s integral theorem. We note that even when
$\operatorname{var}(k)<n-1$, the integral can be efficiently computed using
Cauchy’s integral formula.

The proof for the general case of a strict Lee–Yang variety
$X\subseteq(\mathbb{P}^{1})^{n}$ is more technical but it follows the same
line of thoughts as 4.28. Instead of Cauchy’s integral theorem we will use de
Rham’s theorem. To this end, we will realize $X(\operatorname{T})$ as a
submanifold of a smooth complex variety $\tilde{X}$ such that the fundamental
class of $X(\operatorname{T})$ is trivial in homology of $\tilde{X}$. We will
show, by making use of $\operatorname{var}(\log|z|)\geq d$ for all $z\in
X\setminus X(\operatorname{T})$, that the integrands in question are top-
dimensional regular differential forms on $\tilde{X}$ and therefore closed.

First, we need some preparations from algebraic topology and algebraic
geometry. We will use basic notions and results from these areas. As
references we recommend [dold, hatcher, prasolov] and [hartshorne, harris,
mumford] respectively. We start with a lemma from algebraic topology.

######  Proposition 4.29.

Let $f\colon X\to Y$ be a proper (in the classical topology) and surjective
morphism of smooth irreducible quasi-projective varieties of the same
dimension $n=\dim(X)=\dim(Y)$. Further let ${\mathcal{N}}\subseteq Y$ be a
compact connected oriented smooth submanifold of dimension $k$ which is
disjoint from the branch locus of $f$. Then we have

  1. (1)

The preimage ${\mathcal{M}}=f^{-1}({\mathcal{N}})$ is a disjoint union of
finitely many compact connected smooth submanifolds of dimension $k$.

  2. (2)

The restriction $f|_{{\mathcal{M}}}\colon{\mathcal{M}}\to{\mathcal{N}}$ is a
covering map. In particular ${\mathcal{M}}$ has an orientation induced by the
one of ${\mathcal{N}}$.

  3. (3)

If $[{\mathcal{N}}]=0$ in $H_{k}(Y,{\mathbb{C}})$, then $[{\mathcal{M}}]=0$ in
$H_{k}(X,{\mathbb{C}})$.

###### Proof.

First we note that, since $f$ is proper, the set
${\mathcal{M}}=f^{-1}({\mathcal{N}})$ is compact. Let $U\subseteq Y$ the
complement of the branch locus. Then, letting $V=f^{-1}(U)$, we have that
$f|_{V}\colon V\to U$ is a covering map. Since ${\mathcal{N}}\subseteq U$, the
restriction $f_{{\mathcal{M}}}\colon{\mathcal{M}}\to{\mathcal{N}}$ is also a
covering map. This implies $(1)$ and $(2)$. In order to prove $(3)$ we
consider the Poincaré duality isomorphisms for cohomology with compact support
$D_{X}\colon H^{n-k}_{c}(X,{\mathbb{C}})\to H_{k}(X,{\mathbb{C}})$ and
$D_{Y}\colon H^{n-k}_{c}(Y,{\mathbb{C}})\to H_{k}(Y,{\mathbb{C}})$. We let
$\alpha\in H^{n-k}_{c}(X,{\mathbb{C}})$ and $\beta\in
H^{n-k}_{c}(Y,{\mathbb{C}})$ be the preimages of the fundamental classes of
${\mathcal{M}}$ and ${\mathcal{N}}$, respectively. By assumption we have
$\beta=0$ and we need to show that $\alpha=0$. To this end let $\varphi\in
H_{n-k}(X,{\mathbb{C}})$. We first note that $\alpha(\varphi)$ is the
intersection number of $[{\mathcal{M}}]$ with $\varphi$. Now since
${\mathcal{N}}\subseteq U$ and because $f|_{V}\colon V\to U$ is a covering
map, we can compute the intersection number of $[{\mathcal{M}}]$ with
$\varphi$ as the intersection number of $[{\mathcal{N}}]$ with $f_{*}\varphi$.
This shows that $\alpha(\varphi)=\beta(f_{*}\varphi)$ for all $\varphi\in
H_{n-k}(X,{\mathbb{C}})$. Thus $\alpha=f^{*}\beta=0$ and therefore
$[{\mathcal{M}}]=0$. ∎

For the rest of the subsection let $X\subseteq(\mathbb{P}^{1})^{n}$ be a
strict Lee–Yang variety of dimension $c<n$ and let $k\in{\mathbb{Z}}^{n}$ such
that $\operatorname{var}(k)\geq d=n-c$. We can then choose some subset
$J\subseteq[n]$ of $|J|=d+1$ indices on which the entries of $k$ are nonzero
and alternate signs. That is, if the elements of $J$ are $1\leq
j_{0}<\cdots<j_{d}\leq n$, then

$\operatorname{var}(k_{j_{0}},\ldots,k_{j_{d}})=d.$

Let $K=([n]\setminus J)\cup\\{m\\}$ for some $m\in J$ with $k_{m}<0$. (To make
the choice of $K$ depend only on the vector $k$, we can choose $J$ to be the
lexicographically smallest subset among all valid choices and $m$ to be the
minimal $m\in J$ with $k_{m}<0$.) Finally, let

(7) $\pi\colon X\to(\mathbb{P}^{1})^{K}\cong(\mathbb{P}^{1})^{c}$

the projection on the coordinates indexed by $K$. By 4.18 the map $\pi$ is
unramified at $X(\operatorname{T})$. We denote

$B_{{K}}=\\{z\in(\mathbb{P}^{1})^{{K}}\mid|z_{i}|=1\textrm{ for }i\neq
m\textrm{ and }|z_{m}|\leq 1\\}.$

We further denote $B=\pi^{-1}({B_{K}})\subseteq{X}$.

######  Lemma 4.30.

For $z\in B$ either $z\in\operatorname{T}^{n}$ or
$\textnormal{sgn}(\log|z_{j}|)=\textnormal{sgn}(k_{j})$ for all $j\in J$ and
$|z_{i}|=1$ for all $i\not\in J$.

###### Proof.

Let $z\in B$. By construction, $|z_{i}|=1$ for all $j\not\in J$ and
$|z_{m}|\leq 1$. If $|z_{m}|=1$, then $\pi(z)\in\operatorname{T}^{K}$ and so
$z\in\operatorname{T}^{n}$. If $|z_{m}|<1$, then the vector $\log|z|$ is
nonzero. Since $X$ is a strict Lee–Yang variety and $z\in X$, we have
$\operatorname{var}(\log(|z|))\geq d$. Since $\log|z_{j}|=0$ for all $j\not\in
J$ and $|J|=d+1$, we see that the entries $\log|z_{j}|$ for $j\in J$ must be
nonzero and alternate in sign in order to achieve
$\operatorname{var}(\log|z|)\geq d$. Since the entries $(k_{j})_{j\in J}$ also
alternate sign and $\textnormal{sgn}(\log|z_{m}|)=\textnormal{sgn}(k_{m})$, we
see that the signs of $\log|z_{j}|$ and $k_{j}$ must agree for all $j\in J$. ∎

######  Lemma 4.31.

For all $i\in[n]$ the rational differential
$z_{i}^{-k_{i}}\cdot\frac{dz_{i}}{z_{i}}$ is regular on $B$.

###### Proof.

We go through the three cases $k_{i}<0$, $k_{i}>0$ and $k_{i}=0$.

If $k_{i}<0$, then this differential is regular except when $z_{i}=\infty$. In
this case, by 4.30, we have $|z_{i}|\leq 1$ for all $z\in B$ which thus shows
that our differential is regular on $B$.

If $k_{i}>0$, then our differential is regular except when $z_{i}=0$. In this
case, by 4.30, we have $|z_{i}|\geq 1$ for all $z\in B$ which thus shows that
our differential is regular on $B$.

If $k_{i}=0$, then our differential is regular except when $z_{i}=0$ or
$z_{i}=\infty$. In this case we have $i\not\in J$, so by 4.30, we have
$|z_{i}|=1$ for all $z\in B$ which thus shows that our differential is regular
on $B$. ∎

######  Theorem 4.32.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a strict Lee–Yang variety of dimension
$c=n-d$. Let $I=\\{i_{1},\ldots,i_{c}\\}\subseteq[n]$ such that $1\leq
i_{1}<\cdots<i_{c}\leq n$. Then for all $k\in{\mathbb{Z}}^{n}$ with
$\operatorname{var}(k)\geq d$ we have

$\int_{X(\operatorname{T})}z^{-k}\cdot\frac{dz_{i_{1}}\wedge\cdots\wedge
dz_{i_{c}}}{z_{i_{1}}\cdots z_{i_{c}}}=0.$

###### Proof.

We denote $\omega_{I}=\frac{dz_{i_{1}}\wedge\cdots\wedge
dz_{i_{c}}}{z_{i_{1}}\cdots z_{i_{c}}}$. Let $\rho\colon\tilde{X}\to X$ be a
resolution of singularities. This means that $\tilde{X}$ a smooth variety and
$\rho$ is a proper birational surjective morphism which restricts to an
isomorphism on $\rho^{-1}({X}_{\rm sm})$ where ${X}_{\rm sm}$ is the smooth
part of $X$. Such a resolution of singularities exists by [hironaka]*Main
Theorem I. Now let $U\subseteq X$ be the maximal open subset on which the
rational differential $c$-form $z^{-k}\omega_{I}$ is regular and let
$Z={X}\setminus U$ its complement. Let $Y=(\mathbb{P}^{1})^{K}\setminus\pi(Z)$
where $\pi$ is the map from Equation 7. By 4.31 we have

$\operatorname{T}^{K}\subseteq{B_{K}}\subseteq Y$

and letting $\tilde{Y}=(\pi\circ\rho)^{-1}(Y)\subseteq\tilde{X}$ the $c$-form
$z^{-k}\omega_{I}$ is regular on $\tilde{Y}$. We let $f\colon\tilde{Y}\to Y$
the restriction of $\pi\circ\rho$ to $\tilde{Y}$. Since $X(\operatorname{T})$
is contained in the smooth part of $X$ and because
$X(\operatorname{T})=\pi^{-1}(\operatorname{T}^{K})$, we can identify
$X(\operatorname{T})$ with the subset $f^{-1}(\operatorname{T}^{K})$ of
$\tilde{Y}$. Thus we have to prove

$\int_{f^{-1}(\operatorname{T}^{K})}z^{-k}\omega_{I}=0.$

Because $z^{-k}\omega_{I}$ is a holomorphic $c$-form on $\tilde{Y}$ and
$\dim(\tilde{Y})=c$, it is closed. On the other hand, since
$\operatorname{T}^{K}$ is the boundary of ${B_{K}}\subseteq Y$, we have that
$\operatorname{T}^{K}$ is homologous to zero in $Y$ and thus by 4.29 also
$f^{-1}(\operatorname{T}^{K})$ is homologous to zero in $\tilde{Y}$.
Therefore, it follows from Stokes’ direction of de Rham’s theorem that the
integral in question is zero. ∎

##  5\. Delone sets

We remind the reader that a set $A\subseteq{\mathbb{R}}^{d}$ is called
_uniformly discrete_ if there exists $r>0$ such that $\|x-x^{\prime}\|\geq r$
for every pair of distinct points in $A$. It is called _relatively dense_ if
there exists $R>0$ such that $B_{R}(x)\cap A\neq\emptyset$ for every
$x\in{\mathbb{R}}^{d}$. Finally, one says that $A$ is _Delone_ if it is both
both relatively dense and uniformly discrete. The goal of this section is to
prove the following theorem.

######  Theorem 5.1.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a Lee–Yang variety of dimension
$c=n-d$. Let $X_{\rm sm}$ denote its smooth part, and let
$L\in\mathbb{R}^{n\times d}$ with range
$V=L\mathbb{R}^{d}\in\operatorname{Gr}_{+}(d,n)$. Then the following
statements hold:

  1. (1)

If $X(\operatorname{T})\subseteq X_{\rm sm}$, then $\Lambda(X,L)$ is Delone.

  2. (2)

Conversely, if $x\mapsto\exp(2\pi iLx)$ has a dense image in
$\operatorname{T}^{n}$, then $\Lambda(X,L)$ being Delone implies
$X(\operatorname{T})\subseteq X_{\rm sm}$.

######  Remark 5.2.

Notice that if the $d\times d$ minors of $L$ are positive then
$V\in\operatorname{Gr}_{+}(d,n)$ holds (by definition), and if these minors
are ${\mathbb{Q}}$-linearly independent then the assumption that the image of
$x\mapsto\exp(2\pi iLx)$ is dense in $\operatorname{T}^{n}$ holds as well, by
6.3. Finally $X(\operatorname{T})\subseteq X_{\rm sm}$ holds when $X$ is a
strict Lee–Yang variety.

Throughout this section we assume

  * •

The variety $X\subseteq(\mathbb{P}^{1})^{n}$ is a Lee–Yang variety of
dimension $c=n-d$.

  * •

The real matrix $L\in{\mathbb{R}}^{n\times d}$ has range
$V=L{\mathbb{R}}^{d}\subseteq{\mathbb{R}}^{n}$ that lies in
$\operatorname{Gr}_{+}(d,n)$.

Furthermore, as in Section 4.3, we let

$Y=\\{z\in{\mathbb{C}}^{n}\mid\exp(2\pi iz)\in X\\},$

and $Y({\mathbb{R}})=Y\cap{\mathbb{R}}^{n}$. Note that $Y({\mathbb{R}})$ is a
$c$-dimensional real analytic, ${\mathbb{Z}}^{n}$-periodic subvariety of
${\mathbb{R}}^{n}$, and $\Lambda(X,L)$, as defined in Equation 1, can be
written as

$\Lambda(X,L)=\\{x\in{\mathbb{R}}^{d}\mid Lx\in Y({\mathbb{R}})\\}.$

######  Remark 5.3.

Notice that $x\mapsto Lx$ is an isomorphism between ${\mathbb{R}}^{d}$ and $V$
that sends $\Lambda(X,L)$ to $V\cap Y({\mathbb{R}})$.

We prove the two parts of 5.1 in two different subsections.

###  5.1. Proof of 5.1, part (1)

Assuming that $X(\operatorname{T})\subseteq X_{\rm sm}$, we prove that
$\Lambda(X,L)$ is Delone. First, we show that $\Lambda_{y}$ is infinite for
every $y\in{\mathbb{R}}^{n}$, where

$\Lambda_{y}=\\{x\in{\mathbb{R}}^{d}\mid\exp(2\pi i(Lx+y))\in
X\\}=\\{x\in{\mathbb{R}}^{d}\mid Lx+y\in Y({\mathbb{R}})\\}.$

Suppose that $\tilde{L}\in{\mathbb{Q}}^{n\times d}$ has range
$\tilde{V}=\tilde{L}{\mathbb{R}}^{d}\in\text{Gr}_{+}(d,n)$, and let $y\in
Y({\mathbb{R}})$. Then
$\tilde{\Lambda}_{y}=\\{x\in{\mathbb{R}}^{d}\mid\tilde{L}x+y\in
Y({\mathbb{R}})\\}$ contains the lattice $m{\mathbb{Z}}^{n}$, where $m$ is the
smallest common denominator of the $L_{ij}$ since $Y({\mathbb{R}})$ is
${\mathbb{Z}}^{n}$ periodic. Therefore,
$\tilde{V}\cap(Y({\mathbb{R}})-y)=\tilde{L}\tilde{\Lambda}_{y}$ is infinite.
Since $X(\operatorname{T})\subseteq X_{\rm sm}$, 4.21 says that the map
$x\mapsto\exp(2\pi i(Lx+y))$ is transversal to $X$ for every
$y\in{\mathbb{R}}^{n}$ and every $L$ in the open set of matrices in
${\mathbb{R}}^{n\times d}$ with positive $d\times d$ minors. This means that
$(Y({\mathbb{R}})-y)$ is a smooth manifold666in fact a real analytic manifold
that intersects $\tilde{V}$ transversally for every choice of
$(y,\tilde{V})\in{\mathbb{R}}^{n}\times\text{Gr}_{+}(d,n)$. Since
${\mathbb{R}}^{n}\times\text{Gr}_{+}(d,n)$ is connected, and
$\tilde{V}\cap(Y({\mathbb{R}})-y)$ is infinite for some choice of
$(y,\tilde{V})$, it follows that $(\tilde{V}-y)\cap Y({\mathbb{R}})$ is
infinite for all $(y,\tilde{V})\in{\mathbb{R}}^{n}\times\text{Gr}_{+}(d,n)$.
In particular, $V\cap(Y({\mathbb{R}})-y)$ is infinite for all $y$, and it is
also discrete because
$\dim(Y({\mathbb{R}}))+\dim(V-y)=\dim({\mathbb{R}}^{n})$. Since $x\mapsto Lx$
is an isomorphism that sends $\Lambda_{y}$ to $V\cap(Y({\mathbb{R}})-y)$, then
$\Lambda_{y}$ is discrete and infinite.

We now prove that $\Lambda=\Lambda_{0}$ is Delone, by introducing two
functions, which are continuous due to the transversality:

$\displaystyle r_{0}\colon[0,1]^{n}\to{\mathbb{R}}_{\geq 0},$
$\displaystyle\quad r_{0}(y)=\min\\{\|x\|\mid
x\in\Lambda_{y}\\}=\mathrm{dist}(0,\Lambda_{y}),$ $\displaystyle r_{1}\colon
Y\cap[0,1]^{n}\to{\mathbb{R}}_{+},$ $\displaystyle\quad
r_{1}(y)=\min\\{\|x\|\mid
x\in\Lambda_{y}\setminus\\{0\\}\\}=\mathrm{dist}(0,\Lambda_{y}\setminus\\{0\\}).$

Notice that $r_{1}$ is positive and $r_{0}$ is non-negative and is not the
zero function. Since $[0,1]^{n}$ is compact, this means that $r_{1}$ has a
positive minimum and $r_{0}$ has a positive maximum, say $R_{min},R_{max}$
such that $r_{0}(y)\leq R_{max}$ for all $y\in[0,1]$ and $r_{1}(y)\geq
R_{min}$ for all $y\in Y\cap[0,1]^{n}$. Also notice that for every
$x\in{\mathbb{R}}^{d}$, there exists $y_{x}\in[0,1]^{n}$ that satisfies
$y_{x}-Lx=0\mod{1}$ such that $\Lambda-x=\Lambda_{y_{x}}$. This means that for
all $x\in{\mathbb{R}}^{d}$, the distance between $x$ and $\Lambda$ is bounded
by

$\mathrm{dist}(x,\Lambda)=\mathrm{dist}(0,\Lambda_{y_{x}})=r_{0}(y_{x})\leq
R_{\max}.$

We conclude that $\Lambda$ is relatively dense. To see that it is also
uniformly discrete, let $x$ and $x^{\prime}$ be two distinct points in
$\Lambda$, so that

$\|x-x^{\prime}\|\leq\mathrm{dist}(x,\Lambda\setminus\\{x\\})=\mathrm{dist}(0,\Lambda_{y_{x}}\setminus\\{0\\})=r_{1}(y_{x}),$

where $x\in\Lambda$ implies that $y_{x}\in Y\cap[0,1]^{n}$. Hence $\Lambda$ is
uniformly discrete. ∎

###  5.2. Proof of 5.1, part (2)

From here on we assume, in addition to the assumptions made at the beginnign
of this section, that $x\mapsto\exp(2\pi iLx)$ has dense image in
$\operatorname{T}^{n}$, and that $\Lambda(X,L)$ is Delone. We need to show
that $X\subseteq X_{\rm sm}$. We first prove a series of necessary lemmas.

We denote the orthogonal complement of $V$ in ${\mathbb{R}}^{n}$ by
$V^{\perp}$, and define $\pi_{1},\pi_{2}$ to be the projections from
${\mathbb{R}}^{n}=V\oplus V^{\perp}$ onto $V$ and $V^{\perp}$ respectively. We
will also denote by $\pi_{1},\pi_{2}$ the projections from ${\mathbb{C}}^{n}$
onto the ${\mathbb{C}}$-spans $V_{\mathbb{C}}$ and $V_{\mathbb{C}}^{\perp}$ of
$V$ and $V^{\perp}$ respectively.

######  Lemma 5.4.

If $x\in Y$ and $\pi_{2}(x)$ is real, then $x\in Y({\mathbb{R}})$.

###### Proof.

We can write $x=Ly+x_{2}$ in the ${\mathbb{C}}=V_{\mathbb{C}}\oplus
V_{\mathbb{C}}^{\perp}$ decomposition, with $y\in{\mathbb{C}}^{d}$ and
$x_{2}=\pi_{2}(x)$ which is real by assumption. It suffices to show that $y$
is also real. Let $a=\exp(-2\pi ix_{2})$ and
$m_{a}\colon\operatorname{T}^{n}\to\operatorname{T}^{n}$ defined by
$m_{a}(\exp(2\pi iz)):=\exp(2\pi i(z-x_{2}))$. Then $m_{a}(X)$ is a Lee–Yang
variety by 4.3 and $x\in Y$ is equivalent to $\exp(2\pi ix)\in X$, which
implies $\exp(2\pi iLy)\in m_{a}(X)$, and so $y\in{\mathbb{R}}^{d}$ by 4.4. ∎

######  Lemma 5.5.

For $y\in Y_{\rm sm}({\mathbb{R}})$ we have $T_{y}Y({\mathbb{R}})\cap
V=\\{0\\}$ and $T_{y}Y\cap V_{\mathbb{C}}=\\{0\\}$.

###### Proof.

By part (2) of 4.17 the orthogonal complement of $T_{y}Y({\mathbb{R}})$ is in
$\operatorname{Gr}_{\geq}(d,n)$. Thus by 3.4 we have $T_{y}Y({\mathbb{R}})\cap
V=\\{0\\}$. The second claim follows then from part (1) of 4.17. ∎

######  Corollary 5.6.

Let $y\in Y({\mathbb{R}})$ and $U\subseteq{\mathbb{C}}^{n}$ be an open
neighbourhood of $y$. Then $\pi_{2}(U\cap Y)$ contains an open subset of
$V_{\mathbb{C}}^{\perp}$.

###### Proof.

By 4.8, $Y({\mathbb{R}})$ is the closure of $Y_{\rm sm}({\mathbb{R}})$, so
$Y_{\rm sm}({\mathbb{R}})\cap U\neq\emptyset$ and so there is a point
$y^{\prime}\in Y_{\rm sm}({\mathbb{R}})\cap U$. Applying 5.5 to $y^{\prime}$
allows us to use the inverse function theorem and conclude that $\pi_{2}(O\cap
Y)$ is open for some small enough $O\subseteq U$ neighborhood of $y^{\prime}$.
∎

######  Lemma 5.7.

Let $U\subseteq{\mathbb{C}}^{a+b}$ be a connected open subset and $Z$ an
analytic subvariety of $U$. Let
$\pi\colon{\mathbb{C}}^{a+b}\to{\mathbb{C}}^{a}$ be the projection onto the
first $a$ coordinates and assume

(8) $\forall z\in Z\colon\pi(z)\in{\mathbb{R}}^{a}\Rightarrow
z\in{\mathbb{R}}^{a+b}.$

Then, every point $y\in Z\cap{\mathbb{R}}^{a+b}$ has a connected open
neighborhood $U^{\prime}\subseteq U$ such that $\pi(Z\cap U^{\prime})$ is an
analytic subvariety of the open set
$\pi(U^{\prime})\subseteq{\mathbb{C}}^{a}$.

###### Proof.

We prove the claim by induction on $b$. For $b=0$ the claim is trivial. Thus
let $b>0$ and assume that it is true for $b-1$. Let
$\pi^{\prime}\colon{\mathbb{C}}^{a+b}\to{\mathbb{C}}^{a+b-1}$ be the
projection onto the first $a+b-1$ coordinates and
$\pi^{\prime\prime}\colon{\mathbb{C}}^{a+b-1}\to{\mathbb{C}}^{a}$ be the
projection onto the first $a$ coordinates so that
$\pi=\pi^{\prime\prime}\circ\pi^{\prime}$.

Now let $y\in Z\cap{\mathbb{R}}^{a+b}$ be an arbitrary point. We first prove
that there exists an open polydisc $U_{0}\subseteq{\mathbb{C}}^{a+b}$ around
$y$ such that $Z^{\prime}:=\pi^{\prime}(U_{0}\cap Z)$ is an analytic
subvariety of $\pi^{\prime}(U_{0})\subseteq{\mathbb{C}}^{a+b-1}$. To this end,
observe that $y$ lies in the one-dimensional fiber

$G=\\{(\pi^{\prime}(y),\lambda)\in U\mid\lambda\in{\mathbb{C}}\\}.$

The intersection $G\cap Z$ is a complex subvariety of complex dimension at
most one, so its dimension over the reals is either $0$ or $2$. Since $y$ is
real, (8) implies that

$G\cap Z\subseteq\\{(\pi^{\prime}(y),\lambda)\in
U\mid\lambda\in{\mathbb{R}}\\}.$

In particular, this shows that $G\cap Z$ is at most one-dimensional over
${\mathbb{R}}$ which means it must be a zero dimensional analytic variety,
i.e. a discrete set points. By the projection theorem for analytic varieties,
see e.g. [rudin]*14.2.4, there exists an open polydisc
$U_{0}\subseteq{\mathbb{C}}^{a+b}$ around $y$ such that
$Z^{\prime}:=\pi^{\prime}(U_{0}\cap Z)$ is an analytic subvariety of the open
polydisc $\pi^{\prime}(U_{0})\subseteq{\mathbb{C}}^{a+b-1}$.

Next, we will apply the induction hypothesis to the projection
$\pi^{\prime\prime}$ and the analytic subvariety $Z^{\prime}$ of the connected
open subset $\pi^{\prime}(U_{0})$ of ${\mathbb{C}}^{a+b-1}$. To this end, we
need to show that $\pi^{\prime\prime}$ satisfies (8). Thus let $z^{\prime}\in
Z^{\prime}$ such that $\pi^{\prime\prime}(z^{\prime})\in{\mathbb{R}}^{a}$.
There exists $z\in U_{0}\cap Z$ such that $z^{\prime}=\pi^{\prime}(z)$.
Because $\pi$ satisfies (8) and since
$\pi(z)=\pi^{\prime\prime}(z^{\prime})\in{\mathbb{R}}^{a}$, we have
$z\in{\mathbb{R}}^{a+b}$ and thus
$z^{\prime}=\pi^{\prime}(z)\in{\mathbb{R}}^{a+b-1}$. Therefore, by induction
hypothesis there exists a connected open neighborhood
$U_{1}\subseteq\pi^{\prime}(U_{0})$, containing $\pi^{\prime}(y)$, such that
$Z^{\prime\prime}=\pi^{\prime\prime}(Z^{\prime}\cap U_{1})$ is an analytic
subvariety of the open set
$\pi^{\prime\prime}(U_{1})\subseteq{\mathbb{C}}^{a}$. Finally, we consider the
open neighborhood $U^{\prime}=(\pi^{\prime})^{-1}(U_{1})\cap
U_{0}\subseteq{\mathbb{C}}^{a+b}$ of $y$. Because $U_{0}$ is a polydisc, this
set is connected. We have

$\pi(Z\cap
U^{\prime})=\pi^{\prime\prime}(\pi^{\prime}(Z\cap(\pi^{\prime})^{-1}(U_{1})\cap
U_{0}))=\pi^{\prime\prime}(Z^{\prime}\cap U_{1})=Z^{\prime\prime}$

which is an analytic subvariety of
$\pi^{\prime\prime}(U_{1})=\pi(U^{\prime})$. This proves the claim. ∎

######  Proposition 5.8.

Let $y\in Y({\mathbb{R}})$.

  1. (1)

If $U\subseteq{\mathbb{C}}^{n}$ is an open neighbourhood of $y$, then there
exists an open neighbourhood $U^{\prime}\subseteq U$ of $y$ such that
$\pi_{2}(Y\cap U^{\prime})=\pi_{2}(U^{\prime})$.

  2. (2)

If $U\subseteq{\mathbb{R}}^{n}$ is an open neighbourhood of $y$, then there
exists an open neighbourhood $U^{\prime}\subseteq U$ of $y$ such that
$\pi_{2}(Y({\mathbb{R}})\cap U^{\prime})=\pi_{2}(U^{\prime})$.

###### Proof.

Let $U_{1}\subseteq{\mathbb{C}}^{n}$ be a connected open subset containing $y$
and such that $U_{1}\subseteq U$. By 5.4 and 5.7 there exists a connected open
subset $U^{\prime}\subseteq U_{1}$ with $y\in U^{\prime}$ such that
$\pi_{2}(Y\cap U^{\prime})$ is an analytic subvariety of
$\pi_{2}(U^{\prime})$. On the other hand, by 5.6, the set $\pi_{2}(Y\cap
U^{\prime})$ contains an open subset of $V_{\mathbb{C}}^{\perp}$. This shows
that $\pi_{2}(Y\cap U^{\prime})=\pi_{2}(U^{\prime})$. Part (2) follows from
part (1) and 5.4. ∎

######  Proposition 5.9.

There exists an open neighbourhood $B\subseteq V$ of the origin such that for
every $y\in Y({\mathbb{R}})$ we have $(y+B)\cap Y({\mathbb{R}})=\\{y\\}$.

###### Proof.

Because $\Lambda$ is Delone, there exists $r>0$ such that every pair of
distinct points in $\Lambda$ has distance at least $r$. Let $B=LB_{r/2}(0)$ be
the image under $L$ of the open ball of radius $\frac{r}{2}$ around the origin
in ${\mathbb{R}}^{d}$. We claim that $B$ has the desired properties. Assume
for the sake of a contradiction that there exist distinct $y_{1},y_{2}\in
Y({\mathbb{R}})$ such that

$y_{2}\in(y_{1}+B).$

Let $U_{1},U_{2}\subseteq{\mathbb{R}}^{n}$ be open neighbourhoods of $y_{1}$
and $y_{2}$ such that $U_{1}\cap U_{2}=\emptyset$ and $\pi_{1}(U_{i})\subseteq
y_{1}+B$. By 5.8 there exists an open neighbourhood $U_{i}^{\prime}\subseteq
U_{i}$ of $y_{i}$ such that $\pi_{2}(Y({\mathbb{R}})\cap
U_{i}^{\prime})=\pi_{2}(U_{i}^{\prime})$ for $i=1,2$. Consider the open subset
$W=\pi_{2}(U_{1}^{\prime})\cap\pi_{2}(U_{2}^{\prime})$ of $V^{\perp}$. Because
$\pi_{2}(y_{1})=\pi_{2}(y_{2})\in W$ the set $W_{i}=\pi_{2}^{-1}(W)\cap
U_{i}^{\prime}$ is an open neighbourhood of $y_{i}$ in ${\mathbb{R}}^{n}$ for
$i=1,2$. We have $\pi_{2}(W_{i}\cap Y({\mathbb{R}}))=W$. The assumption on $L$
implies that $V+{\mathbb{Z}}^{n}$ is dense in ${\mathbb{R}}^{n}$. Then

$\pi_{2}(V+{\mathbb{Z}}^{n})=\pi_{2}({\mathbb{Z}}^{n})$

is dense in $V^{\perp}$. Thus there exists $m\in{\mathbb{Z}}^{n}$ such that
$\pi_{2}(m)\in W$ and we obtain $y_{i}^{\prime}\in W_{i}\cap Y({\mathbb{R}})$
such that $\pi_{2}(y_{i}^{\prime})=\pi_{2}(m)$, i.e. $y_{i}^{\prime}-m\in V$,
for $i=1,2$. We can therefore write $y_{i}^{\prime}-m=Lx_{i}$ for some
$x_{i}\in{\mathbb{R}}^{d}$. We have

$\exp(2\pi iLx_{i})=\exp(2\pi iy_{i}^{\prime}-2\pi im)=\exp(2\pi
iy_{i}^{\prime})\in X(\operatorname{T}).$

Thus $x_{1},x_{2}\in\Lambda$. Finally, we have

$L(x_{1}-x_{2})=y_{1}^{\prime}-y_{2}^{\prime}=\pi_{1}(y_{1}^{\prime})-\pi_{1}(y_{2}^{\prime}).$

By our choice of the $U_{i}$ and since $W_{i}\subseteq U_{i}$, we have
$y_{1}^{\prime}\neq y_{2}^{\prime}$, which implies $x_{1}\neq x_{2}$, and

$\pi_{1}(y_{1}^{\prime})-\pi_{1}(y_{2}^{\prime})\in B-B\subseteq 2B$

which implies that $\|x_{1}-x_{2}\|<r$ contradicting our choice of $r$. ∎

######  Corollary 5.10.

For every $y\in Y({\mathbb{R}})$ we can find an open neighbourhood
$U\subseteq{\mathbb{C}}^{n}$ of $y$ such that

$\psi\colon U\cap Y\to\pi_{2}(U),\,y\mapsto\pi_{2}(y)$

is surjective and $|\psi^{-1}(x)|=1$ for every real $x\in V^{\perp}$.

###### Proof.

Let $B\subseteq V$ as in 5.9 and let $U_{1}\subseteq{\mathbb{C}}^{n}$ a
neighbourhood of $y$ such that $\pi_{1}(U)\subseteq y+\frac{1}{2}B$. Then
every $x\in V^{\perp}$ has at most one preimage in $U\cap Y({\mathbb{R}})$
under $\pi_{2}$. By 5.4 we even have that such $x\in V^{\perp}$ has at most
one preimage in $U\cap Y$ under $\pi_{2}$. Using 5.8 we can achieve
surjectivity by further shrinking $U$. ∎

We are now in position to prove part (2) of 5.1.

###### Proof of 5.1 part (2).

Assume that $\Lambda(X,L)$ is Delone. We need to show that
$Y({\mathbb{R}})\subseteq Y_{\rm sm}$. Thus let $y\in Y({\mathbb{R}})$ and
assume that $y\not\in Y_{\rm sm}$. By 5.10 we can find an open neighbourhood
$U\subseteq{\mathbb{C}}^{n}$ of $y$ such that

$\psi\colon U\cap Y\to\pi_{2}(U),\,y\mapsto\pi_{2}(y)$

is surjective and $|\psi^{-1}(x)|=1$ for every real $x\in V^{\perp}$. Let
$v\in V^{\perp}$ be a real nonzero vector which is not contained in the image
of the tangent cone at $y$ of the singular locus of $Y$. This ensures that $y$
is an isolated singularity of $Y^{\prime}=\psi^{-1}(G)$ where
$G=\pi_{2}(y)+{\mathbb{C}}v$. Thus by 5.5 the preimage under
$\psi|_{Y^{\prime}}$ of every real point on $G$ in a neighbourhood of $y$,
except for $y$, has cardinality one even when counted with multiplicities.
Moreover, because $y$ a singular point of $Y^{\prime}$, the preimage of $y$
under $\psi|_{Y^{\prime}}$ has higher multiplicity. On the other hand, the map
$\psi|_{Y^{\prime}}$ is finite and flat at $y$. Indeed, it is finite by
[fischer]*Lemma in §3.2 and it is flat because the local ring of $Y^{\prime}$
at $y$ is a finitely generated torsion-free module over the local ring of $G$
at $\pi_{2}(y)$ which is a principal ideal domain, hence the former is free
over the latter. Thus by [fischer]*Corollary to Proposition 3.13 the fiber
cardinality (counting multiplicities) of $\psi|_{Y^{\prime}}$ is constant in a
neighbourhood of $\psi(y)$ contradicting our observation above. ∎

##  6\. Non-periodicity and almost periodicity

In this section we will prove that, under certain conditions, our construction
results in Bohr almost periodic sets that do not contain any periodic set.

######  Definition 6.1.

A Delone set $A\subseteq{\mathbb{R}}^{d}$ is called _Bohr almost periodic_ if
for every $\epsilon>0$ there is a relatively dense set
$T(\epsilon)\subseteq{\mathbb{R}}^{d}$ such that for every $\tau\in
T(\epsilon)$, the Hausdorff distance between $A$ and $A+\tau=\\{x+\tau:x\in
A\\}$ is at most $\epsilon$:

$\mathrm{dist}_{H}(A,A+\tau)\leq\epsilon.$

######  Theorem 6.2.

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a strict Lee–Yang variety of dimension
$c=n-d<n$ and $L$ be a real $n\times d$ matrix all of whose $d\times d$ minors
are positive. The set $\Lambda(X,L)$ is Bohr almost periodic.

###### Proof.

By 4.24 there exists a positive constant $C>0$ such that for every
$y\in\left[-\frac{1}{2},\frac{1}{2}\right]^{n}$ the Hausdorff distance of
$\Lambda$ and

$\Lambda_{y}=\\{x\in{\mathbb{R}}^{d}\mid\exp(2\pi i(Lx+y))\in X\\}$

can be bounded by $\mathrm{dist}(\Lambda_{0},\Lambda_{y})\leq C\|y\|$. Given
$\epsilon>0$ small enough let

$T(\epsilon)=\\{\tau\in{\mathbb{R}}^{d}\mid\mathrm{dist}(L\tau,{\mathbb{Z}}^{n})<\frac{\epsilon}{C}\\}.$

Then for each $\tau\in T(\epsilon)$ there is
$y\in\left[-\frac{1}{2},\frac{1}{2}\right]^{n}$ such that $\exp(2\pi
iy)=\exp(2\pi iL\tau)$ and $C\|y\|\leq\epsilon$. Since

$\Lambda+\tau=\\{x+\tau\mid\exp(2\pi iLx)\in
X(\operatorname{T})\\}=\\{x\mid\exp(2\pi iLx-2\pi iL\tau)\in
X(\operatorname{T})\\}=\Lambda_{-y},$

our choice of $C$ gives $\mathrm{dist}_{H}(\Lambda,\Lambda+\tau)\leq\epsilon$.
To see that $T(\epsilon)$ is relatively dense we will show that it contains a
product $T_{1}(\delta)\times T_{2}(\delta)\times\ldots\times T_{d}(\delta)$
such that each $T_{j}(\delta)\subseteq{\mathbb{R}}$ is relatively dense. Let
$L_{1},\ldots,L_{d}$ be the columns of $L$ and let $\delta>0$ such that
$\mathrm{dist}(L\tau,{\mathbb{Z}}^{n})<\frac{\epsilon}{C}$ whenever
$\mathrm{dist}(\tau_{j}L_{j},{\mathbb{Z}}^{n})<\delta$ for all $j$. Define
$T_{j}(\delta)\subseteq{\mathbb{R}}$ as the set of $\tau_{j}$ for which
$\mathrm{dist}(\tau_{j}L_{j},{\mathbb{Z}}^{n})<\delta$. The fact that
$T_{j}(\delta)$ is a relatively dense set follows from Kronecker’s Theorem,
see [almostperiodic]*Corollary on page 34 for example. ∎

In order to prove non-periodicity statements, we shall assume that the
$d\times d$ minors of $L$ are ${\mathbb{Q}}$-linearly independent. We record
some immediate consequences of this assumption.

######  Lemma 6.3.

If $L\in{\mathbb{R}}^{n\times d}$, with $n>d\geq 1$, such that the $d\times d$
minors of $L$ are ${\mathbb{Q}}$-linearly independent, then the following
holds:

  1. (1)

For every non-zero $v\in L{\mathbb{R}}^{d}$ we have

$\dim_{{\mathbb{Q}}}(v):=\dim(\mathrm{span}_{{\mathbb{Q}}}\\{v_{1},\ldots,v_{n}\\})>n-d.$

Equivalently, $\det(AL)\neq 0$ for every matrix $A\in{\mathbb{Z}}^{d\times n}$
of rank $d$.

  2. (2)

The map ${\mathbb{R}}^{d}\to\operatorname{T}^{n},\,x\mapsto\exp(2\pi iLx)$ is
injective with dense image.

  3. (3)

The map ${\mathbb{Z}}^{n}\to{\mathbb{R}}^{d},\,k\mapsto L^{t}k$ is injective
with dense image.

###### Proof.

For part (1), assume, by means of contradiction, that there exists
$y\in{\mathbb{R}}^{d}$ such that $v=Ly\neq 0$ and $\dim_{{\mathbb{Q}}}(v)\leq
n-d$. This means the entries of $v$ satisfy at least $d$ linear relations
$\langle A_{j},v\rangle=0$ with $A_{j}\in{\mathbb{Z}}^{n}$ and
$(A_{1},\ldots,A_{d})$ linearly independent. Taking the matrix
$A\in{\mathbb{Z}}^{d\times n}$ whose rows are these $A_{j}$’s provides an
integer matrix of rank $d$ such that $Av=ALy=0$. In particular, we have
$\det(AL)=\sum_{I}A_{I}L_{I}=0$, summing over $I\in\binom{[n]}{d}$, by Cauchy-
Binet. Since $A$ has full rank, at least one minor is non-zero $A_{I}\neq 0$,
so this is a ${\mathbb{Q}}$-linear relation on the minors $L_{I}$,
contradicting the assumption.

For parts (2) and (3), the map in (2) is a continuous group homomorphism whose
kernel consists of all $x\in{\mathbb{R}}^{d}$ such that
$v=Lx\in{\mathbb{Z}}^{n}$. Because $\dim_{{\mathbb{Q}}}(v)=1\leq n-d$, part
(1) shows that $v=Lx=0$ and thus $x=0$ since $L$ has full rank. This proves
that the map in (2) is injective. By Pontryagin duality this is equivalent to
the map in (3) having dense image, see e.g. [stroppel]*Proposition 23.2. The
map in (3) being injective directly follows from
$\det(L^{t}A^{t})=\det(AL)\neq 0$ for every matrix $A\in{\mathbb{Z}}^{d\times
n}$ of rank $d$. As before, this is equivalent to the map in (2) having dense
image. ∎

######  Remark 6.4.

Let $S\in{\mathbb{R}}^{d\times d}$ be an invertible matrix. The matrix
$L\in{\mathbb{R}}^{n\times d}$ has ${\mathbb{Q}}$-linearly independent
$d\times d$ minors if and only if $LS$ does.

######  Remark 6.5.

There exist explicit positive parametrizations of the positive Grassmannian
$\operatorname{Gr}_{+}(d,n)$. See e.g. [Postnikov]. For example,
$\operatorname{Gr}_{+}(2,4)$ can be parameterized by
$(w,x,y,z)\in{\mathbb{R}}_{+}^{4}$, defining the corresponding point in
$\operatorname{Gr}_{+}(2,4)$ to be the range of
$L=L(w,x,y,z):=\begin{pmatrix}1&0&-wy&-w\\\ 0&1&wxy+z&wx\end{pmatrix}$. The
$2\times 2$ minors $(L_{I})_{I\in\binom{[4]}{2}}$ are

$L_{\\{1,2\\}}=1,\ L_{\\{1,3\\}}=wxy+z,\ L_{\\{1,4\\}}=wx,\ L_{\\{2,3\\}}=wy,\
L_{\\{2,4\\}}=w,\ L_{\\{3,4\\}}=wz.$

In particular, the minors are positive if and only if
$(w,x,y,z)\in{\mathbb{R}}_{+}^{4}$, and if we take $w,x,y,z$ to be
algebraically independent over ${\mathbb{Q}}$, or even just square-roots of
distinct primes, then these minors will ${\mathbb{Q}}$-linearly independent.

######  Definition 6.6 (Algebraic-torus cosets).

We say that a subgroup $H\subseteq({\mathbb{C}}^{*})^{n}$ is an _algebraic
subtorus of dimension $m\leq n$_ if there is an $n\times m$ matrix with
integer coefficients $A\in{\mathbb{Z}}^{n\times m}$ of rank $m$ such that

$H=\\{\exp(2\pi iAy)\mid y\in{\mathbb{C}}^{m}\\}.$

Given a point $z=\exp(2\pi ix)\in({\mathbb{C}}^{*})^{n}$ the _algebraic-torus
coset_ $zH$ is given by

$zH=\\{\exp(2\pi i(x+Ay)\mid y\in{\mathbb{C}}^{d}\\}.$

######  Theorem 6.7.

Let $X\subseteq({\mathbb{C}}^{*})^{n}$ be an algebraic variety of dimension
$c=n-d$, of total degree $D$, and assume that none of the irreducible
components of $X$ is an algebraic-torus coset in $({\mathbb{C}}^{*})^{n}$. Let
$L\in{\mathbb{R}}^{n\times d}$ with ${\mathbb{Q}}$-linearly independent
$d\times d$ minors. Then $\Lambda(X,L)$ intersects every $m$-dimensional
${\mathbb{Q}}$-vector space $W\subseteq{\mathbb{R}}^{d}$ in at most $r^{m+1}$
points, with $r=e^{(6D\binom{n+D}{D})^{(5D\binom{n+D}{D})}}$

6.7 is a consequence of [Evertse00]*Theorem 1.2, which is Evertse’s proof of a
conjecture by Lang, commonly known as Lang’s $G_{m}$ conjecture:

######  Theorem 6.8 ([Evertse00]*Theorem 1.2).

Let $V\subseteq({\mathbb{C}}^{*})^{n}$ be an algebraic variety of total degree
$D$ and let $G$ be a multiplicative subgroup of $({\mathbb{C}}^{*})^{n}$ of
finite rank $m$. Let $\overline{G}:=\\{z\in({\mathbb{C}}^{*})^{n}\mid\exists
j\in{\mathbb{N}}:~{}z^{j}\in G\\}$ be its division group. Then
$\overline{G}\cap V$ is contained in a union of at most $R$ algebraic-torus
cosets $z_{j}H_{j}\subseteq V$ for $R\leq
e^{(m+1)(6D\binom{n+D}{D})^{(5D\binom{n+D}{D})}}$.

We will need the following lemma.

######  Lemma 6.9.

If $L\in{\mathbb{R}}^{n\times d}$ has ${\mathbb{Q}}$-linearly independent
$d\times d$ minors, and $zH$ is an algebraic-torus coset of dimension $m<n-d$,
then there is at most one point $y\in{\mathbb{R}}^{d}$ with $\exp(2\pi iLy)\in
zH$.

###### Proof.

Assume that there are two points, $y\neq y^{\prime}$, such that both
$\exp(2\pi iLy)\in zH$ and $\exp(2\pi iLy^{\prime})\in zH$, then $\exp(2\pi
iL(y-y^{\prime}))\in H$, that is,

$v=L(y-y^{\prime})=Au+2\pi k$

for some $u\in{\mathbb{R}}^{m}$ and $k\in{\mathbb{Z}}^{n}$, where
$A\in{\mathbb{Z}}^{n\times m}$. It follows that

$\dim_{{\mathbb{Q}}}(v)\leq\dim_{{\mathbb{Q}}}(Au)+1\leq\dim_{{\mathbb{Q}}}(u)+1\leq
m+1\leq n-d$

in contradiction to part (1) of 6.3. ∎

We can now prove 6.7.

###### Proof of 6.7.

Let $\Lambda=\Lambda(X,L)$. Let
$W=\mathrm{span}_{{\mathbb{Q}}}(y_{1},\ldots,y_{m})\subseteq{\mathbb{R}}^{d}$
be an $m$-dimensional ${\mathbb{Q}}$-linear subspace, then $\exp(2\pi
iW)=\bar{G}$ is the division group of the group
$G\subseteq({\mathbb{C}}^{*})^{n}$ generated by $\exp(2\pi
iy_{1}),\ldots,\exp(2\pi iy_{m})$. According to Lang’s $G_{m}$ conjecture
(i.e. 6.8), there are $R=R(W,X)$ algebraic-torus cosets in $X$, say
$z_{j}H_{j}\subseteq X$, such that $y\in W\cap\Lambda\Rightarrow\exp(2\pi
iy)\in\cup_{j=1}^{R}z_{j}H_{j}$. That is
$W\cap\Lambda\subseteq\cup_{j=1}^{M}\Lambda_{j}$ where $\Lambda_{j}$ is the
set of points $y\in\Lambda$ with $\exp(2\pi iy)\in z_{j}H_{j}$. By assumption,
$X$ has dimension $n-d$ and no irreducible component which is an algebraic-
torus cosets, so the algebraic-torus cosets contained in $X$ must have lower
dimension, $\dim(z_{j}H_{j})<n-d$ for all $j$. It follows from 6.9 that each
$\Lambda_{j}$ contains at most one point, so $W\cap\Lambda$ contains at most
$R$ points. 6.8 also provides a bound on $R$ in terms of $m$ and $D$ the total
degree of $X$, $R\leq r^{m+1}$, with
$r=e^{(6D\binom{n+D}{D})^{(5D\binom{n+D}{D})}}$. ∎

######  Corollary 6.10 (Non-periodicity).

Let $X\subseteq(\mathbb{P}^{1})^{n}$ be a strict Lee–Yang variety of dimension
$c=n-d<n$ such that no irreducible component of $X$ is an algebraic-torus
coset. Let $L$ be a real $n\times d$ matrix all of whose $d\times d$ minors
are positive and linearly independent over ${\mathbb{Q}}$. Then the following
holds:

  1. (1)

$\Lambda(X,L)\subseteq{\mathbb{R}}^{d}$ intersects every lattice in at most
$r^{d+1}$ points, and every discrete periodic set in finitely many points.

  2. (2)

$\Lambda(X,L)\subseteq{\mathbb{R}}^{d}$ intersects every set obtained by the
“cut and project” method from a lattice in
${\mathbb{R}}^{d}\times{\mathbb{R}}^{N}$, for some $N\in{\mathbb{N}}$, in at
most $r^{d+N+1}$ points.

Here $r$ is the constant introduced in 6.7.

###### Proof.

The ${\mathbb{Q}}$-span of a lattice in ${\mathbb{R}}^{d}$ has dimension $d$
as a ${\mathbb{Q}}$-vector space. Thus by 6.7 it can intersect $\Lambda(X,L)$
in at most $r^{d+1}$ points. By the same argument $\Lambda(X,L)$ intersects
the translate of a lattice in at most $r^{d+2}$ points. Because every discrete
periodic set is a finite union of translates of lattices, this proves part
(1).

For the same reason, every projection of a lattice from ${\mathbb{R}}^{d+N}$
to ${\mathbb{R}}^{d}$ will have $\dim_{{\mathbb{Q}}}\leq N+d$. Now recall that
a set $A\subseteq{\mathbb{R}}^{d}$ is called a “cut and project” set, if there
is a lattice ${\mathcal{L}}\subseteq{\mathbb{R}}^{d}\times{\mathbb{R}}^{N}$
for some $N\in{\mathbb{N}}$ and a bounded set $O\subseteq{\mathbb{R}}^{N}$
with non-empty interior such that $A=\\{x\in{\mathbb{R}}^{d}\mid\exists y\in
O\ \text{s.t.}\ (x,y)\in{\mathcal{L}}\\}$. This construction implies that
$\dim_{{\mathbb{Q}}}(A)\leq\dim_{{\mathbb{Q}}}({\mathcal{L}})\leq d+N$. Now
the claim follows from 6.7. ∎

##  7\. Proof of 2.3 and 2.6

In this section, we will prove 2.3 and 2.6. We fix a strict Lee–Yang variety
$X\subseteq(\mathbb{P}^{1})^{n}$ of dimension $c=n-d$ and multidegree
$d({X})=(d_{J})_{J\in\binom{[n]}{d}}$ and a matrix $L\in{\mathbb{R}}^{n\times
d}$ with positive $d\times d$ minors. We let $\Lambda=\Lambda(X,L)$. As in
Section 4.3 we consider

$Y({\mathbb{R}})=\\{y\in{\mathbb{R}}^{n}\mid\exp(2\pi iy)\in X\\}$

which is a ${\mathbb{Z}}^{n}$-periodic real analytic manifold of dimension
$c=n-d$.

######  Definition 7.1.

($m_{I}$) For any $I=\\{i_{1},\ldots,i_{c}\\}$, $1\leq i_{1}<\cdots<i_{c}\leq
n$, define the (periodic) measure $m_{I}$ on $Y({\mathbb{R}})$ given by the
$c$-dimensional volume in the $I$ coordinates
$dm_{I}(x)=|dx_{i_{1}}\wedge\ldots\wedge dx_{i_{c}}|$, and its Fourier
transform $\widehat{m}_{I}\colon{\mathbb{Z}}^{n}\to{\mathbb{C}}$,

$\widehat{m}_{I}(k)=\int_{Y({\mathbb{R}})\cap[0,1]^{n}}e^{-2\pi i\langle
x,k\rangle}dm_{I}(x).$

By 4.20 there is an orientation of $Y({\mathbb{R}})$ such that the measure
$m_{I}$ can be computed by integrating against the differential form
$s(I)\cdot dx_{i_{1}}\wedge\cdots\wedge dx_{i_{c}}$. Moreover, the Fourier
transform $\widehat{m}_{I}(k)$ can be expressed as

(9) $\widehat{m}_{I}(k)=\int_{Y({\mathbb{R}})\cap[0,1]^{n}}e^{-2\pi i\langle
x,k\rangle}dm_{I}(x)=\frac{s(I)}{(2\pi
i)^{c}}\int_{X(\operatorname{T})}z^{-k}\frac{dz_{i_{1}}\wedge\ldots\wedge
dz_{i_{c}}}{z_{i_{1}}\cdot\ldots\cdot z_{i_{c}}}$

with the orientation on $X(\operatorname{T})$ as in 4.20.

######  Lemma 7.2.

The support of $\widehat{m}_{I}$ is contained in
${\mathbb{Z}}^{n}_{\operatorname{var}<d}=\\{k\in{\mathbb{Z}}^{n}\mid\operatorname{var}(k)<d\\}$,
and $\widehat{m}_{I}(0)=d_{[n]\setminus I}$.

###### Proof.

By Equation 9 the claim follows from 4.32 and Equation 4. ∎

######  Definition 7.3.

For every Borel set $A\subseteq Y({\mathbb{R}})$ we define:

$m_{L}(A):=\lim_{\epsilon\to
0}\frac{1}{\epsilon^{d}}\operatorname{Vol}_{n}\left(A+\epsilon LQ\right)$

where $\mathrm{Vol}_{n}$ is the Lebesgue measure in ${\mathbb{R}}^{n}$,
$Q=[0,1]^{d}\subseteq{\mathbb{R}}^{d}$ and $A+\epsilon LQ$ is the Minkowski
sum of $A$ and $\epsilon LQ$ which is the re-scaled image of $Q$ under $L$.

######  Lemma 7.4.

The measure $m_{L}$ depends linearly on the $d\times d$ minors of $L$,

$m_{L}=\sum_{I\in\binom{[n]}{d}}L_{I}\cdot m_{[n]\setminus I}.$

In particular,
$\widehat{m}_{L}(k)=\sum_{I\in{\binom{[n]}{d}}}L_{I}\cdot\widehat{m}_{[n]\setminus
I}(k)$ is supported inside ${\mathbb{Z}}^{n}_{\operatorname{var}<d}$, namely
$\widehat{m}_{L}(k)=0$ when $\operatorname{var}(k)\geq d$. Moreover, it is
uniformly bounded by the zero coefficient

$|\widehat{m}_{L}(k)|\leq\widehat{m}_{L}(0)=\sum_{I\in{\binom{[n]}{d}}}L_{I}\cdot
d_{I}.$

###### Proof.

If we assume that $m_{L}=\sum_{I\in\binom{[n]}{d}}L_{I}m_{[n]\setminus I}$
then we can use the triangle inequality to get $|m_{[n]\setminus
I}(k)|\leq\int_{Y({\mathbb{R}})\cap[0,1]^{n}}dm_{[n]\setminus
I}=\widehat{m}_{[n]\setminus I}(0)=d_{I}$ where we used 7.2 in the last
equality, from which we get

$|\widehat{m}_{L}(k)|\leq\widehat{m}_{L}(0)=\sum_{I\in{\binom{[n]}{d}}}L_{I}d_{I}.$

To show that $m_{L}=\sum_{I\in\binom{[n]}{d}}L_{I}m_{[n]\setminus I}$, we need
to calculate the density $dm_{L}(x_{0})$ at every $x_{0}\in Y({\mathbb{R}})$.
Given $I=\\{i_{1},\ldots,i_{c}\\}$ with $0<i_{1}<\ldots<i_{c}\leq n$ consider
the $c$-form $dx_{I}=dx_{i_{1}}\wedge\ldots,dx_{i_{c}}$ and the sign
$s(I)=(-1)^{\sum_{i\in I}i-\sum_{i\notin I}i}$ so that

$dm_{I}(x)=s(I)dx_{I}.$

Let $A$ an open neighborhood of $x_{0}\in Y({\mathbb{R}})$ such that there
exists an open subset $O\subseteq{\mathbb{R}}^{c}$ and a smooth map

$\varphi\colon O\to Y({\mathbb{R}})$

parametrizing $A$ such that $\varphi(0)=x_{0}$. We denote the standard
coordinates on ${\mathbb{R}}^{c}$ by $y=(y_{1},\ldots,y_{c})$. The tangent
space $T_{x_{0}}Y({\mathbb{R}})$ is transversal to the image of $L$ by 4.17.
Thus, after shrinking $A$ if necessary we can parameterize $A+\epsilon LQ$ by

$\psi\colon O\times(\epsilon Q)\to A+\epsilon LQ,\,(y,z)\mapsto\varphi(y)+Lz$

where $z=(z_{1},\ldots,z_{d})$ are the standard coordinates on
${\mathbb{R}}^{d}$. Using the notations $dy=dy_{1}\wedge\ldots\wedge dy_{c}$
and $dz=dz_{1}\wedge\ldots\wedge dz_{d}$ we can calculate the Jacobian of
$\psi$,

$\displaystyle\det(D\psi(y,z))dy\wedge dz=$
$\displaystyle\det\begin{pmatrix}D\varphi(y)&L\end{pmatrix}dy\wedge dz$
$\displaystyle=$ $\displaystyle\sum_{I\in{\binom{[n]}{c}}}s([n]\setminus
I)L_{[n]\setminus
I}\det\left(\frac{\partial(\varphi_{i_{1}},\ldots,\varphi_{i_{c}})}{\partial(y_{1},\ldots,y_{c})}\right)dy\wedge
dz$ $\displaystyle=$ $\displaystyle\sum_{I\in{\binom{[n]}{c}}}s([n]\setminus
I)L_{[n]\setminus I}dx_{I}\wedge dz,$

where in the second line we use the Laplace expansion along the rows
corresponding to $[c]$, and in the last line we used the chain rule for forms.
Since the Jacobian is independent of $z$, by taking $\epsilon>0$ small enough
so that $\psi$ is injective we get

$\displaystyle\epsilon^{-d}\operatorname{Vol}_{n}(A+\epsilon LQ)$
$\displaystyle=\epsilon^{-d}\int_{z\in\epsilon Q}\int_{x\in
A}\sum_{I\in{\binom{[n]}{c}}}s([n]\setminus I)L_{[n]\setminus I}dx_{I}\wedge
dz$ $\displaystyle=$ $\displaystyle\int_{x\in
A}\sum_{I\in{\binom{[n]}{c}}}s([n]\setminus I)L_{[n]\setminus
I}dx_{I}=\sum_{I\in{\binom{[n]}{c}}}L_{[n]\setminus I}m_{I}(A).\qed$

Let us now prove that the needed summation formula holds, from which we will
be able to conclude 2.3 and 2.6.

######  Proposition 7.5 (Summation formula).

For every $f\in\mathcal{S}({\mathbb{R}}^{d})$,

$\sum_{x\in\Lambda}\hat{f}(x)=\sum_{k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d}}\widehat{m}_{L}(k)f(L^{t}k).$

###### Proof.

5.1 shows that $\Lambda$ is Delone, so the left sum is absolutely converging.
Using 3.6, 3.5, and $|\widehat{m}_{L}(k)|\leq\widehat{m}_{L}(0)$, the right
sum is also absolutely converging777We elaborate on that and show that
$\sum_{k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d};|L^{t}k|<R}|\widehat{m}_{L}(k)|=O(R^{n})$
in the proof of 2.6 which follows. To prove the summation formula we
approximate $m_{L}$ and the counting measure of $\Lambda$ with smooth
compactly supported functions. We consider $m_{L}$ as a singular measure on
${\mathbb{R}}^{n}$ that is supported on $Y({\mathbb{R}})$, i.e.
$m_{L}(A):=m_{L}(A\cap Y({\mathbb{R}}))$ for every Borel set
$A\subseteq{\mathbb{R}}^{n}$. For small enough $\epsilon>0$ let
$g_{\epsilon}\colon{\mathbb{R}}^{n}\to[0,\epsilon^{-d}]$ be a smooth
${\mathbb{Z}}^{n}$-periodic function supported in an $\epsilon^{2}$
neighborhood of $Y({\mathbb{R}})+\epsilon LQ$ that is equal to $\epsilon^{-d}$
on $Y({\mathbb{R}})+\epsilon LQ$. Then $g_{\epsilon}d\operatorname{Vol}_{n}$
converges in the vague topology to $m_{L}$, i.e. for every continuous and
compactly supported $h\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$,

$\lim_{\epsilon\to
0}\int_{{\mathbb{R}}^{n}}hg_{\epsilon}d\operatorname{Vol}_{n}=\int_{Y({\mathbb{R}})}hdm_{L}.$

Since $g_{\epsilon}$ is ${\mathbb{Z}}^{n}$-periodic it has a Fourier series
$g_{\epsilon}(y)=\sum_{k\in{\mathbb{Z}}^{n}}\hat{g}_{\epsilon}(k)e^{2\pi
i\langle k,y\rangle}$ with $\hat{g}_{\epsilon}(k)=\int_{[0,1]^{n}}e^{-2\pi
i\langle x,k\rangle}g_{\epsilon}(x)d\operatorname{Vol}_{n}$, and since
$g_{\epsilon}$ is smooth, $\hat{g}_{\epsilon}(k)$ is fast decaying as
$|k|\to\infty$ for fixed $\epsilon$. On the other hand, for every fixed
$k\in{\mathbb{Z}}^{n}$,

(10) $\lim_{\epsilon\to 0}\hat{g}_{\epsilon}(k)=\widehat{m}_{L}(k).$

Similarly, the function $x\mapsto g_{\epsilon}(Lx)$ is supported in an
$O(\epsilon^{2})$ neighborhood of $\Lambda+\epsilon Q$ and is equal to
$\epsilon^{-d}=\operatorname{Vol}_{d}(\epsilon Q)$ on $\Lambda+\epsilon Q$. So
for every $f\in\mathcal{S}({\mathbb{R}}^{d})$,

$\lim_{\epsilon\to
0}\int_{{\mathbb{R}}^{d}}f(x)g_{\epsilon}(Lx)dx=\sum_{x\in\Lambda}f(x).$

The Fourier series of $x\mapsto g_{\epsilon}(Lx)$ is induced from that of
$g_{\epsilon}$,

$g_{\epsilon}(Lx)=\sum_{k\in{\mathbb{Z}}^{n}}\hat{g}_{\epsilon}(k)e^{2\pi
i\langle k,Lx\rangle}=\sum_{k\in{\mathbb{Z}}^{n}}\hat{g}_{\epsilon}(k)e^{2\pi
i\langle L^{t}k,x\rangle}.$

For fixed $\epsilon>0$ and for every $f\in{\mathcal{S}}({\mathbb{R}}^{d})$ we
have

$\displaystyle\int_{{\mathbb{R}}^{d}}\hat{f}(x)g_{\epsilon}(Lx)dx=$
$\displaystyle\int_{{\mathbb{R}}^{d}}\hat{f}(x)\sum_{k\in{\mathbb{Z}}^{n}}\hat{g}_{\epsilon}(k)e^{2\pi
i\langle L^{t}k,x\rangle}dx$ $\displaystyle=$
$\displaystyle\sum_{k\in{\mathbb{Z}}^{n}}\hat{g}_{\epsilon}(k)\int_{{\mathbb{R}}^{d}}\hat{f}(x)e^{2\pi
i\langle L^{t}k,x\rangle}dx$ $\displaystyle=$
$\displaystyle\sum_{k\in{\mathbb{Z}}^{n}}\hat{g}_{\epsilon}(k)f(L^{t}k),$

where we used that $f\in{\mathcal{S}}({\mathbb{R}}^{d})$ and
$\hat{g}_{\epsilon}(k)$ is fast decaying in $k$ to interchange summation and
integration. Using the dominant converging theorem, and $\lim_{\epsilon\to
0}\hat{g}_{\epsilon}(k)=\widehat{m}_{L}(k)$, we can take the limit,

$\sum_{k\in{\mathbb{Z}}^{n}}\widehat{m}_{L}(k)f(L^{t}k)=\lim_{\epsilon\to
0}\sum_{k\in{\mathbb{Z}}^{n}}\hat{g}_{\epsilon}(k)f(L^{t}k)=\lim_{\epsilon\to
0}\int_{{\mathbb{R}}^{d}}\hat{f}(x)g_{\epsilon}(Lx)dx=\sum_{x\in\Lambda}\hat{f}(x).\qed$

######  Remark 7.6.

It is worth noting that $\widehat{m}_{L}$ is in
$\ell_{\infty}({\mathbb{Z}}^{n})$ but, as seen in the proof above, not in
$\ell_{2}({\mathbb{Z}}^{n})$. By Parseval’s theorem,
$\sum_{k\in{\mathbb{Z}}^{n}}|\hat{g}_{\epsilon}(k)|^{2}=\int_{[0,1]^{n}}|g_{\epsilon}|^{2}d\operatorname{Vol}_{n}\approx\epsilon^{-d}\widehat{m}_{L}(0)\to\infty$
as $\epsilon\to 0$.

We are now in position to prove 2.3 and 2.6

###### Proof of 2.3.

The set $\Lambda$ is real by 4.4. The set $\Lambda$ is Delone by 5.1. 3.6
states that the set

(11) $\Lambda^{\prime}=\\{L^{t}k\mid k\in{\mathbb{Z}}^{n},\
\operatorname{var}(k)<d\\}$

is discrete. Moreover, 3.6 shows that, for every $\xi\in\Lambda^{\prime}$, the
set of all $k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d}$ with $L^{t}k=\xi$ is
finite. Therefore, we can define the coefficient

(12) $c_{\xi}=\sum_{k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d},\
L^{t}k=\xi}\widehat{m}_{L}(k)$

for every $\xi\in\Lambda^{\prime}$. Now 7.5 implies that for every Schwartz
function $f\in\mathcal{S}({\mathbb{R}}^{d})$, we have

$\sum_{x\in\Lambda}\hat{f}(x)=\sum_{\xi\in\Lambda^{\prime}}c_{\xi}f(\xi).$

This proves that $\Lambda$ has property (1) of 1.1 of a Fourier quasicrystal.
It remains to prove the polynomial growth bounds from part (2) of 1.1. Because
$\Lambda$ is Delone, we have $|\Lambda\cap B_{R}(0)|=O(R^{d})$. Furthermore,
using that $|\widehat{m}_{L}(k)|\leq\int|e^{-2\pi i\langle
x,k\rangle}|dm_{L}(x)=\widehat{m}_{L}(0)$, 3.6 gives

$\sum_{\xi\in\Lambda^{\prime}\cap
B_{R}(0)}|c_{\xi}|\leq\widehat{m}_{L}(0)\cdot|\\{k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d}\mid|L^{t}k|<R\\}|=O(R^{n}).$

This proves that $\Lambda$ is a Fourier quasicrystal with spectrum contained
in $\Lambda^{\prime}$. ∎

###### Proof of 2.6.

The set $\Lambda$ is Bohr almost periodic by 6.2. The set
$\Lambda^{\prime}\subseteq{\mathbb{R}}^{d}$ and the coefficients
$c_{\xi}\in{\mathbb{C}}$ for $\xi\in\Lambda^{\prime}$ are given as above in
Equation 11 and Equation 12, respectively. 3.5 implies that $k=0$ is the only
solution to $L^{t}k=0$ with $k\in{\mathbb{Z}}^{n}_{\operatorname{var}<d}$, so
$c_{0}=\widehat{m}_{L}(0)$. Therefore, 7.4 shows that

$c_{0}=\sum_{I\in{\binom{[n]}{d}}}L_{I}\cdot d_{I}.$

The remaining statement of part (1) in 2.6, namely that $c_{0}$ is the density
of $\Lambda$, will be shown in 10.1 using that $\Lambda$ is a Fourier
quasicrystal by 2.3. For part (2) note that by definition of
$\Lambda^{\prime}$ it is clear that $\dim_{{\mathbb{Q}}}(\Lambda^{\prime})\leq
n$. The statement on its growth rate follows from 3.6. For part (3) we let
$c_{L,k}=\widehat{m}_{L}(k)$ for every $k\in{\mathbb{Z}}^{n}$. Equation 12 and
7.4 imply the formula for $c_{\xi}$; the sum in question being finite was
already observed in the proof of 2.3 above. The stated dependence of $c_{L,k}$
on $L$ also follows 7.4. Finally

$|c_{L,k}|=|\widehat{m}_{L}(k)|\leq\int|e^{-2\pi i\langle
x,k\rangle}|dm_{L}(x)=\widehat{m}_{L}(0)=c_{0}$

gives the desired bound. ∎

##  8\. Reduction to strict Lee–Yang varieties

This section is motivated by [AlonCohenVinzant], whose main result can be
stated as follows: If $X\subseteq({\mathbb{C}}^{*})^{n}$ is an algebraic
hypersurface, i.e. of pure codimension one, and $\ell\in{\mathbb{R}}^{n}$ such
that $\Lambda(X,\ell)$ is real, then there is a Lee–Yang hypersurface
$\tilde{X}$ and a vector $\tilde{\ell}$ with positive entries, such that
$\Lambda(X,\ell)=\Lambda(\tilde{X},\tilde{\ell})$. The main result of this
section, 8.4, is of similar flavor. Namely, it gives sufficient criteria on an
algebraic variety $X$ and a matrix $L$ under which there is a strict Lee–Yang
variety $\tilde{X}$ and a matrix $\tilde{L}$ with positive maximal minors such
that $\Lambda(X,L)=\Lambda(\tilde{X},\tilde{L})$. Before we formulate the
precise statement, we prove some preparatory lemmas.

######  Lemma 8.1.

Let $U\subseteq{\operatorname{Gr}}(d,n)$ be a nonempty open subset. Let
$L\in{\mathbb{R}}^{n\times d}$ be a real $n\times d$ matrix whose range is in
$U$, $L{\mathbb{R}}^{d}\in U$. There exists a non-singular $n\times n$ matrix
$S$ with entries in ${\mathbb{Z}}$ and the following two properties:

  1. (1)

The range of $SL$ is in $\operatorname{Gr}_{+}(d,n)$.

  2. (2)

The range of $S^{-1}L^{\prime}$ is in $U$ for every $n\times d$ matrix
$L^{\prime}$ whose range is in $\operatorname{Gr}_{\geq}(d,n)$.

###### Proof.

If a matrix $S$ satisfies (1) and (2), then every multiple of $S$ by a non-
zero scalar also satisfies (1) and (2). Thus, it suffices to find such a
matrix $S$ with rational entries. Furthermore, since (1) and (2) are open
conditions on $S$, it even suffices to find such a matrix $S$ with entries in
${\mathbb{R}}$. Throughout the proof we will consider $\operatorname{Gr}(d,n)$
as a subset of $\mathbb{P}(\wedge^{d}{\mathbb{R}}^{n})$ via the Plücker
embedding. Since $\operatorname{{\mathbf{G}L}}_{n}({\mathbb{R}})$ acts
transitively on $\operatorname{Gr}(d,n)$, we can assume without loss of
generality that the range of $L$ is spanned by $e_{1},\ldots,e_{d}$. We
observe that it suffices to prove the claim for any open neighbourhood of the
range of $L$ that is contained in $U$. Hence without loss of generality, we
can assume that $U$ is contained in the open affine chart of
$\mathbb{P}(\wedge^{d}{\mathbb{R}}^{n})$ where the Plücker coordinate
corresponding to $[d]$ is not zero. This chart is naturally homeomorphic to
${\mathbb{R}}^{N}$ where $N=\binom{n}{d}-1$ and the range of $L$ corresponds
to the origin of ${\mathbb{R}}^{N}$. Thus by further shrinking $U$ we can
assume that $U$ is the intersection of $\operatorname{Gr}(d,n)$ with an open
$\epsilon$-ball $B$ around the origin in this chart. Under the natural
projection

$\wedge^{d}{\mathbb{R}}^{n}\setminus\\{0\\}\to\mathbb{P}(\wedge^{d}{\mathbb{R}}^{n})$

the ball $B$ is the image of the open convex cone

$C=\left\\{\sum_{I\in\binom{[n]}{d}}\lambda_{I}e_{I}\mid\sum_{I\neq[d]}\lambda_{I}^{2}<\epsilon\cdot\lambda_{[d]}^{2},\,\lambda_{[d]}>0\right\\}\subseteq\wedge^{d}{\mathbb{R}}^{n}$

where $e_{I}=e_{i_{1}}\wedge\cdots\wedge e_{i_{d}}$ if
$I=\\{i_{1},\ldots,i_{d}\\}$ and $1\leq i_{1}<\cdots<i_{d}\leq n$.

Let $A$ be a $d\times n$ matrix all of whose $d\times d$ minors are positive.
Let $A_{1}$ and $A_{2}$ be the $d\times(n-d)$ and $d\times d$ matrices such
that $A=\left(\begin{array}[]{@{}c|c@{}}A_{1}&A_{2}\end{array}\right)$, and
define the $n\times n$ matrix $T_{\nu}$, in the block decomposition
${\mathbb{R}}^{n}={\mathbb{R}}^{n-d}\times{\mathbb{R}}^{d}$, as follows

$T_{\nu}=\left(\begin{array}[]{@{}c|c@{}}A_{1}&A_{2}\\\ \hline\cr 0_{n-d\times
d}&\nu\cdot I_{n-d}\end{array}\right),$

where $I_{n-d}$ is the $(n-d)\times(n-d)$ identity matrix and $0_{n-d\times
d}$ the $n-d\times d$ zero matrix. Clearly, for $\nu\neq 0$, the $n\times n$
matrix $T_{\nu}$ is invertible. On the other hand, for every
$I\in\binom{[n]}{d}$ the linear subspace spanned by all $T_{0}(e_{i})$ for
$i\in I$ is just the range of $L$ and therefore in $B$ for every
$I\in\binom{[n]}{d}$. Hence for small enough $\nu>0$ the linear subspace
spanned by all $T_{\nu}(e_{i})$ for $i\in I$ is also in $B$. By our positivity
assumption on $A$, we even have that the induced linear map

$\wedge^{d}T_{\nu}\colon\wedge^{d}{\mathbb{R}}^{n}\to\wedge^{d}{\mathbb{R}}^{n}$

sends $e_{I}$ to $C$ for all $I\in\binom{[n]}{d}$. As this is an open
condition, it is also satisfied by

$S=\left(\begin{array}[]{@{}c|c@{}}A_{1}&A_{2}\\\ \hline\cr R&\nu\cdot
I_{n-d}\end{array}\right),$

where $R$ is an $(n-d)\times d$ matrix sufficiently close to the zero matrix
so that the span of the first $d$ columns of $S$ is in
$\operatorname{Gr}_{+}(d,n)$. Such a matrix exists because the closure of
$\operatorname{Gr}_{+}(d,n)$ is $\operatorname{Gr}_{\geq}(d,n)$ [Karp]*Theorem
3.15(ii). Since $C$ is a convex cone, every nontrivial nonnegative linear
combination of the $e_{I}$ is also mapped to $C$ by $\wedge^{d}S$. This
implies in particular that $\operatorname{Gr}_{\geq}(d,n)$ is mapped to a
subset of $U$. Hence the range of $SL$ is in $\operatorname{Gr}_{+}(d,n)$ and
$S^{-1}L^{\prime}\in U$ for every $d\times n$ matrix whose range is in
$\operatorname{Gr}_{\geq}(d,n)$. ∎

We apply such coordinate changes to the _amoeba_ of the variety $X$. Formally,
the _amoeba_ of an algebraic variety $X\subseteq({\mathbb{C}}^{*})^{n}$,
denoted ${\mathcal{A}}(X)$, is the image of $X$ under the map
$({\mathbb{C}}^{*})^{n}\to{\mathbb{R}}^{n}$ taking $(z_{1},\ldots,z_{n})$ to
$(\log|z_{1}|,\ldots,\log|z_{n}|)$. That is,
${\mathcal{A}}(X)=\left\\{a\in{\mathbb{R}}^{n}\mid\exists
b\in{\mathbb{R}}^{n}\ \text{s.t.}\ \exp(a+ib)\in X\right\\}$. See [GKZ,
MikhalkinAmoeba] for more detail.

######  Lemma 8.2.

Let $X\subseteq({\mathbb{C}}^{*})^{n}$ be a closed equidimensional algebraic
subvariety of dimension $c=n-d$ which is invariant under the coordinate-wise
involution $z\mapsto 1/\bar{z}$ and
${\mathcal{A}}(X)\subseteq{\mathbb{R}}^{n}$ be its amoeba. If
$S\in{\mathbb{Z}}^{n\times n}$ is a nonsingular integer valued $n\times n$
matrix, then the set $X^{\prime}=\\{\exp(2\pi iSx)\mid
x\in{\mathbb{C}}^{n},\exp(2\pi ix)\in X\\}$ is a closed equidimensional
algebraic subvariety of $({\mathbb{C}}^{*})^{n}$ of dimension $c=n-d$ which is
invariant under the coordinate-wise involution $z\mapsto 1/\bar{z}$. In
particular, we have ${\mathcal{A}}(X^{\prime})=S{\mathcal{A}}(X)$.

###### Proof.

Let $S=(s_{ij})_{ij}$ with $s_{ij}\in{\mathbb{Z}}$. We consider the morphism

$f_{S}\colon({\mathbb{C}}^{*})^{n}\to({\mathbb{C}}^{*})^{n},\,(z_{1},\ldots,z_{n})\mapsto(z_{1}^{s_{11}}\cdots
z_{n}^{s_{1n}},\ldots,z_{1}^{s_{n1}}\cdots z_{n}^{s_{nn}}).$

The set $X^{\prime}=f_{S}(X)$ is invariant under the coordinate-wise
involution $z\mapsto 1/\bar{z}$ and clearly satisfies $\exp(2\pi ix)\in
X\Leftrightarrow\exp(2\pi iSx)\in X^{\prime}$ for all $x\in{\mathbb{C}}^{n}$.
Hence it suffices to show that $f_{S}$ is a finite map. We can write $S$ in
Smith normal form $S=RDT$ where $R,T$ are invertible integer matrices and $D$
is a diagonal integer matrix. The morphisms $f_{R}$ and $f_{T}$ are
automorphisms of $({\mathbb{C}}^{*})^{n}$ so in particular finite. The
morphism $f_{D}$ can be decomposed into morphisms that take the $m$-th power
of the $i$-th entry and leave the remaining entries unchanged. These are
clearly finite. Finally, the statement that
${\mathcal{A}}(X^{\prime})=S{\mathcal{A}}(X)$ follows directly from $\exp(2\pi
ix)\in X\Leftrightarrow\exp(2\pi iSx)\in X^{\prime}$ by taking the logarithm
of the absolute value of both sides. ∎

######  Lemma 8.3.

Let $X\subseteq({\mathbb{C}}^{*})^{n}$ be a closed algebraic subvariety of
dimension $c=n-d$ which is invariant under the coordinate-wise involution
$z\mapsto 1/\bar{z}$ and ${\mathcal{A}}(X)\subseteq{\mathbb{R}}^{n}$ be its
amoeba. If there exists an open subset $U\subseteq\operatorname{Gr}(d,n)$ such
that

  1. (i)

$\operatorname{Gr}_{\geq}(d,n)\subseteq U$, and

  2. (ii)