# Exponential Sums Equations and Tropical Geometry

Francesco Paolo Gallinaro

###### Abstract

Zilber’s Exponential-Algebraic Closedness Conjecture states that algebraic
varieties in $\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ intersect the
graph of complex exponentiation, unless that contradicts the algebraic and
transcendence properties of exp. We establish a case of the conjecture,
showing that it holds for varieties which split as the product of a linear
subspace of the additive group and an algebraic subvariety of the
multiplicative group. This amounts to solving certain systems of exponential
sums equations, and it generalizes old results of Zilber, which required the
linear subspace to either be defined over a generic subfield of the real
numbers, or it to be any subspace defined over the reals assuming unproved
conjectures from Diophantine geometry and transcendence theory. The proofs use
the theory of amoebas and tropical geometry.

††Keywords: Exponential-Algebraic Closedness, exponential sums equations,
tropical geometry.††2020 MSC: 03C65, 11L99, 14T90.

##  1 Introduction

It is well-known in model theory that the field $\mathbb{C}$ of complex
numbers is strongly minimal: all its subsets that are definable in first-order
logic in the language of rings are finite or cofinite. This is essentially due
to the Chevalley-Tarski theorem which asserts that projections of
constructible sets in algebraically closed fields are constructible: that
implies that definable subsets of $\mathbb{C}$ are solution sets of boolean
combinations of polynomial equations.

If we add a symbol for the complex exponential to the language, however,
strong minimality does not hold anymore: the kernel of exp is a countable
definable subset of $\mathbb{C}$. In fact, the ring of integers is definable
as the multiplicative stabilizer of the kernel, and this leads to the failure
of good model-theoretic properties. The structure obtained in this way (the
complex numbers in a language $\\{+,-,\cdot,0,1,\textnormal{exp}\\}$) is
denoted $\mathbb{C}_{\textnormal{exp}}$.

Nonetheless, it is true that there is no obvious way to define (in first-order
logic using only exponentials and polynomials) the real numbers in
$\mathbb{C}_{\textnormal{exp}}$. A conjecture of Zilber, the Quasiminimality
Conjecture, suggests that all definable sets should be countable or
cocountable.

######  Conjecture 1.1 (Quasiminimality Conjecture, [Zil97]).

The structure $\mathbb{C}_{\textnormal{exp}}$ is quasiminimal, i.e., all
definable subsets of $\mathbb{C}_{\textnormal{exp}}$ are countable or
cocountable.

The methods which have been developed to attack the quasiminimality
conjecture, such as the technique of quasiminimal excellence (see for example
[Bay+14]) are expected to yield a good structure theory of definable sets also
in higher dimension: a proof of the Quasiminimality Conjecture would likely
give that exponential-algebraic varieties, definable in powers of $\mathbb{C}$
using polynomials and exponentials, behave overall similarly to algebraic
varieties, provided we avoid some exceptional cases like $\mathbb{Z}$.

As an approach to his conjecture, Zilber proved in [Zil05] that there is a
quasiminimal exponential field $\mathbb{B}_{\textnormal{exp}}$ which is very
similar to $\mathbb{C}_{\textnormal{exp}}$, and conjectured them to be
isomorphic. There are two big obstructions on the way to the proof of this
isomorphism: Schanuel’s Conjecture and Exponential-Algebraic Closedness. The
former is a famous problem in transcendental number theory, which predicts a
lower bound for transcendence of the exponential function.

######  Conjecture 1.2 (Schanuel, see [Lan66, p. 30]).

Let $z_{1},\dots,z_{n}\in\mathbb{C}$ be linearly independent over
$\mathbb{Q}$. Then

$\textnormal{trdeg}(z_{1},\dots,z_{n},\textnormal{exp}(z_{1}),\dots,\textnormal{exp}(z_{n}))\geq
n.$

Very little is known about Schanuel’s Conjecture: it is true for $n=1$, but
already for $n=2$ it is considered an extremely hard problem, as it would for
example imply algebraic independence of $e$ and $\pi$ (as it implies
$\textnormal{trdeg}(1,i\pi,e,-1)\geq 2$).

The Exponential-Algebraic Closedness Conjecture, which will be stated
precisely below (Conjecture 3.2) predicts sufficient conditions for an
algebraic variety $V\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ to
intersect the graph of (a Cartesian power of) the exponential function. In a
sense, it is a generalization of a dual form of Schanuel’s Conjecture:
Schanuel’s Conjecture states that algebraic varieties of small dimension
defined over the rationals cannot contain points on the graph of exp, unless
these satisfy a $\mathbb{Q}$-linear relation in the first coordinates.
Exponential-Algebraic Closedness, on the other hand, predicts that all
algebraic varieties which “should”, in a precise sense, intersect the graph of
exp do so.

This conjecture became particularly relevant after the following result of
Bays and Kirby was proved.

######  Theorem 1.3 ([BK18, Theorem 1.5]).

If the Exponential-Algebraic Closedness Conjecture holds, then
$\mathbb{C}_{\textnormal{exp}}$ is quasiminimal.

In other words, they were able to remove the dependence of the Quasiminimality
Conjecture on Schanuel’s Conjecture; the former was therefore seen as more
accessible. This motivated several results around Exponential-Algebraic
Closedness which have been established in the last few years, see for example
[AKM22], [BM17], [DFT21], [Kir19], [MZ16].

In this paper we establish Exponential-Algebraic Closedness for a specific
class of subvarieties of $\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$,
those that split as the product of a linear subspace of $\mathbb{C}^{n}$ and
an algebraic subvariety of $(\mathbb{C}^{\times})^{n}$. More precisely, our
main theorem is as follows.

######  Theorem (Theorem 8.8).

Let $L\times W\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ be an
additively free, rotund algebraic variety, with $L\leq\mathbb{C}^{n}$ a linear
space and $W\subseteq(\mathbb{C}^{\times})^{n}$ an algebraic variety.

Then $L\times W\cap\Gamma_{\textnormal{exp}}\neq\varnothing$.

Here $\Gamma_{\textnormal{exp}}$ denotes the graph

$\\{(z_{1},\dots,z_{n},w_{1},\dots,w_{n})\in\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}\mid\textnormal{exp}(z_{j})=w_{j}\,\forall
j=1,\dots,n\\}$

of the exponential function. Here additively free means that $L$ is not
contained in a translate of a subspace of $\mathbb{C}$ which is defined over
$\mathbb{Q}$. The precise definition of rotund will be given later (Definition
3.1): it requires certain dimension inequalities to be satisfied, such as
$\dim L+\dim W\geq n$.

It is very likely that this theorem implies quasiminimality of a reduct of
$\mathbb{C}_{\textnormal{exp}}$, namely the two-sorted structure
$\mathbb{C}^{\mathbb{C}}=(\mathbb{C},\textnormal{exp},\mathbb{C})$ in which
the domain of exp is $\mathbb{C}$ seen as a $\mathbb{C}$-vector space and the
codomain $\mathbb{C}$ with the full field structure.

The systems of equations which are solved by Theorem 8.8 are systems of
exponential sums equations, i.e. equations that take the form
$\sum_{j=1}^{k}\textnormal{exp}(\varphi_{j}(z))$ where
$\varphi:\mathbb{C}^{n}\rightarrow\mathbb{C}$ is a linear function. Thus, for
example, the theorem implies solvability of easy equations such as
$\textnormal{exp}(z)+\textnormal{exp}(\sqrt{2}z)+1=0$ and
$\textnormal{exp}(z)+\textnormal{exp}(iz)+1=0$, which by abusing notations and
allowing for multivalued raising to powers operators on the complex numbers
correspond to the equations $w+w^{\sqrt{2}}+1=0$ and $w+w^{i}+1=0$. By
$w^{\sqrt{2}}$ (resp. $w^{i}$) here we mean any determination of
$\textnormal{exp}(\sqrt{2}\log w)$ (resp. $\textnormal{exp}(i\log w)$). We
will use these examples to present the different geometric behaviours that
this kind of systems may take, see for instance Example 3.4 below.

Theorem 8.8 generalizes results of Zilber from [Zil02] and [Zil15], which
required strong geometric assumptions to prove the conjecture for a smaller
class of varieties. The precise statement of Zilber’s theorem is given below
(Theorem 3.3); however, let us note here that his result needed the linear
subspace of $\mathbb{C}^{n}$ to be defined over the real numbers, while we are
able to prove the theorem for arbitrary complex subspaces. His approach was to
use a theorem of Khovanskii to find solutions to almost all systems and then
use some transcendence statement to make sure that the approximate solutions
obtained in this way form sequences that converge to actual solutions. The
theorem of Khovanskii, which we also need, is part of the theory of
fewnomials, a precursor of tropical geometry: our argument improves Zilber’s
in that it uses tropical geometry more systematically, together with some easy
complex analysis, to weaken the conditions needed on an approximate solution
to yield an actual solution.

Our proofs use techniques from tropical geometry and from the theory of
amoebas. Specifically, in the case in which $L$ is defined over the reals we
also apply the theorem of Khovanskii, but we use amoebas to simplify and
extend to every such $L$ the process of obtaining solutions from the
approximations. In the case in which $L$ is not defined over the reals we use
tropical geometry to compare the behaviour of $\textnormal{exp}(L)$ and $W$ as
their points approach $0$ or $\infty$; in particular, we use Tevelev’s
tropical compactifications to replace $W$ by a complete variety whose
behaviour at $0$ and $\infty$ is well understood.

The paper is structured as follows. In Section 2 we gather some results from
complex geometry which are used in the rest of the paper, specifically the
Open Mapping Theorem and the Proper Mapping Theorem which are necessary to
describe, locally and globally, the images of complex analytic functions. In
Section 3 we state precisely the Exponential-Algebraic Closedness Conjecture
and deduce some consequences of a theorem of Kirby which will be important
later on.

In Section 4 we introduce amoebas and tropical geometry, and in Section 5 the
framework of exponential sums equations which is needed to interpret the
question of intersecting varieties of the form $L\times W$ with the graph of
exp.

Section 6 contains the first main theorem of the paper, Theorem 6.16: this is
Theorem 8.8 with the further assumption that $L$ is defined over the reals.

Section 7 introduces more technical tools from tropical geometry, due to
Kazarnovskii, that are necessary to treat the case in which $L$ is not defined
over the reals. Finally, Section 8 contains the proof of the main theorem.

Acknowledgements. The author wishes to thank his supervisor, Vincenzo Mantova,
for suggesting to work on this problem, for many valuable comments, and for
spotting (at least) a mistake in an early version of this paper. Thanks are
due to Jonathan Kirby for spotting some inaccuracies and to the referee for
many useful comments. This research was done as part of the author’s PhD
project, supported by a scholarship of the School of Mathematics at the
University of Leeds.

##  2 Some Preliminaries from Complex Geometry

In this short section we gather some results from complex geometry which will
be used later on. For more details on the theory of complex analytic sets we
refer the reader to [Chi12] (especially Chapter 1 and Appendix 2).

######  Definition 2.1.

A continuous map $f:X\rightarrow Y$ of topological spaces is proper if for
every compact set $K\subseteq Y$, $f^{-1}(K)$ is compact.

It is interesting to see when the subset of a Cartesian product has proper
projection on one of the coordinates.

######  Proposition 2.2 ([Chi12, 3.1]).

Let $X$ and $Y$ be locally compact, Hausdorff topological spaces, $D\subseteq
X$ and $G\subseteq Y$ open subsets with $\overline{G}$ compact.

Let $A$ be a relatively closed subset of $D\times G$. The projection
$\pi:A\rightarrow D$, $(x,y)\mapsto x$, is proper if and only if $A$ does not
have limit points on $D\times\partial G$.

###### Proof.

$(\Rightarrow)$ Suppose the projection is proper, and there is a sequence
$\\{a_{j}\\}_{j\in\omega}\subseteq A$ such that $a=\lim_{j}a_{j}\in
D\times\partial G$. Since $D$ and $G$ are open, $a\notin D\times G$ (and
therefore $a\notin A$); however, since $a\in D\times\partial G$ the sequence
$\\{\pi(a_{j})\\}_{j\in\omega}$ converges to some $b\in D$.

Then consider a compact neighbourhood $U$ of $b$ in $D$. We have that
$\pi^{-1}(U)$ is not compact, because $a\notin\pi^{-1}(U)$ (it is not in the
domain $A$ of $\pi$) but it is the limit of the $a_{j}$’s, which lie in $A$.

$(\Leftarrow)$ Suppose $A$ does not have limit points on $D\times\partial G$,
and let $K\subseteq D$ be a compact set. Then $\pi^{-1}(K)$ is compact because
any sequence in it that has a limit in $D$ has a limit in $A$, given that $A$
is closed in $D\times G$ without limit points on $D\times\partial G$. ∎

Proper maps are important in complex analysis because they preserve
analyticity of complex analytic sets. As usual, the dimension of the fibres of
a holomorphic map is relevant when we want to study the image of the map.

######  Definition 2.3.

Let $f:A\rightarrow B$ be a holomorphic map between complex analytic sets. For
any $z\in A$, let $\dim_{z}f$ denote the codimension of the fibre

$\dim_{z}f:=\dim A-\dim f^{-1}(f(z))$

and $\dim f$ the maximal such value,

$\dim f:=\max_{z\in A}\dim_{z}f.$

######  Theorem 2.4 (Remmert’s Proper Mapping Theorem, [Chi12, 5.8]).

Let $A$ be a complex analytic set and $Y$ a complex manifold, and suppose
$f:A\rightarrow Y$ is proper.

Then $f(A)$ is an analytic subset of $Y$, and

$\dim f(A)=\dim f$

When the map is not proper there is not much that can be said about its image;
the best one can hope for is the following result.

######  Proposition 2.5 ([Chi12, 3.8]).

Let $A$ be a complex analytic set and $Y$ a complex manifold, and suppose
$f:A\rightarrow Y$ is holomorphic.

Then $f(A)$ is contained in a countable union of analytic subsets of $Y$, of
dimension not exceeding $\dim f$.

Of a similar flavour to the Proper Mapping Theorem is the Open Mapping
Theorem, which says that if the fibres of a map have the “right” dimension
then the map is open. The reader may recall the one variable version of this
fact from a basic course in complex analysis: in that case, the statement is
that any non-constant holomorphic map
$f:U\subseteq\mathbb{C}\rightarrow\mathbb{C}$ is open.

######  Theorem 2.6 (Open Mapping Theorem, [Chi12, Appendix 2, Theorem 2]).

Let $A$ be a complex analytic set and $Y$ a complex manifold. A holomorphic
map $f:A\rightarrow Y$ is open if and only if $\dim f^{-1}(z)=\dim A-\dim Y$
for every $z\in f(A)$.

An immediate corollary of the Open Mapping Theorem is the following.

######  Corollary 2.7.

Let $A$ be a complex analytic set, $Y$ a complex manifold, $f:A\rightarrow Y$
holomorphic and suppose $f^{-1}(f(z))$ has dimension $\dim A-\dim Y$.

Then there is an open neighbourhood $U$ of $z$ such that $f$ is open on $z$.

##  3 Exponential-Algebraic Closedness

In this section we introduce the exponential-algebraic closedness problem, and
the special case that we will deal with in this paper.

Before we begin, let us state that we will always assume our algebraic
varieties to be irreducible. Recall moreover that by a $K$-linear subspace of
$\mathbb{C}^{n}$, for $K\subseteq\mathbb{C}$ a subfield, we mean a subspace of
$\mathbb{C}^{n}$ that is defined over $K$.

The problem is about finding solutions to systems of equations in exponentials
and polynomials, and is interpreted geometrically. More precisely, it is about
finding sufficient conditions for an algebraic subvariety of
$\mathbb{C}\times(\mathbb{C}^{\times})^{n}$ to intersect the graph of the
exponential function. By an algebraic subvariety of
$\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ we mean a subset of
$\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ that is defined by finitely
many polynomial equations for polynomials in the ring
$\mathbb{C}[z_{1},\dots,z_{n},w_{1}^{\pm 1},\dots,w_{n}^{\pm 1}]$.

The conditions conjectured to be sufficient by Zilber for such a $V$ to
intersect the graph of exp are the following.

######  Definition 3.1.

Suppose $V\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ is an
algebraic subvariety. Let $\pi_{1}$ and $\pi_{2}$ be the projections from
$\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ onto $\mathbb{C}^{n}$ and
$(\mathbb{C}^{\times})^{n}$ respectively.

We say that $V$ is additively free if $\pi_{1}(V)$ is not contained in a
translate of a $\mathbb{Q}$-linear subspace of $\mathbb{C}^{n}$, that it is
multiplicatively free if $\pi_{2}(V)$ is not contained in a translate of an
algebraic subgroup of $(\mathbb{C}^{\times})^{n}$, and that it is free if it
is both additively and multiplicatively free.

Let $L\leq\mathbb{C}^{n}$ be a $\mathbb{Q}$-linear subspace of
$\mathbb{C}^{n}$; let

$\pi_{L}:\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}\twoheadrightarrow\mathbb{C}^{n}/L\times(\mathbb{C}^{\times})^{n}/(\textnormal{exp}(L))$

be the algebraic quotient map. We say that $V$ is rotund if for every $L$,
$\dim(\pi_{L}(V))\geq n-\dim L$.

The conjecture then reads as follows.

######  Conjecture 3.2 (Zilber, [Zil05]).

Suppose $V\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ is an
algebraic variety; let
$\Gamma_{\textnormal{exp}}\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$
denote the graph of (the $n$-th cartesian power of) the exponential function.
If $V$ is free and rotund, then
$V\cap\Gamma_{\textnormal{exp}}\neq\varnothing$.

Some partial results concerning this conjecture can be found in [AKM22],
[BM17], [DFT21], [Kir19], [MZ16], [Mar06] (some of these were already cited in
the Introduction).

We are particularly interested in the results of Zilber in [Zil02] and
[Zil15]; here, and in the rest of the paper, by an algebraic subvariety of
$(\mathbb{C}^{\times})^{n}$ we mean a subset of $(\mathbb{C}^{\times})^{n}$
that is defined by finitely many polynomial equations for polynomials in
$\mathbb{C}[w_{1}^{\pm 1},\dots,w_{n}^{\pm 1}]$.

######  Theorem 3.3 (Zilber).

Suppose $L\times W$ is a free rotund variety, where
$W\subseteq(\mathbb{C}^{\times})^{n}$ is an algebraic subvariety and and $L$
is either:

  * 1.

If the Conjecture on Intersection with Tori and the Uniform Schanuel
Conjecture hold, any $\mathbb{R}$-linear space;

  * 2.

A $K$-linear space, for a subfield $K$ of the reals that is generated by a
tuple that is exponentially-algebraically independent in the sense of [BKW10].

Then $L\times W\cap\Gamma_{\textnormal{exp}}\neq\varnothing$.

The assumptions in Theorem 3.3 should be explained. The theorem was first
proved, under the assumption (1), in [Zil02, Theorem 5]: the Conjecture on
Intersections with Tori is now studied in its own right as one of the main
open problems in Diophantine geometry. It is a case of what is now known as
the Zilber-Pink Conjecture which is stated in the more general setting of
Shimura varieties.

The theorem under the assumption (2) was proved in [Zil15, Theorem 7.2]:
there, Zilber used a result of transcendence for real powers established by
Bays, Kirby and Wilkie in [BKW10, Theorem 1.3] to get rid of the Conjecture on
Intersections with Tori; however, the Bays-Kirby-Wilkie theorem only holds for
subfields of the reals which are generated over $\mathbb{Q}$ by a finite tuple
of numbers which satisfies a notion of independence called exponential
transcendence. It is worth noting that almost all tuples of complex numbers
(in a measure-theoretic sense) satisfy this assumptions.

Our goal in this paper is to establish a stronger result than Theorem 3.3,
which does not need any additional assumptions on free rotund varieties of the
form $L\times W$ in which $L$ is linear. We begin by examining two examples.

######  Example 3.4.

Let

$L_{\sqrt{2}}:=\left\\{\left(z_{1},z_{2}\right)\mid
z_{2}=\sqrt{2}z_{1}\right\\},$ $L_{i}:=\left\\{\left(z_{1},z_{2}\right)\mid
z_{2}=iz_{1}\right\\}$

and

$W:=\\{(w_{1},w_{2})\in(\mathbb{C}^{\times})^{2}\mid w_{1}+w_{2}+1=0\\}.$

The varieties $L_{\sqrt{2}}\times W$ and $L_{i}\times W$ will serve as our
recurring examples for this paper.

It is clear that both varieties are free: $L_{\sqrt{2}}$ and $L_{i}$ are not
contained in $\mathbb{Q}$-linear subspaces of $\mathbb{C}^{2}$, and $W$ is not
contained in an algebraic subgroup of $(\mathbb{C}^{\times})^{2}$. They are
also both rotund (in fact, for subvarieties
$V\subseteq\mathbb{C}^{2}\times(\mathbb{C}^{\times})^{2}$, freeness implies
rotundity: if $\pi_{1}(V)$ is not contained in any 1-dimensional
$\mathbb{Q}$-linear subspace $L\leq\mathbb{C}^{2}$, then $\dim\pi_{L}(V)\geq
1$).

Suppose we want to find an intersection between $L_{\sqrt{2}}\times W$ and
$\Gamma_{\textnormal{exp}}$, the graph of the exponential. Then we want to
find a point $(z_{1},z_{2},\textnormal{exp}(z_{1}),\textnormal{exp}(z_{2}))$
in $L\times W$, which thus has to satisfy $z_{2}=\sqrt{2}z_{1}$ and
$\textnormal{exp}(z_{1})+\textnormal{exp}(z_{2})=1$; this is equivalent to
finding a complex number $z\in\mathbb{C}$ such that
$\textnormal{exp}(z)+\textnormal{exp}(\sqrt{2}z)+1=0$. By abusing notation,
and writing $w^{\sqrt{2}}$ to mean any determination of
$\textnormal{exp}(\sqrt{2}(\log w))$, we can see this as looking for a point
$w\in\mathbb{C}^{\times}$ such that $w+w^{\sqrt{2}}+1=0$.

The same reasoning, of course, can be applied to $L_{i}\times W$, and thus we
can see intersecting that with $\Gamma_{\textnormal{exp}}$ to be equivalent to
solving $w+w^{i}+1=0$.

The two situations are quite different from a geometric perspective, as we now
see. For a given number $w\in\mathbb{C}^{\times}$, all the determinations of
$w^{\sqrt{2}}$ form a dense subset of the set
$\\{z\in\mathbb{C}^{\times}\mid|z|=|w|^{\sqrt{2}}\\}$. In fact, if
$w=\rho(\cos\theta+i\sin\theta)\in\mathbb{C}^{\times}$ for some
$\rho\in\mathbb{R}^{>0}$ and $\theta\in[0,2\pi[$ then

$\textnormal{exp}^{-1}(w)=\\{x+iy\in\mathbb{C}\mid e^{x}=\rho\wedge
y\in\theta+2\pi\mathbb{Z}\\}.$

Since $\sqrt{2}\mathbb{Z}+\mathbb{Z}$ is dense in $\mathbb{R}/\mathbb{Z}$, the
set

$\textnormal{exp}(\sqrt{2}\textnormal{exp}^{-1}(w))=\\{\textnormal{exp}(\sqrt{2}x+i\sqrt{2}y)\in\mathbb{C}^{\times}\mid
e^{x}=\rho\wedge y\in\theta+2\pi\mathbb{Z}\\}=$
$=\left\\{\rho^{\sqrt{2}}(\cos(\sqrt{2}(\theta+2k\pi))+i\sin(\sqrt{2}(\theta+2k\pi)))\in\mathbb{C}^{\times}\mid
k\in\mathbb{Z}\right\\}$

is dense in
$\\{z\in\mathbb{C}^{\times}\mid|z|=|w|^{\sqrt{2}}=\rho^{\sqrt{2}}\\}$ as we
wanted. We see then that $\textnormal{exp}(L_{\sqrt{2}})$ is dense in

$\textnormal{exp}(L_{\sqrt{2}})\cdot\mathbb{S}_{1}^{2}=\left\\{(w_{1},w_{2})\in(\mathbb{C}^{\times})^{2}\mid|w_{2}|=|w_{1}|^{\sqrt{2}}\right\\}$

where $\mathbb{S}_{1}$ denotes the unit circle $\\{z\in\mathbb{C}\mid|z|=1\\}$
and the operation $|w_{1}|^{\sqrt{2}}$ is well-defined, not multivalued, as it
is a real power of a real number.

On the other hand,

$\textnormal{exp}(i\textnormal{exp}^{-1}(w))=\\{\textnormal{exp}(-y+ix)\in\mathbb{C}^{\times}\mid
e^{x}=\rho\wedge y\in\theta+2\pi\mathbb{Z}\\}=$
$=\left\\{e^{\theta+2k\pi}(\cos(\rho)+i\sin(\rho))\in\mathbb{C}^{\times}\mid
k\in\mathbb{Z}\right\\}$

is an infinite discrete subset of $\mathbb{C}^{\times}$. Thus
$\textnormal{exp}(L_{i})$ is a closed, complex analytic subgroup of
$(\mathbb{C}^{\times})^{2}$.

These different behaviours are shown in Figure 1.

$(a)$$i\mathbb{R}$$0$$2\pi i$$\mathbb{R}$$2\pi
i+\mathbb{R}$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$(b)$$i\mathbb{R}$$0$$2\pi
i$$\mathbb{R}$$2\pi i+\mathbb{R}$••••••••• Figure 1: The exponential map
identifies the multiplicative group $\mathbb{C}^{\times}$ with the strip
$\\{z\in\mathbb{C}\mid 0\leq\textnormal{Im}(z)\leq 2\pi\\}$, glued with itself
joining the outer lines: in other words, the multiplicative group is a
cylinder. With this interpretation, the sets of all determinations of
$1^{\sqrt{2}}$ and $1^{i}$ are shown in the figure: in $(a)$ we see the dense
sets of points obtained as $(2\pi i\mathbb{Z})\cdot\sqrt{2}+\mathbb{Z}$, in
$(b)$ the discrete set $-2\pi\mathbb{Z}$.

An important ingredient in our proofs will be a result by Kirby which combines
the Ax-Schanuel theorem (a result on atypical intersections of algebraic
subvarieties of $\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ with the graph
of exp) with the Remmert open mapping theorem from complex analysis. We are
going to apply it to the following function.

######  Definition 3.5 ($\delta$-map of a variety).

Let $V$ be an algebraic subvariety of
$\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$. The $\delta$-map of $V$ is
the function

$\delta:V\rightarrow(\mathbb{C}^{\times})^{n}$

which maps $(v_{1},v_{2})\in V$ to $\frac{v_{2}}{\textnormal{exp}(v_{1})}$.

The following fact was established by Kirby in [Kir19, Proposition 6.2 and
Remark 6.3].

######  Fact 3.6.

Suppose the variety $V\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$
is rotund. Then there is a Zariski-open dense subset $V^{\circ}\subseteq V$
such that the $\delta$-map of $V$ is open on $V^{\circ}$.

For varieties of the form $L\times W$, actually, something stronger holds and
we can say more about the structure of the set $V^{\circ}$.

######  Proposition 3.7.

Suppose $L\times W$ is a rotund algebraic subvariety of
$\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$, with $L$ a linear subspace of
$\mathbb{C}^{n}$ and $W$ an algebraic subvariety of
$(\mathbb{C}^{\times})^{n}$. Then there is a Zariski-open dense subset
$W^{\circ}\subseteq W$ such that the $\delta$-map of $L\times W$ is open on
$L\times W^{\circ}$.

###### Proof.

Suppose $(l_{0},w_{0})\in L\times W$ is a point around which $\delta$ is open;
let $U_{L}$ be a neighbourhood of $l_{0}$ and $U_{W}$ be a neighbourhood of
$w_{0}$ such that $\delta_{|U_{L}\times U_{W}}$ is open. Let $l$ be any point
in $L$, and $V_{L}=(l-l_{0})+U_{L}$ a neighbourhood of $l$ that is a translate
of $U_{L}$. Then any open subset $O_{V}$ of $V_{L}\times U_{W}$ is a translate
by $((l-l_{0}),1)$ of an open subset $O_{U}$ of $U_{L}\times U_{W}$. This
implies that $\delta(O_{V})$ is a translate by $(\textnormal{exp}(l-l_{0}))$
of $\delta(O_{U})$, so an open set.

Therefore if $\delta$ is open around $(l_{0},w_{0})$, then it is open around
any point of $L\times\\{w_{0}\\}$. Thus the Zariski-open dense set of Fact 3.6
must be of the form $L\times W^{\circ}$ for some Zariski-open dense subset
$W^{\circ}$ of $W$. ∎

Moreover, we have that openness of the $\delta$-map at a single point is
sufficient to prove rotundity.

######  Proposition 3.8.

Let $L\times W$ be an algebraic subvariety of
$\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$, with $L$ a linear subspace of
$\mathbb{C}^{n}$ and $W$ an algebraic subvariety of
$(\mathbb{C}^{\times})^{n}$.

If there is a point at which the $\delta$-map of $L\times W$ is open, then the
variety is rotund.

###### Proof.

Suppose $Q$ is a linear subspace of $\mathbb{C}^{n}$. Let
$\pi_{Q}:\mathbb{C}^{n}\twoheadrightarrow\mathbb{C}^{n}/Q$ and
$\pi_{\textnormal{exp}(Q)}:(\mathbb{C}^{\times})^{n}\twoheadrightarrow(\mathbb{C})^{\times}/\textnormal{exp}(Q)$
denote the projections; $\delta$ the $\delta$-map of $L\times W$; $\delta_{Q}$
the $\delta$-map of $\pi_{Q}(L)\times\pi_{\textnormal{exp}(Q)}(W)$.

Let $(l,w)\in L\times W$. We see that

$\delta_{Q}(\pi_{Q}(l),\pi_{\textnormal{exp}(Q)}(w))=\frac{\pi_{\textnormal{exp}(Q)}(w)}{\textnormal{exp}(\pi_{Q}(l))}=$
$=\frac{w\cdot\textnormal{exp}(Q)}{\textnormal{exp}(l)\cdot\textnormal{exp}(Q)}=\frac{w}{\textnormal{exp}(l)}\cdot\textnormal{exp}(Q)=\pi_{\textnormal{exp}(Q)}(\delta(l,w))$

and therefore the square

$L\times
W$$(\mathbb{C}^{\times})^{n}$$(\mathbb{C}^{\times})^{n}/\textnormal{exp}(Q)$$\pi_{Q}(L)\times\pi_{\textnormal{exp}(Q)}(W)$$\delta$$\pi_{\textnormal{exp}(Q)}$$\pi_{Q}\times\pi_{\textnormal{exp}(Q)}$$\delta_{Q}$

commutes.

Now let $(l_{0},w_{0})\in L\times W$ be the point at which $\delta$ is open,
so that there are neighbourhoods $U_{L}\subseteq L$ of $l_{0}$ and
$U_{W}\subseteq W$ of $w_{0}$ such that $\delta_{|U_{L}\times U_{W}}$ is an
open map. Then we see that

$\delta_{Q}(\pi_{Q}(U_{L})\times\pi_{\textnormal{exp}(Q)}(U_{W}))=\pi_{\textnormal{exp}(Q)}(\delta(U_{L}\times
U_{W}))$

and the set on the right-hand side is the projection of an open set, and thus
it has to be open. Thus the set on the left-hand side is open too, and that is
only possible if $\dim L+\dim W\geq n-\dim Q$ as we wanted. ∎

Using this fact together with a common procedure known as the Rabinowitsch
trick we can make a very useful reduction, proving that we can assume without
loss of generality that if $L\times W$ is rotund then the $\delta$-map is open
everywhere.

######  Lemma 3.9.

Suppose $L\times W\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ is
an additively free, rotund algebraic subvariety, with $L$ a linear subspace of
$\mathbb{C}^{n}$ and $W$ an algebraic subvariety of
$(\mathbb{C}^{\times})^{n}$. Then there is an additively free, rotund
subvariety $L^{\prime}\times W^{\prime}$ of
$\mathbb{C}^{n+1}\times(\mathbb{C}^{\times})^{n+1}$ such that the $\delta$-map
of $L^{\prime}\times W^{\prime}$ is open on all the domain, and if
$\textnormal{exp}(L^{\prime})\cap W^{\prime}\neq\varnothing$ then
$\textnormal{exp}(L)\cap W\neq\varnothing$.

###### Proof.

Since the set of points where $\delta$ is open is of the form $L\times
W^{\circ}$, there is an algebraic function $F:W\rightarrow\mathbb{C}$ such
that $\delta$ is open around each point $(l,w)$ for which $F(w)\neq 0$. Let
$L^{\prime}:=L\times\mathbb{C}$ and

$W^{\prime}:=\left\\{(w_{1},\dots,w_{n+1})\in(\mathbb{C}^{\times})^{n}\mid(w_{1},\dots,w_{n})\in
W\wedge F(w_{1},\dots,w_{n})=w_{n+1}\right\\}.$

Consider then the variety $L^{\prime}\times W^{\prime}$. It is clear that
$L^{\prime}\times W^{\prime}$ is additively free.

For rotundity we use Proposition 3.8. Given a point in $L^{\prime}\times
W^{\prime}$, which therefore has the form
$(l_{1},\dots,l_{n+1},w_{1},\dots,w_{n+1})$ there are neighbourhoods
$U_{L}\subseteq L$ and $U_{W}\subseteq W$ of $(l_{1},\dots,l_{n})$ and
$(w_{1},\dots,w_{n})$ respectively such that the $\delta$-map of $L\times W$
is open on $U_{L}\times U_{W}$. Let $U_{L^{\prime}}:=U_{L}\times\mathbb{C}$,
and

$U_{W^{\prime}}:=\\{(w_{1},\dots,w_{n+1})\in
W^{\prime}\mid(w_{1},\dots,w_{n})\in U_{W}\\}.$

The image of $U_{L^{\prime}}\times U_{W^{\prime}}$ under the $\delta$-map of
$L^{\prime}\times W^{\prime}$ is then the Cartesian product of an open subset
of $(\mathbb{C}^{\times})^{n}$ by $\mathbb{C}$, i.e. an open subset of
$(\mathbb{C}^{\times})^{n+1}$: therefore the $\delta$-map of $L^{\prime}\times
W^{\prime}$ is open at all of its points, and therefore the variety must be
rotund.

Finally, it is clear that if
$(w_{1},\dots,w_{n+1})\in\textnormal{exp}(L^{\prime})\cap W^{\prime}$ then
$(w_{1},\dots,w_{n})\in\textnormal{exp}(L)\cap W$. ∎

We introduce another assumption which simplifies the proofs: we may take $\dim
L=\textnormal{codim}W$, so that $\dim L\times W=n$ when $L\times
W\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$.

######  Lemma 3.10.

Let $L\times W$ be an additively free, rotund variety in
$\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$. Then there is a space
$L^{\prime\prime}\subseteq L$ such that $L^{\prime\prime}\times W$ is
additively free and rotund, and $\dim L^{\prime\prime}+\dim W=n$.

###### Proof.

Let $\delta$ be the $\delta$-map of $L\times W$; by rotundity, $\delta$ is
open around some point $(0,w)\in L\times W$. This implies that there is a
point $z\in\log W$, with $\textnormal{exp}(z)=w$, which lies in an irreducible
component of the set $\log W\cap z+L$ of dimension $\dim L+\dim W-n$, by
Theorem 2.6. For a sufficiently generic subspace $H\leq\mathbb{C}^{n}$, of
dimension $2n-\dim L-\dim W$, we will have then that

$\dim(\log W\cap z+L\cap z+H)=\dim L+\dim W-n+2n-\dim L-\dim W-n=0$

and thus $z$ is an isolated point in it. Therefore, the $\delta$-map of
$(L\cap H)\times W$ is open at the point $(0,w)$: the variety $(L\cap H)\times
W$ is then rotund by Proposition 3.8 and since $H$ is generic we may also take
it to be additively free. Therefore $L^{\prime\prime}:=L\cap H$ satisfies the
lemma. ∎

Thus in what follows we will, when necessary, assume freely that $\dim
L=\textnormal{codim}W$. This will not affect the generality of our statements.

##  4 Amoebas and Tropical Geometry

In this section we introduce the basics on amoebas and tropical geometry. In
the next sections we will show how this ties to the exponential-algebraic
closedness question for varieties of the form $L\times W$. Amoebas will be
more important in the case in which $L$ is defined over the reals, as then we
will see that finding a point in $\textnormal{exp}(L)\cap W$ is as hard as
finding a point in the intersection of the real part of $L$ with the amoeba of
$W$. Tropicalizations will play a more important role in the case in which $L$
is not defined over the reals, as then we will need a precise understanding of
the behaviour of $W$ as its points approach $0$ or $\infty$, which will be
given by tropical geometry.

###  4.1 Amoebas

Amoebas were introduced in [GKZ94, Chapter 6] as a tool to analyse the
behaviour near 0 and $\infty$ of subvarieties of $(\mathbb{C}^{\times})^{n}$.
A good survey on their properties is [Mik04].

Denote by
$\textnormal{Log}:(\mathbb{C}^{\times})^{n}\rightarrow\mathbb{R}^{n}$ the map

$(z_{1},\dots,z_{n})\mapsto\left(\log|z_{1}|,\dots,\log|z_{n}|\right).$

######  Definition 4.1.

Let $W\subseteq(\mathbb{C}^{\times})^{n}$ be an algebraic variety. The amoeba
$\mathcal{A}_{W}$ of $W$ is the image of $W$ under the map Log.

Figure 2 shows a picture of an amoeba.

An amoeba is a closed proper subset of $\mathbb{R}^{n}$. Much of the theory of
amoebas has been carried out for amoebas of hypersurfaces; however, we will
use amoebas of varieties of arbitrary codimension. For now, we only state the
following theorem, which will help us later on to establish the tie between
amoebas and tropical varieties.

######  Theorem 4.2 ([Pur08, Corollary 5.2]).

Let $W\subseteq(\mathbb{C}^{\times})^{n}$ be an algebraic variety, and $I$ be
the ideal of polynomials which vanish on $W$. Then

$\mathcal{A}_{W}=\bigcap_{f\in I}\mathcal{A}_{f}$

where $\mathcal{A}_{f}$ denotes the amoeba of the hypersurface cut out by $f$.

Figure 2: The amoeba of the algebraic variety $W$ defined by
$w_{1}+w_{2}+1=0$. We see that the amoeba has three “tentacles”: the diagonal
one corresponds to the behaviour of $W$ when $w_{1}$ and $w_{2}$ are both very
big, and thus their absolute values are roughly the same; the vertical one
corresponds to points for which $w_{2}$ is very close to 0 (and thus its
logarithm to $-\infty$) and $w_{1}$ to $-1$; the horizontal one to points with
$w_{1}$ close to 0 and $w_{2}$ to $-1$.

###  4.2 Polyhedral Geometry

In this subsection we review a few basic facts on polyhedral geometry, so that
we have all the tools to discuss tropical varieties later on.

We start by recalling some of the definitions, starting with the basic notion
of polyhedron.

######  Definition 4.3.

A polyhedron is a subset of $\mathbb{R}^{n}$ of the form

$\left\\{x\in\mathbb{R}^{n}\mid Ax\leq b\right\\}$

where $A$ is a $d\times n$ matrix with real entries and $b\in\mathbb{R}^{d}$
(the inequality is componentwise). If the matrix has entries in $\mathbb{Q}$
and $b\in\mathbb{Q}^{d}$ we will say the polyhedron is rational.

A bounded polyhedron is called a polytope.

Given a polyhedron $P\subseteq\mathbb{R}^{n}$, the affine span
$\textnormal{aff}(P)$ is the smallest affine subspace of $\mathbb{R}^{n}$
which contains $P$.

A face of a polyhedron $P$ with respect to some $w\in\mathbb{R}^{n}$ is a
subset of $P$ of the form

$\textnormal{face}_{w}(P):=\\{x\in P\mid w\cdot x\geq w\cdot y\,\,\,\forall
y\in P\\}.$

We will denote by $\textnormal{aff}(P)$ the affine span of a polyhedron, and
by $\textnormal{relint}(P)$ the relative interior of $P$ in
$\textnormal{aff}(P)$.

$(0,0)$$(1,0)$$(0,1)$ Figure 3: This triangle is a polyhedron defined by
$x_{1}+x_{2}\leq 1$, $x_{1}\geq 0$ and $x_{2}\geq 0$. The three sides are
faces induced by the vectors $(1,1)$, $(-1,0)$ and $(0,-1)$.

We will deal with sets of polyhedra that enjoy good coherence properties.

######  Definition 4.4.

A polyhedral complex $\Sigma$ is a set of polyhedra such that:

  * 1.

If $P\in\Sigma$ then every face of $P$ is in $\Sigma$;

  * 2.

If $P_{1},P_{2}\in\Sigma$, then $P_{1}\cap P_{2}$ is either empty or a face of
both (and thus an element of $\Sigma$).

The union of all polyhedra of $\Sigma$ is called the support of $\Sigma$ and
denoted $|\Sigma|$. We will often abuse notation and write $x\in\Sigma$ for
$x\in\mathbb{R}^{n}$ to mean that $x\in|\Sigma|$, so there is $P\in\Sigma$
such that $x\in P$; when $P$ is a polyhedron, $P\in\Sigma$ will literally mean
that $P$ is one of the polyhedra of $\Sigma$.

The polyhedra in a polyhedral complex are called the cells of the complex;
those which are not contained in any larger polyhedra are the facets of the
complex and the faces of a facet which are not contained in any larger
polyhedron (other than the facet itself) are called the ridges of the complex.
Obviously, by a rational polyhedral complex we will mean a polyhedral complex
all of whose polyhedra are rational.

The type of polyhedral complex that we will mostly be interested in is the
normal fan of a polytope.

######  Definition 4.5.

A cone in $\mathbb{R}^{n}$ is a polyhedron $P$ for which there exist
$v_{1},\dots,v_{k}\in\mathbb{R}^{n}$ such that

$P=\left\\{\sum_{i=1}^{k}\lambda_{i}v_{i}\mid\lambda_{1},\dots,\lambda_{k}\in\mathbb{R}_{\geq
0}\right\\}.$

A fan is a polyhedral complex $\Sigma$ such that all polyhedra in $\Sigma$ are
cones.

######  Definition 4.6.

Let $P$ be a polytope. The normal fan of $P$ is the polyhedral complex which
contains for every face $F$ of $P$ the polyhedron obtained as the closure (in
the Euclidean topology on $\mathbb{R}^{n}$) of the set

$\left\\{w\in\mathbb{R}^{n}\mid F=\textnormal{face}_{w}(P)\right\\}.$

The $(n-1)$-skeleton of the normal fan is the polyhedral complex obtained by
removing the polyhedra of dimension $n$ from the normal fan.

$A$$B$$C$$c$$a$$b$$B^{\vee}$$A^{\vee}$$C^{\vee}$$a^{\vee}$$b^{\vee}$$c^{\vee}$
Figure 4: (The support of) the normal fan of the polytope of Figure 3: the
normal cone to the full triangle, which is the face of $(0,0)$, is the origin;
the normal cones to the sides of the triangles are the three half-lines; and
the normal cones to the three vertices of the triangles are the portions of
space in between the half-lines. Each of the normal cones (except the origin)
has been labelled to show which face of the polytope it is normal to.
Considering only the origin and the three half-lines, we obtain the
$1$-skeleton of the complex.

We will see in the next subsection that tropicalizations of algebraic
varieties are polyhedral complexes which enjoy strong structural properties.

We are also interested in the notion of mixed volume of a collection of
polytopes. First we recall how to perform standard operations on subsets of
Euclidean space.

######  Definition 4.7.

Let $A,B\subseteq\mathbb{R}^{n}$. The Minkowski sum of $A$ and $B$ is the set

$A+B:=\\{a+b\in\mathbb{R}^{n}\mid a\in A,\,b\in B\\}.$

If $\lambda\in\mathbb{R}$, then

$\lambda A:=\\{\lambda a\in\mathbb{R}^{n}\mid a\in A\\}.$

We define the normalized volume of a polytope $P\subseteq\mathbb{R}^{n}$ to be
the standard Euclidean volume multiplied by $n!$. This is so that the smallest
simplex with integer vertices in $\mathbb{R}^{n}$ has volume 1.

We then use the following fact. It is proved in [MS15, Proposition 4.6.3],
under the stronger assumption that the polytopes have integer vertices, to
draw the stronger conclusion that the resulting polynomial has integer
coefficients; however, the same proof will yield the statement that we give
here.

######  Proposition 4.8.

Let $P_{1},\dots,P_{r}$ be polytopes in $\mathbb{R}^{n}$. The normalized
volume of the polytope $\lambda_{1}P_{1}+\dots+\lambda_{r}P_{r}$ is a
homogeneous polynomial in $\lambda_{1},\dots,\lambda_{r}$ of degree $n$.

As an example, consider the polytopes $P_{1}$ and $P_{2}$ where $P_{1}$ is the
segment with vertices $(0,0)$ and $(0,1)$ and $P_{2}$ the segment with
vertices $(0,0)$ and $(1,0)$: the volume of the Minkowski sum
$\lambda_{1}P_{1}+\lambda_{2}P_{2}$ is then clearly $\lambda_{1}\lambda_{2}$.

This allows us to give the following definition of mixed volume.

######  Definition 4.9.

Let $P_{1},\dots,P_{n}$ be polytopes in $\mathbb{R}^{n}$. The mixed volume of
the polytopes, denoted $\textnormal{MV}(P_{1},\dots,P_{n})$ is the coefficient
of the unique square-free monomial $\lambda_{1}\cdots\lambda_{n}$ in the
polynomial obtained in Proposition 4.8.

We will see later on how to relate the mixed volumes of a collection of
polytopes to the intersection of the $(n-1)$-skeletons of their normal fans.

###  4.3 Tropicalizations

Tropical geometry is the heart of this section. Here we will introduce
tropical varieties and show how to interpret them as limits of amoebas; we
will also state part of the Structure Theorem, a fundamental result in
tropical geometry which establishes structural properties of tropical
varieties.

There are several equivalent ways to define tropical varieties. We choose the
approach based on initial forms of polynomials, as it is going to be the most
convenient one to discuss tropical compactifications later on (in Section 8).

The initial forms of a Laurent polynomial with complex coefficients can be
thought of as limit forms that the polynomial takes when it is evaluated at
some point $w\in(\mathbb{C}^{\times})^{n}$, some of whose coordinates approach
$0$ or $\infty$.

In the next definition we use multi-index notation: for $k\in\mathbb{Z}^{n}$,
we use $w^{k}$ to mean the monomial $w_{1}^{k_{1}}\cdots w_{n}^{k_{n}}$.

######  Definition 4.10.

Let $x\in\mathbb{R}^{n}$, and let $f$ be a Laurent polynomial in $n$
variables, so $f\in\mathbb{C}[w_{1}^{\pm 1},\dots,w_{n}^{\pm 1}]$. Write $f$
as

$f:=\sum_{k\in S}c_{k}w^{k}$

for some finite subset $S\subseteq\mathbb{Z}^{n}$. The initial form of $f$
with respect to $x$ is the polynomial

$\textnormal{in}_{x}(f):=\sum_{k\in S_{x}}c_{k}w^{k}$

where $S_{x}$ is the set

$S_{x}:=\\{k\in S\mid x\cdot k\geq y\cdot k\,\,\,\forall y\in S\\}.$

If $I$ is an ideal in the ring of Laurent polynomials, then
$\textnormal{in}_{x}(I)$ denotes the set of initial forms of polynomials in
$f$.

######  Example 4.11.

To see an example of how to take initial forms, consider the polynomial
$f:=w_{1}+w_{2}+1$. For this, the set $K$ is the set
$\\{(1,0),(0,1),(0,0)\\}$. Hence, it is easy to see that:

  * 1.

If $x$ has $x_{1},x_{2}<0$, then $x\cdot(0,0)$ is bigger than $x\cdot(1,0)$
and $x\cdot(0,1)$; thus the initial form of $f$ is 1.

  * 2.

If $x_{1}-x_{2}>0$ and $x_{1}>0$, then the largest scalar product is
$x\cdot(1,0)$; thus the initial form of $f$ is $w_{1}$.

  * 3.

Similarly, if $x_{1}-x_{2}<0$ and $x_{2}>0$ then we obtain $w_{2}$ as an
initial form.

  * 4.

If $x_{1}=x_{2}$, and both are positive, the maximum is obtained twice as
$x\cdot(1,0)=x\cdot(0,1)$. Therefore the initial form is $w_{1}+w_{2}$.

  * 5.

If $x_{1}=0$ and $x_{2}<0$ then $x\cdot(1,0)=x\cdot(0,0)$ and both are larger
than $x\cdot(0,1)$, so the initial form is $w_{1}+1$.

  * 6.

In the same way, if $x_{2}=0$ and $x_{1}<0$ then the initial form is
$w_{2}+1$.

  * 7.

Finally, if $x=(0,0)$ then the initial form of $f$ is $f$ itself as all scalar
products are the same.

It is clear that the regions we chose form a partition of $\mathbb{R}^{2}$,
and that they are in fact the relative interiors of the polyhedra that we see
in Figure 4.

The idea of initial forms is that if we plug in very large or very small
values for some of the variables of a polynomial, then the value of the
polynomial has no hope of being zero unless there are at least two monomials
that are roughly of the same size. Thus, most of the time, the value of the
polynomial will be decided simply by the fact that one of the monomials takes
a much larger value than the others. This motivates the following definition
of tropical variety.

######  Definition 4.12.

Let $W\subseteq(\mathbb{C}^{\times})^{n}$ be an algebraic variety, and let $I$
be the ideal of Laurent polynomials which vanish on $W$. Then the
tropicalization $\textnormal{Trop}(W)$ of $W$ is the subset of
$\mathbb{R}^{n}$ defined as

$\\{x\in\mathbb{R}^{n}\mid\textnormal{in}_{x}(I)\neq\langle 1\rangle\\}.$

######  Remark 4.13.

As the ideal is taken in the ring of Laurent polynomials, an ideal is the
whole ring if and only if it is generated by monomials, as these are the
invertible elements there. Hence the tropical variety of $W$ can be defined as
the set of $x$’s for which the initial ideal of $I$ is not a monomial ideal.

######  Example 4.14.

It is clear at this point that the tropicalization of the algebraic variety

$W:=\\{(w_{1},w_{2})\in(\mathbb{C}^{\times})^{2}\mid w_{1}+w_{2}+1=0\\}$

is the $(n-1)$-skeleton of the polyhedral complex shown in Figure 4.

As we mentioned above, there are several ways to define tropical varieties,
the equivalence of which is known as the Fundamental Theorem of Tropical
Algebraic Geometry ([MS15, Theorem 3.2.5]). Here we state just part of it.

######  Theorem 4.15 ([MS15, Theorem 3.2.5]).

Let $W\subseteq(\mathbb{C}^{\times})^{n}$ be an algebraic variety, $I$ the
ideal of Laurent polynomials which vanish on $W$. Then

$\textnormal{Trop}(W)=\bigcap_{f\in I}\textnormal{Trop}(V(f))$

where $V(f)$ denotes the hypersurface cut out in $(\mathbb{C}^{\times})^{n}$
by $f$.

Theorem 4.15 is obviously very similar to the corresponding result for amoebas
(Theorem 4.2). The comparison between the two allows us to make precise the
idea of tropical varieties being “nonstandard amoebas”, in the sense that they
can be thought of as amoebas where the base of the logarithm is infinite.

First, we recall what we mean by Hausdorff metric.

######  Definition 4.16.

Let $A,B\subseteq\mathbb{R}^{n}$ be closed sets, and let $d$ denote the
standard Euclidean metric on $\mathbb{R}^{n}$. The Hausdorff distance between
$A$ and $B$ is defined as

$d_{\textnormal{Haus}}(A,B):=\max\left\\{\sup_{a\in A}d(a,B),\sup_{b\in
B}d(b,A)\right\\}$

where $d(x,Y)$ denotes the usual Euclidean distance between the point $x$ and
the set $Y$.

We will call Hausdorff topology the topology induced on the space of closed
sets of $\mathbb{R}^{n}$ by this metric, and we will say that a set $S$ is the
Hausdorff limit of a sequence $\\{S_{j}\\}_{j\in\omega}$ if the sequence
converges to $S$ in the Hausdorff topology. It is clear by the definition that
$d_{\textnormal{Haus}}$ is a metric, not a pseudometric, and therefore the
topology is $T_{2}$: Hausdorff limits are unique.

For $t\in\mathbb{R}$, denote by
$\textnormal{Log}_{t}:(\mathbb{C}^{\times})^{n}\rightarrow\mathbb{R}^{n}$ the
map

$(w_{1},\dots,w_{n})\mapsto(\log_{t}|w_{1}|,\dots,\log_{t}|w_{n}|)$

and for an algebraic variety $W$ denote by $\mathcal{A}_{W}^{t}$ the image of
$W$ under $\textnormal{Log}_{t}$ (so that the usual map Log coincides with
$\textnormal{Log}_{e}$ and the usual amoeba $\mathcal{A}_{W}$ with
$\mathcal{A}_{W}^{e}$).

The limit of the amoebas in base $t$ of a variety, for $t$ going to infinity,
is the tropicalization.

######  Theorem 4.17.

For $t\rightarrow\infty$, the sets $\mathcal{A}_{W}^{t}$ converge to
$\textnormal{Trop}(W)$ in the Hausdorff metric.

###### Proof.

An easy proof for hypersurfaces is given in [Mik04a, Corollar 6.4]. For
varieties of arbitrary codimension it is a harder problem, see [Jon16, Theorem
A]. ∎

We finish this section by reviewing a few more results in tropical geometry
which describe structural properties of tropical varieties.

######  Definition 4.18.

A polyhedral complex is pure if all its facets (i.e. all the polyhedra which
are not faces of larger polyhedra in the complex) have the same dimension.

Thus for pure complexes it makes sense to talk about the dimension of the
complex, meaning the dimension of the facets. Part of the Structure Theorem
asserts that the tropicalization of an algebraic variety of (complex)
dimension $d$ is a pure polyhedral complex of (real) dimension $d$.

######  Theorem 4.19 (Part of the Structure Theorem, [MS15, Theorem 3.3.6]).

Let $W\subseteq(\mathbb{C}^{\times})^{n}$ be an algebraic variety of dimension
$d$. Then $\textnormal{Trop}(W)$ is a pure polyhedral complex of dimension
$d$.

######  Definition 4.20.

Let $\Sigma$ be a polyhedral complex, $\tau$ a polyhedron in $\Sigma$. The
star $\textnormal{star}_{\tau}(\Sigma)$ is the polyhedral complex formed by
the polyhedra of the form

$\sigma^{\prime}:=\\{\lambda(x-y)\mid\lambda\geq 0,x\in\tau,y\in\sigma\\}$

for each polyhedron $\sigma\in\Sigma$ that $\tau$ is a face of (including
$\tau=\textnormal{face}_{0}(\tau)$).

######  Example 4.21.

If $\Sigma$ is the 1-skeleton of the polyhedral complex in Figure 4, then the
star of each half-line is the line that contains it.

We recall the result which states that the star of a cell in
$\textnormal{Trop}(W)$ is the tropicalization of an initial variety of $W$ and
review some of its consequences.

######  Lemma 4.22 ([MS15, Lemma 3.3.7]).

Let $\Sigma$ be the polyhedral fan supported on $\textnormal{Trop}(W)$.
Suppose $\tau\in\Sigma$ is a face, and $w\in\textnormal{relint}(\tau)$. Then
$\textnormal{star}_{\tau}(\Sigma)=\textnormal{Trop}(\textnormal{in}_{w}(W))$.

As the star depends on the face, and not on the point, this implies that the
initial variety $\textnormal{in}_{w}(W)$ is constant as $w$ varies in the
relative interior of a face. Therefore, we can give the following definition.

######  Definition 4.23.

Let $\Sigma$ be the polyhedral fan supported on $\textnormal{Trop}(W)$,
$\tau\in\Sigma$. We denote by $W_{\tau}$ the initial variety
$\textnormal{in}_{w}(W)$ for every $w\in\textnormal{relint}(\tau)$.

We should remark that initial varieties of irreducible varieties are not
necessarily irreducible.

Note that if $\tau$ is a facet of $\Sigma$ then by definition the star of
$\tau$ in $\Sigma$ is a linear space (the linear span of $\tau$) and therefore
the tropicalization of the initial variety $W_{\tau}$ is a subspace of
$\mathbb{R}^{n}$. This implies that $W_{\tau}$ is a finite union of cosets of
an algebraic subgroup of $(\mathbb{C}^{\times})^{n}$.

######  Example 4.24.

Thinking back about Example 4.11, we see how in that case the four non-
monomial initial forms of the polynomial correspond to the four polyhedra in
the $1$-skeleton of the fan (three half-lines and the origin).

Finally, we consider the following proposition, according to which the initial
variety of $W\subseteq(\mathbb{C}^{\times})^{n}$ with respect to the face
$\tau$ is invariant under multiplication by elements in the image under exp of
the complex space generated by $\tau$.

######  Proposition 4.25.

Let $W\subseteq(\mathbb{C}^{\times})^{n}$ be an algebraic variety,
$\tau\in\textnormal{Trop}(W)$ a face of its tropicalization,
$\tau_{\mathbb{C}}$ the complex subspace of $\mathbb{C}^{n}$ generated by
$\tau$ (seen as a subset of $\mathbb{R}^{n}\subseteq\mathbb{C}^{n}$).

Then $W_{\tau}$ is invariant under translation by elements of
$\textnormal{exp}(\tau_{\mathbb{C}})$.

###### Proof.

This is a consequence of [MS15, Corollary 3.2.13], which introduces a notion
of tropicalization for monomial maps and shows that monomial maps commute with
tropicalizations. Since quotienting by an algebraic subgroup of
$(\mathbb{C}^{\times})^{n}$ can be seen as a monomial map, this result implies
that the dimension of $W_{\tau}/\textnormal{exp}(\tau_{\mathbb{C}})$ is equal
to the dimension of $W_{\tau}$, which is therefore invariant under translation
by elements of $\textnormal{exp}(\tau_{\mathbb{C}})$. ∎

##  5 Exponential Sums Equations

In this section we introduce the framework of exponential sums equations. This
is a natural way to interpret the question of exponential-algebraic closedness
for varieties of the form $L\times W$, and allows us to talk about Newton
polytopes and related objects.

Let

$(\mathbb{C}^{n})^{\vee}:=\left\\{\varphi:\mathbb{C}^{n}\rightarrow\mathbb{C}\mid\varphi\textnormal{
is linear}\right\\}$

denote the usual dual space of the complex vector space $\mathbb{C}^{n}$. Note
that we can easily talk about convexity in this space: given two functions
$\varphi_{1},\varphi_{2}$, the segment between $\varphi_{1}$ and $\varphi_{2}$
is the set $\\{(1-t)\varphi_{1}+t\varphi_{2}\mid t\in[0,1]\\}$, and a set is
convex if it contains the segment between any two of its elements. As usual,
the convex hull of a set is the smallest convex set which contains it.

######  Definition 5.1.

An exponential sum is a function $f:\mathbb{C}^{n}\rightarrow\mathbb{C}$ of
the form

$z\mapsto\sum_{\varphi\in S}c_{\varphi}\textnormal{exp}(\varphi(z))$

where $S\subseteq(\mathbb{C}^{n})^{\vee}$ is a finite set, and
$c_{\varphi}\in\mathbb{C}$ for each $\varphi$.

The Newton polytope of the exponential sum $f$ is the convex hull of $S$ in
$(\mathbb{C}^{n})^{\vee}$.

Of course systems of exponential sums equations can take very different forms,
but we are only interested in the ones we can attach to varieties of the form
$L\times W$.

A subvariety $W\subseteq(\mathbb{C}^{\times})^{n}$ is defined by a system of
Laurent polynomial equations, so equations of the form

$\sum_{j\in S}c_{j}w^{j}=0$

where $S\subseteq\mathbb{Z}^{n}$ is a finite subset. It is clear that any
Laurent polynomial can be seen as an exponential sum of the form

$\sum_{j\in S}c_{j}\textnormal{exp}(j\cdot z)=0$

where $j\cdot z=j_{1}z_{1}+\dots+j_{n}z_{n}$ denote the usual scalar product.
The system of exponential sums obtained from the Laurent polynomials which
define $W$ clearly defines the complex analytic subset $\log W$ of
$\mathbb{C}^{n}$.

It is well-known (see for example [Mil08, Aside 3.55]) that for every point
$w$ of an irreducible affine variety $W$ there is a Zariski-open neighbourhood
of $w$ in which $W$ is a complete intersection, that is, it is defined by
exactly $\textnormal{codim}W$ polynomial. As we are working in the
multiplicative group, we can then apply the Rabinowitsch trick and assume
without loss of generality that $W$ is a complete intersection.

As for the linear space $L\leq\mathbb{C}^{n}$, this is defined by linear
equations such as $\lambda_{1}z_{1}+\dots+\lambda_{n}z_{n}=0$. Clearly, the
function
$\varphi_{\lambda}:(z_{1},\dots,z_{n})\mapsto\lambda_{1}z_{1}+\dots+\lambda_{n}z_{n}$
is an element of $(\mathbb{C}^{n})^{\vee}$, and therefore, $L$ is the unique
irreducible component containing 0 of the complex analytic set defined by the
exponential sums

$\textnormal{exp}(\varphi_{\lambda}(z))-1=0.$

This larger set is a countable union of translates of $L$. Note that it is not
the set $L+2\pi i\mathbb{Z}^{n}$: to see this, note that
$\textnormal{exp}(\varphi_{\lambda}(z))-1=0$ defines a complex analytic sets,
which therefore is always closed, while in general $L+2\pi i\mathbb{Z}^{n}$ is
not, for example in the case in which $L$ is defined by
$z_{2}-\sqrt{2}z_{1}=0$.

######  Definition 5.2.

Let $L\leq\mathbb{C}^{n}$ be a linear space defined by equations
$\varphi_{\lambda}(z)=0$, $W\subseteq(\mathbb{C}^{\times})^{n}$ an algebraic
variety defined by equations $\sum_{j\in S}c_{j}w^{j}$.

The system of exponential sums attached to $L\times W$ is the system defined
by the corresponding equations of the form
$\textnormal{exp}(\varphi_{\lambda}(z))=1$ and $\sum_{j\in
S}c_{j}\textnormal{exp}(j\cdot z)=0$.

It is useful to associate a system of exponential sums to the variety $L\times
W$ because it allows us to give a characterization of rotundity of the pair in
terms of the Newton polytope of the system; this is similar to [Zil02, Lemma
3] (with its converse) but we state it in a slightly different way.

######  Definition 5.3.

Let $V$ be a vector space of dimension $n$, and let $A_{1},\dots,A_{n}$ be
finite subsets of $V$. We say that $A_{1},\dots,A_{n}$ satisfy the Rado
property if there is a basis $\\{v_{1},\dots,v_{n}\\}$ of $V$ such that
$v_{j}\in A_{j}$ for each $j$.

This property is named after Rado’s Theorem on Independent Transversals:

######  Theorem 5.4 ([Rad42, Theorem 1]).

Let $A_{1},\dots A_{n}$ be finite subsets of a vector space $V$ of dimension
$n$. Then the span of $\bigcup_{j\in J}A_{j}$ has dimension at least $|J|$ for
each subset $J\subseteq\\{1,\dots,n\\}$ if and only if $A_{1},\dots,A_{n}$
satisfy the Rado property.

For a subspace $L\leq\mathbb{C}^{n}$, we denote as usual by $L^{\perp}$ the
subspace of the dual space defined as

$L^{\perp}:=\\{\varphi\in(\mathbb{C}^{n})^{\vee}\mid\varphi(z)=0\,\forall z\in
L\\}.$

For a polytope $P\subseteq(\mathbb{C}^{n})^{\vee}$, we denote by $v(P)$ the
finite set of its vertices (its 0-dimensional faces).

######  Lemma 5.5.

Let $L\leq\mathbb{C}^{n}$ be a linear space of dimension $d$, and
$W\subseteq(\mathbb{C}^{\times})^{n}$ a variety of codimension $d$ defined by
polynomials $f_{1},\dots,f_{d}$ with Newton polytopes $P_{1},\dots,P_{d}$. Let
$\pi_{L^{\perp}}:(\mathbb{C}^{n})^{\vee}\twoheadrightarrow(\mathbb{C}^{n})^{\vee}/L^{\perp}$
denote the projection.

Then the variety $L\times W$ is rotund if and only if
$\pi_{L^{\perp}}(v(P_{1})),\dots\pi_{L^{\perp}}(v(P_{d}))$ satisfy the Rado
property.

###### Proof.

$(\Rightarrow)$ Suppose the projections of the vertex sets do not satisfy the
Rado property. Then by Theorem 5.4, after renumbering the polytopes there is
$k\leq d$ such that $\pi_{L^{\perp}}(P_{1}),\dots,\pi_{L^{\perp}}(P_{k})$ are
contained in a subpace $T\leq(\mathbb{C}^{n})^{\vee}/L^{\perp}$ of dimension
$\dim T<k$. We may take $T$ to be $\pi_{L^{\perp}}(S)$, with $S$ the span of
$P_{1},\dots,P_{k}$ in $(\mathbb{C}^{n})^{\vee}$. Since all the sets
$v(P_{j})$ are sets of integer vectors, $S$ is defined over $\mathbb{Q}$.

Therefore, as $\dim(\pi_{L^{\perp}}(S))<k$, we must have $\dim
S-\dim(L^{\perp}\cap S)<k$. As is well-known, taking the annihilator of
subspaces in $(\mathbb{C}^{n})^{\vee}$ yields spaces which are isomorphic to
spaces in $\mathbb{C}^{n}$ \- namely, we canonically have that
$(L^{\perp})^{\perp}\cong L$ and $S^{\perp}$ is isomorphic to (and will then
be identified with) a rational subspace of $\mathbb{C}^{n}$. Under these
identifications we have $(L^{\perp}\cap S)^{\perp}=L+S^{\perp}$. Then we have

$\dim(\pi_{S^{\perp}}(L))=\dim L-\dim(L\cap S^{\perp})=$ $=n-\dim
L^{\perp}-(n-\dim(L^{\perp}+S))=$ $=\dim L^{\perp}+\dim S-\dim(L^{\perp}\cap
S)-\dim L^{\perp}=$ $=\dim S-\dim(L^{\perp}\cap S)=\dim(\pi_{L^{\perp}}(S)).$

Hence we have $\dim(\pi_{S^{\perp}}(L))=\dim(\pi_{L^{\perp}}(S))<k$.

Consider now the algebraic subgroup $\textnormal{exp}(S^{\perp})$ of
$(\mathbb{C}^{\times})^{n}$. Since $S$ is the span of polytopes
$P_{1},\dots,P_{k}$, the quotient of the variety $W$ under
$\textnormal{exp}(S^{\perp})$ must have codimension at least $k$, as the
quotient of each of the hypersurfaces cut out by $f_{1},\dots,f_{k}$ is still
a hypersurface (cut out by a polynomial with Newton polytope $Q_{j}$ for which
$\pi_{S^{\perp}}^{\vee}(Q_{j})=P_{j}$, where
$\pi_{S^{\perp}}^{\vee}:S\rightarrow(\mathbb{C}^{n})^{\vee}$ is the transpose
of the linear map $\pi_{S^{\perp}}$). Therefore,

$\dim(\pi_{\textnormal{exp}(S^{\perp})}(W))\leq\dim S-k.$

Hence, we find that there is a rational subspace $S^{\perp}\leq\mathbb{C}^{n}$
such that

$\dim(\pi_{S^{\perp}}(L))+\dim(\pi_{\textnormal{exp}(S^{\perp})}(W))<k+\dim
S-k=n-\dim S^{\perp}$

which contradicts the definition of rotundity. Hence if the polytope
projections do not satisfy the Rado property then $L\times W$ is not rotund,
establishing one direction of the lemma.

$(\Leftarrow)$ Now assume the projections of the vertex sets of the polytopes
satisfy the Rado property, and let $S^{\perp}\leq\mathbb{C}^{n}$ be the
annihilator of some rational subspace $S$ of $(\mathbb{C}^{n})^{\vee}$ (we
introduce $S^{\perp}$ as an annihilator rather than by itself to maintain
consistency in the proof - this way, in both directions $S^{\perp}$ is a
subspace of $\mathbb{C}^{n}$ and $S$ of $(\mathbb{C}^{n})^{\vee}$). Consider
the variety $\pi_{\textnormal{exp}(S^{\perp})}(W)$: this has dimension $\dim
S-k$ for some $k\geq 0$.

Replacing, if necessary, the polynomials $f_{1},\dots,f_{k}$ by other elements
of the ideal of polynomials which vanish on $W$ we may assume without loss of
generality that $f_{1},\dots,f_{k}$ define the variety
$W\cdot\textnormal{exp}(S^{\perp})$, and hence that the polytopes
$P_{1},\dots,P_{k}$ are contained in $S$. Hence, by the Rado property of the
sets $\pi_{L^{\perp}}(v(P_{1})),\dots,\pi_{L^{\perp}}(v(P_{k}))$, we have
$\dim({\pi_{L^{\perp}}(S)})\geq k$.

We have proved above that $\dim(\pi_{L^{\perp}}(S))=\dim(\pi_{S^{\perp}}(L))$,
and as a consequence $\dim(\pi_{S^{\perp}}(L))\geq k$. Thus,

$\dim\pi_{S^{\perp}}(L)+\dim\pi_{\textnormal{exp}(S^{\perp})}(W)\geq k+\dim
S-k=\dim S=n-\dim S^{\perp}.$

As $S^{\perp}$ was arbitrary, this establishes rotundity of $L\times W$. ∎

######  Corollary 5.6.

Let $L\times W\subseteq\mathbb{C}^{n}\times(\mathbb{C}^{\times})^{n}$ be a
variety with $L$ linear and $\dim L+\dim W=n$.

Then $L\times W$ is rotund if and only if the Newton polytopes of the
associated system of exponential sums satisfy the Rado property. If $L$ is
defined over the reals, this is equivalent to the polytope having non-zero
mixed volume.

###### Proof.

The first assertion is an immediate consequence of Lemma 5.5. For the second
one apply [MS15, Lemma 4.6.6], which ties mixed volume to the Rado property. ∎

##  6 Raising to Real Powers

In this section we establish the main result of the paper for varieties of the
form $L\times W$, where $L$ is defined over the real numbers; we are going to
do so using a density property of sets of the form $\textnormal{exp}(L)$.
Throughout this section, even when not explicitly stated, we assume that the
space $L$ is defined over $\mathbb{R}$.

Recall that $\mathbb{S}_{1}$ denotes the unit circle
$\\{z\in\mathbb{C}\mid|z|=1\\}$.

######  Proposition 6.1.

Let $L\leq\mathbb{C}^{n}$ be an $\mathbb{R}$-linear space that is not
contained in any $\mathbb{Q}$-linear space. Then $\textnormal{exp}(L)$ is
dense in $\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$.

###### Proof.

Let $\textnormal{Re}(L)$ denote the set
$\\{\textnormal{Re}(l)\in\mathbb{R}^{n}\mid l\in L\\}$. Since $L$ is
$\mathbb{R}$-linear, we have $L=\textnormal{Re}(L)+i\textnormal{Re}(L)$.

Since $L$ is not contained in any $\mathbb{Q}$-linear subspace of
$\mathbb{C}^{n}$, $\textnormal{Re}(L)$ is not contained in any
$\mathbb{Q}$-linear subspace of $\mathbb{R}^{n}$. Therefore,
$\textnormal{exp}(i\textnormal{Re}(L))$ is dense in $\mathbb{S}_{1}^{n}$: to
see this, consider $\mathbb{S}_{1}^{n}$ as
$\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$, and
$\textnormal{exp}(i\textnormal{Re}(L))$ as
$\textnormal{Re}(L)+2\pi\mathbb{Z}^{n}$: as $\textnormal{Re}(L)$ is not
contained in any rational subspace, $\textnormal{Re}(L)+2\pi\mathbb{Z}^{n}$ is
not contained in any proper closed subgroup of
$\mathbb{R}^{n}/2\pi\mathbb{Z}^{n}$, and is therefore dense; hence,
$\textnormal{exp}(i\textnormal{Re}(L))$ is dense in $\mathbb{S}_{1}^{n}$.

Hence given a neighbourhood $U$ of the identity in
$\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$, it contains points of
$\textnormal{exp}(i\textnormal{Re}(L))\subseteq\textnormal{exp}(L)$. As
$\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$ is a group this suffices to prove
density. ∎

The set $\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$ can easily be related to
the material discussed in the previous section.

######  Proposition 6.2.

Let $L$ be an $\mathbb{R}$-linear subspace of $\mathbb{C}^{n}$. Then:

  * 1.

$\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}=\textnormal{Log}^{-1}(\textnormal{Re}(L))$.

  * 2.

If $W$ is an algebraic subvariety of $(\mathbb{C}^{\times})^{n}$, then
$\mathcal{A}_{W}\cap\textnormal{Re}(L)\neq\varnothing$ if and only if
$W\cap\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}\neq\varnothing$.

###### Proof.

Part 1 is straightforward: $w\in\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$ if
and only if there is $l\in L$ such that

$\left(|w_{1}|,\dots,|w_{n}|\right)=\left(|\textnormal{exp}(l_{1})|,\dots,|\textnormal{exp}(l_{n})|\right)=\left(e^{\textnormal{Re}(l_{1})},\dots,e^{\textnormal{Re}(l_{n})}\right)$

if and only if $\textnormal{Log}(w)\in\textnormal{Re}(L)$.

Part 2 follows by a similar argument: both statements are true if and only if
there is $(l,w)\in L\times W$ such that
$\frac{w}{\textnormal{exp}(l)}\in\mathbb{S}_{1}^{n}$. ∎

We will now see that thanks to Lemma 3.9 it is sufficient to intersect
$\mathcal{A}_{W}$ and $\textnormal{Re}(L)$ to find an intersection between
$\textnormal{exp}(L)$ and $W$.

Let $(\ast)$ denote the following assumption:

For every additively free rotund variety of the form $L\times W$,
$\textnormal{Re}(L)\cap\mathcal{A}_{W}\neq\varnothing$.

######  Lemma 6.3.

Suppose assumption $(\ast)$ holds.

Then for every additively free rotund variety of the form $L\times W$,
$\textnormal{exp}(L)\cap W\neq\varnothing$.

###### Proof.

By Lemma 3.9, we can assume without loss of generality that the $\delta$-map
of $L\times W$ is open. By Proposition 6.2 and the assumption $(\ast)$, we
know that $\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}\cap W\neq\varnothing$.
Clearly, this implies that the image $\textnormal{im}(\delta)$ of the
$\delta$-map intersects $\mathbb{S}_{1}^{n}$. As $\textnormal{im}(\delta)$ is
open, this means that actually there is an open subset of $\mathbb{S}_{1}^{n}$
contained in $\textnormal{im}(\delta)$: thus, as
$\textnormal{exp}(i\textnormal{Re}(L))$ is dense in $\mathbb{S}_{1}^{n}$, it
must be the case that
$\textnormal{im}(\delta)\cap\textnormal{exp}(i\textnormal{Re}(L))\neq\varnothing$.

Hence there is a point $(l,w)\in L\times W$ such that
$\frac{w}{\textnormal{exp}(l)}\in\textnormal{exp}(i\textnormal{Re}(L))\leq\textnormal{exp}(L)$:
this implies that $w\in\textnormal{exp}(L)$. ∎

Hence, Lemma 6.3 means that we only need to prove that the assumption $(\ast)$
holds to establish the result.

######  Remark 6.4.

The reader who is familiar with [Kir19] will notice that assumption $(\ast)$
says, in the language of that paper, that all additively free rotund varieties
$L\times W$ intersect the blurring of $\Gamma_{\textnormal{exp}}$ by
$\mathbb{S}_{1}^{n}$, and Lemma 6.3 says that intersecting these varieties
with the blurred graph is equivalent to intersecting them with the actual
graph.

To intersect $\textnormal{Re}(L)$ and $\mathcal{A}_{W}$ we will use a result
of Khovanskii to show that given any additively free rotund variety $L\times
W$ we can find $W^{\prime}$ such that $\textnormal{exp}(L)\cap
W^{\prime}\neq\varnothing$ and $\mathcal{A}_{W^{\prime}}=\mathcal{A}_{W}$.

Recall from Definition 5.2 that if $L\leq\mathbb{C}^{n}$ is a linear space and
$W\subseteq(\mathbb{C}^{\times})^{n}$ an algebraic variety, the system of
exponential sums attached to $L\times W$ is the system of equations:

$\begin{cases}\textnormal{exp}(\lambda_{1,1}z_{1})\cdots\textnormal{exp}(\lambda_{1,n}z_{n})=1\\\
\vdots\\\
\textnormal{exp}(\lambda_{d,1}z_{1})\cdots\textnormal{exp}(\lambda_{d,n}z_{n})=1\\\
\sum_{j\in S_{1}}c_{j}\textnormal{exp}(j\cdot z)=0\\\ \vdots\\\ \sum_{j\in
S_{n-d}}\textnormal{exp}(j\cdot z)=0\\\ \end{cases}$

We recall the definition of a coherent set of faces in a collection of
polyhedra, and of a shortening of a system. These can be found in [Kho91,
Section 3.13] (the set of faces is called concordant there) and in [Zil02,
Section 6].

######  Definition 6.5.

Let $P_{1},\dots,P_{k}\subseteq\mathbb{R}^{n}$ be convex polytopes. A coherent
set of faces of $P_{1},\dots,P_{k}$ is a set of polytopes $Q_{1},\dots,Q_{k}$
for which there is $w\in\mathbb{R}^{n}$ such that

$Q_{j}=\textnormal{face}_{w}(P_{j})$

for each $j$.

As an easy example, let $P_{1}$ denote the segment in $\mathbb{R}^{2}$ with
vertices $(0,0)$ and $(1,0)$ and $P_{2}$ the segment in $\mathbb{R}^{2}$ with
vertices $(0,0)$ and $(0,1)$. Then
$\\{(0,0)\\}=\textnormal{face}_{(-1,0)}(P_{1})$ and
$P_{2}=\textnormal{face}_{(-1,0)}(P_{2})$: so $(0,0)$ and $P_{2}$ form a
coherent set of faces of $P_{1},P_{2}$.

######  Definition 6.6.

Let $f_{1}=\dots=f_{k}=0$ be a system of exponential sums equations. Let, for
each $j$, $S_{j}$ be the set of exponents of the exponential sum $f_{j}$,
$P_{j}$ its Newton polytope (so $P_{j}$ is the convex hull of the discrete set
$S_{j}$), and suppose $Q_{1},\dots,Q_{k}$ is a coherent set of faces of
$P_{1},\dots,P_{k}$.

The shortening of the system associated to $Q_{1},\dots,Q_{k}$ is the system
$g_{1}=\dots=g_{k}=0$, where

$g_{j}=\sum_{\varphi\in S_{j}\cap
Q_{j}}c_{\varphi}\textnormal{exp}(\varphi(z))$

for each $j$.

Note that by definition the Newton polytope of the exponential sum $g_{j}$ is
$Q_{j}$ for each $j$.

######  Example 6.7.

A shortening describes the behaviour of the system of exponential sums as some
of the variables approach infinity. As an example, consider the exponential
sums

$f_{1}(z_{1},z_{2})=\textnormal{exp}(z_{1})+\textnormal{exp}(z_{2})$
$f_{2}(z_{1},z_{2})=\textnormal{exp}(z_{1})+\textnormal{exp}(z_{2})+1.$

Then $P_{1}=\textnormal{conv}\\{(1,0),(0,1)\\}$ and
$P_{2}=\textnormal{conv}\\{(1,0),(0,1),(0,0)\\}$. $P_{1}$ then has three faces
(the two points $(1,0)$ and $(0,1)$ and $P_{1}$ itself), which are obtained as
$\textnormal{face}_{w}(P_{1})$ for $w$ equal to $(-1,0)$, $(0,-1)$ or $(1,1)$
respectively.

It is easy to see that each of these induces a face of $P_{2}$:
$\textnormal{face}_{(-1,0)}(P_{2})=\textnormal{conv}\\{(0,0),(0,1)\\}$,
$\textnormal{face}_{(1,1)}(P_{2})=P_{1}$, and
$\textnormal{face}_{(0,1)}(P_{2})=\textnormal{conv}\\{(0,0),(-1,0)\\}$. This
means that there are three coherent sets of faces of $P_{1}$ and $P_{2}$, and
thus the system has three shortenings:

$\begin{cases}\textnormal{exp}(z_{1})=0\\\ \textnormal{exp}(z_{1})+1=0\\\
\end{cases}$

and

$\begin{cases}\textnormal{exp}(z_{2})=0\\\ \textnormal{exp}(z_{2})+1=0\\\
\end{cases}$

which are obviously inconsistent as
$0\notin\textnormal{im}(\textnormal{exp})$, and

$\begin{cases}\textnormal{exp}(z_{1})+\textnormal{exp}(z_{2})=0\\\
\textnormal{exp}(z_{1})+\textnormal{exp}(z_{2})=0\\\ \end{cases}$

which defines a curve.

######  Definition 6.8.

Let $G\subseteq\mathbb{R}^{n}$ be an open set. A system of exponential sums is
non-degenerate at infinity in the domain $\mathbb{R}^{n}+iG$ if:

  * 1.

All solutions of the system in the domain are isolated;

  * 2.

All shortenings of the system do not have solutions in $\mathbb{R}^{n}+iG$.

Solvability of non-degenerate at infinity systems of exponential sums has been
established long ago. In particular, we have the following result of Zilber.

######  Theorem 6.9 ([Zil02, Theorem 4] and subsequent discussion; see also
[Kho91, Theorem 3.13.1]).

Let $f_{1}=\dots=f_{n}=0$ be a system of exponential sums with exponents in
$\mathbb{R}$. If there are arbitrarily large balls $G\subseteq\mathbb{R}^{n}$
such that the system is non-degenerate at infinity in $\mathbb{R}^{n}+iG$,
then the system has a solution.

The idea of Khovanskii’s theorem is that exponential sums with real exponents
can be approximated by exponential sums with rational exponents, for which it
is easier to find zeros as they are essentially Laurent polynomials. The
condition of non-degeneracy at infinity is needed to make sure that the
approximated solutions do not diverge as the approximating systems converge to
the one we are interested in.

As an example, one can consider the system

$\begin{cases}\textnormal{exp}(z_{1})+\textnormal{exp}(z_{2})+1=0\\\
\textnormal{exp}(\sqrt{2}z_{1})-\textnormal{exp}(z_{2})=0\\\ \end{cases}$

The second equation can be approximated for example by

$\textnormal{exp}(7z_{1})-\textnormal{exp}(5z_{2})=0$

and the system then becomes essentially the system

$\begin{cases}w_{1}+w_{2}+1=0\\\ w_{1}^{7}-w_{2}^{5}=0\\\ \end{cases}$

which can be solved by algebraic means. Improving the precision of the
approximations, we see that solutions cannot escape to infinity, because we
know by looking at the initial variety that if $(a_{1},a_{2})$ is a point
satisfying $a_{1}+a_{2}+1=0$ and $|a_{1}|\gg 1$, then $|a_{1}|$ and $|a_{2}|$
have to be roughly the same and therefore cannot satisfy an equation of the
form $|a_{1}|^{p}=|a_{2}|^{q}$ where $\frac{p}{q}$ is very close to
$\sqrt{2}$.

We are going to show that given a variety $L\times W$ whose $\delta$-map is
open, there is $s\in\mathbb{S}_{1}^{n}$ such that:

  * 1.

$\mathcal{A}_{s\cdot W}=\mathcal{A}_{W}$;

  * 2.

The system of exponential sums associated to $L\times s\cdot W$ is non-
degenerate at infinity in $\mathbb{C}^{n}$.

This will allow us to use Theorem 6.9 to find the intersection between
$\textnormal{Re}(L)$ and $\mathcal{A}_{W}$ that we are looking for.

First of all, for a change, we notice a feature of varieties that are not
rotund. For this, we are going to use a simple property of holomorphic
functions in several variables.

######  Proposition 6.10.

Let $U\subseteq\mathbb{C}^{n}$ be an open set such that
$U\cap\mathbb{R}^{n}\neq\varnothing$, and assume $f:U\rightarrow\mathbb{C}$ is
a non-zero holomorphic function. Then $f$ does not identically vanish on
$U\cap\mathbb{R}^{n}$.

###### Proof.

By induction on $n$. If $n=1$, then it follows from the fact that zeros of
holomorphic functions are isolated.

If $n>1$, assume $f$ vanishes on every point of $U\cap\mathbb{R}^{n}$. Let
$\pi:U\rightarrow\mathbb{C}$ be the projection on the last coordinate, and
$\pi^{\prime}:U:\rightarrow\mathbb{C}^{n-1}$ the projection on the first $n-1$
coordinates. For $r\in\pi(U\cap\mathbb{R}^{n})$ consider the function
$f_{r}:\pi^{\prime}(\pi^{-1}(r))\rightarrow\mathbb{C}$ defined by
$(z_{1},\dots,z_{n-1})\mapsto f(z_{1},\dots,z_{n},r)$. Since $f$ vanishes on
$U\cap\mathbb{R}^{n}$, $f_{r}$ vanishes on
$\pi^{\prime}(\pi^{-1}(r))\cap\mathbb{R}^{n-1}$ for all $r$, and therefore by
the inductive hypothesis all functions $f_{r}$ are identically zero.

Therefore $f$ vanishes on the set
$\bigcup_{r\in\pi(U\cap\mathbb{R}^{n})}\pi^{-1}(r)$. As this has real
codimension 1 in $U$, and the zero-locus of $f$ must be a complex analytic
set, $f$ is identically zero. ∎

######  Corollary 6.11.

Let $f:(\mathbb{C}^{\times})^{n}\rightarrow\mathbb{C}$ be a non-zero
holomorphic function. Then $f$ does not vanish on any open subset of
$\mathbb{S}_{1}^{n}$.

###### Proof.

Apply the previous proposition to the function
$f(\textnormal{exp}(iz)):U\rightarrow\mathbb{C}$ for every
$U\subseteq\mathbb{C}^{n}$ which intersects $\mathbb{R}^{n}$. ∎

######  Lemma 6.12.

Let $\left\\{L_{j}\times W_{j}\mid j\in\omega\right\\}$ be a countable set of
non-rotund varieties, and let $\delta_{j}$ be the $\delta$-map of $L_{j}\times
W_{j}$ for each $j$. Then
$\mathbb{S}_{1}^{n}\nsubseteq\bigcup_{j\in\omega}\textnormal{im}(\delta_{j})$.

###### Proof.

By Proposition 2.5, the image of a complex analytic function $f:A\rightarrow
B$ is contained in a countable union of complex analytic subsets of $B$, of
dimension at most $\dim A-\min\\{\dim f^{-1}(b)\mid b\in B\\}.$

In this case then consider $\delta_{j}:L_{j}\times
W_{j}\rightarrow(\mathbb{C}^{\times})^{n}$. Since $L_{j}\times W_{j}$ is not
rotund, its image has empty interior, and the fibres of $\delta_{j}$ need to
have positive dimension. Thus, the image of each $\delta_{j}$ must be
contained in a countable union of analytic subsets of
$(\mathbb{C}^{\times})^{n}$ of positive codimension; hence in particular it is
contained in a countable union of analytic subsets of
$(\mathbb{C}^{\times})^{n}$ of codimension 1, each of which is defined by an
equation of the form $g(z)=0$ for $g$ a complex analytic function. By
Corollary 6.11, each of the $g$’s does not vanish on open subsets of
$\mathbb{S}_{1}^{n}$. Hence for each $g$, the set
$\\{z\in\mathbb{S}_{1}^{n}\mid g(z)=0\\}$ is a real analytic subset of
$\mathbb{S}_{1}^{n}$, of positive real codimension: therefore the union of
these zero sets cannot cover $\mathbb{S}_{1}^{n}$.

If we consider countably many $\delta$’s at once we still take a countable
union, so the argument still applies. ∎

We recall that for these kind of varieties, $w\notin\textnormal{im}(\delta)$
is equivalent to $\textnormal{exp}(L)\cap w^{-1}\cdot W=\varnothing.$

We now examine shortenings of systems associated to varieties of the form
$L\times W$.

######  Lemma 6.13.

Let $f_{1}=\dots=f_{n}=0$ be a system of exponential sums equations associated
to a variety of the form $L\times W$, so for all $j$ we have that $f_{j}$ can
be rewritten as a Laurent polynomial (all exponents are integers) or it has
only two terms (see Definition 5.2).

Then for any shortening of the system, one of the following holds:

  * 1.

There is $j$ for which the shortened equation $f_{j}^{\prime}$ has the form
$\textnormal{exp}(\varphi(z))=0$, and is thus inconsistent;

  * 2.

The shortened system is associated to $L\times W_{\tau}$ for an initial
variety $W_{\tau}$ of $W$ such that $L\times W_{\tau}$ is not rotund.

###### Proof.

Assume the first condition does not hold; then all faces in the coherent set
$Q_{1},\dots,Q_{n}$ which defines the shortening are positive dimensional. In
particular, all polytopes of the equations defining $L$ appear among the
$Q_{j}$’s, and so the system defines $L\times W^{\prime}$ for some initial
variety $W^{\prime}$ of $W$.

However, all the faces are contained in translates of the same hyperplane, and
therefore the mixed volume $\textnormal{MV}(Q_{1},\dots,Q_{n})$ has to be zero
by [MS15, Lemma 4.6.6]. So the variety is not rotund by Corollary 5.6. ∎

Thus we may prove the following result.

######  Lemma 6.14.

Let $L\times W$ be an additively free rotund variety whose $\delta$-map is
open. Then there is $s\in\mathbb{S}_{1}^{n}$ such that the system defining
$L\times s\cdot W$ is non-degenerate at infinity in $\mathbb{C}^{n}$.

###### Proof.

Since the $\delta$-map is open, all of its fibres are discrete: therefore all
intersections $L\cap z+\log W$ are isolated, and the first condition in the
definition of non-degneracy at infinity is satisfied for all systems
associated to varieties of the form $L\times w\cdot W$ for
$w\in(\mathbb{C}^{\times})^{n}$.

Consider now all shortenings of the system corresponding to $L\times W.$ We
saw in Lemma 6.13 that there are two kinds; let us focus on the second kind,
namely the one associated to $L\times W_{\tau}$ for some initial variety
$W_{\tau}$ of $W.$

This system defines

$\left(\bigcup_{t\in T}t+L\right)\cap\log W_{\tau}$

for a countable set $T$. Each translate $t+L$ intersects $\log W_{\tau}$ if
and only if $\textnormal{exp}(t)\in\textnormal{im}(\delta_{\tau})$ where
$\delta_{\tau}$ is the $\delta$-map of $L\times W_{\tau}$.

By Lemma 6.12, there is $s\in\mathbb{S}_{1}^{n}$ such that for each $t\in T$,
$s\cdot\textnormal{exp}(t)$ does not lie in the image of $\delta_{\tau}$ for
each shortening of this kind: therefore, $\textnormal{exp}(t)$ does not lie in
the image of the $\delta$-map of $L\times s^{-1}\cdot W_{\tau}$ for every
$t\in T$ and for every shortening.

Since $s^{-1}\cdot W_{\tau}=(s^{-1}\cdot W)_{\tau}$, this means that all
shortenings of the second kind of the system are inconsistent. The shortenings
of the first kind are always inconsistent, as they equate the exponential of
something to 0, and therefore the system associated to $L\times s^{-1}\cdot W$
is non-degenerate at infinity in $\mathbb{C}^{n}$. ∎

It is immediate that if $s\in\mathbb{S}_{1}^{n}$ then $\mathcal{A}_{s\cdot
W}=\mathcal{A}_{W}$. Therefore, we obtain the desired point in
$\textnormal{Re}(L)\cap\mathcal{A}_{W}$.

######  Lemma 6.15.

Let $L\times W$ be a variety whose $\delta$-map is open. Then
$\textnormal{Re}(L)\cap\mathcal{A}_{W}\neq\varnothing$.

###### Proof.

By Lemma 6.14, there is $s\in\mathbb{S}_{1}^{n}$ such that the system
associated to $L\times s\cdot W$ is non-degenerate at infinity in
$\mathbb{C}^{n}$; therefore by Theorem 6.9 it has a solution.

If a point $z$ solves this system, then
$\textnormal{Re}(z)\in\textnormal{Re}(L)$: the exponential sums associated to
$L$ take the form $\textnormal{exp}(\varphi_{\lambda}(z))=1$, and if
$\varphi_{\lambda}$ is a real function then it needs to be the case that
$\lambda\cdot\textnormal{Re}(z)=0$. We have already noticed that
$\textnormal{Re}(z)\in\mathcal{A}_{s\cdot W}=\mathcal{A}_{W}$, and therefore
$\textnormal{Re}(z)\in\textnormal{Re}(L)\cap\mathcal{A}_{W}$. ∎

######  Theorem 6.16.

Let $L\times W$ be an additively free rotund variety, $L\leq\mathbb{C}^{n}$
linear defined over the reals. Then $\textnormal{exp}(L)\cap
W\neq\varnothing$.

###### Proof.

It follows from the combination of Lemmas 6.3 and 6.15. ∎

The Rabinowitsch trick (which we have already used to show that the
$\delta$-map can be assumed to be open) actually implies that the intersection
between $\textnormal{exp}(L)$ and $W$ is Zariski-dense in $W$.

######  Corollary 6.17.

Let $L\times W$ be an additively free rotund variety, $L\leq\mathbb{C}^{n}$
defined over the reals. Then $\textnormal{exp}(L)\cap W$ is Zariski-dense in
$W$.

###### Proof.

Let $F:W\rightarrow\mathbb{C}$ be an algebraic function: then $\\{w\in W\mid
F(w)\neq 0\\}$ is a Zariski-open subset of $W$. Define

$W^{\prime}:=\\{(w_{1},\dots,w_{n+1})\in\mathbb{C}^{n}\mid(w_{1},\dots,w_{n})\in\mathbb{C}^{n}\wedge
w_{n+1}=F(w_{1},\dots,w_{n})\\}.$

Then $(L\times\mathbb{C})\times W^{\prime}$ is an additively free rotund
subvariety of $\mathbb{C}^{n+1}\times(\mathbb{C}^{\times})^{n+1}$, and
therefore there is a point in $\textnormal{exp}(L\times\mathbb{C})\cap
W^{\prime}$, i.e. a point $(w,w_{n+1})$ with $w\in\textnormal{exp}(L)\cap W$
and $0\neq w_{n+1}=F(w)$. Hence, $w\in\textnormal{exp}(L)\cap\\{w\in W\mid
F(w)\neq 0\\}$, as we wanted. ∎

##  7 Complex Tropical Geometry

If we try to extend our result to all varieties of the form $L\times W$ with
$L$ linear (so without any assumptions on the field of definition of $L$) we
run into some obvious issues - the first of which is the fact that, for such
an $L$, $\textnormal{exp}(L)$ is not dense in
$\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$. The easiest example of this is
the space $L=\\{(z_{1},z_{2})\in\mathbb{C}^{2}\mid z_{2}=iz_{1}\\}$: in this
case $\textnormal{exp}(L)$ is a closed 1-dimensional analytic subgroup of
$(\mathbb{C}^{\times})^{2}$, while
$\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{2}=(\mathbb{C}^{\times})^{2}$. This
obviously makes it harder to find good approximated solutions.

However, a second aspect to be taken into consideration is the fact that if
$L$ is not defined over $\mathbb{R}$ then
$L\lneq\textnormal{Re}(L)+i\textnormal{Re}(L)$; and in particular,
$\dim_{\mathbb{C}}L<\dim_{\mathbb{R}}\textnormal{Re}(L)$. Thus, if we are
given an additively free rotund variety $L\times W$, with $\dim
L=\textnormal{codim}W$, then we find that
$\textnormal{Re}(L)\cap_{st}\textnormal{Trop}(W)$ must be a positive
dimensional polyhedral complex. Vaguely, this can be interpreted to mean that
“there are ways to make $\textnormal{exp}(L)$ and $W$ approach infinity in the
same direction”: in such a direction, $W$ will resemble one of its initial
varieties $W_{\tau}$, and if $L\times W_{\tau}$ is itself rotund (and
therefore satisfies the open mapping property) then we have good chances to
use points in $\textnormal{exp}(L)\cap W_{\tau}$ as approximated solutions of
our system.

###  7.1 Stable Intersections

Stable intersections are intersections between polyhedral complexes which are
preserved under small perturbations. Our approach is slightly different from
the one in [MS15, Section 3.6], as they also care about the multiplicities of
the polyhedra in a complex, which we do not define and will not need.

######  Definition 7.1.

Let $\Sigma_{1}$, $\Sigma_{2}$ be polyhedral complexes in $\mathbb{R}^{n}$.
The stable intersection of $\Sigma_{1}$ and $\Sigma_{2}$, denoted
$\Sigma_{1}\cap_{st}\Sigma_{2}$, is the polyhedral complex consisting of
polyhedra of the form $\sigma_{1}\cap\sigma_{2}$, where
$\sigma_{i}\in\Sigma_{i}$ for $i=1,2$ and $\dim(\sigma_{1}+\sigma_{2})=n$.

$\Sigma$$l$ Figure 5: The stable intersection of the usual polyhedral complex
$\Sigma$ with the line $l$ is $\\{0\\}$: each facet intersects the line
transversely. The stable intersection of $\Sigma$ with, say, the $x$-axis is
still $\\{0\\}$: although the $x$-axis intersects the horizontal facet, the
Minkowski sum of the facet and the axis does not have dimension $2$, and so
the intersection does not count towards the stable intersection.

In “classical” tropical geometry, the main interest in stable intersections
comes from this fact, which ties the combinatorial definition to its geometric
meaning.

######  Fact 7.2 ([MS15, Theorem 3.6.1]).

Let $W_{1},W_{2}\subseteq(\mathbb{C}^{\times})^{n}$ be algebraic varieties.
Then
$\textnormal{Trop}(W_{1})\cap_{st}\textnormal{Trop}(W_{2})\neq\varnothing$ if
and only if the set

$\\{w\in(\mathbb{C}^{\times})^{n}\mid w\cdot W_{1}\cap
W_{2}\neq\varnothing\\}$

is Zariski-open dense in $(\mathbb{C}^{\times})^{n}$.

We conclude this section by stating a theorem which goes in that direction.
Note that stable intersection is associative (see [MS15, Remark 3.6.14].)

######  Theorem 7.3 ([MS15, Theorem 4.6.9]).

Suppose $P_{1},\dots,P_{r}$ are polytopes in $\mathbb{R}^{n}$ with integer
vertices and $\Sigma_{1},\dots,\Sigma_{r}$ are the $(n-1)$-skeletons of their
normal fans. Fix $w\in\mathbb{R}^{n}$, and let
$Q_{i}:=\textnormal{face}_{w}(P_{i})$.

Then $w\in\Sigma_{1}\cap_{st}\dots\cap_{st}\Sigma_{r}$ if and only if for
every $J\subseteq\\{1,\dots,r\\}$ we have $\dim\left(\sum_{j\in
J}Q_{j}\right)\geq|J|$, if and only if the $r$-dimensional mixed volume of the
faces $\textnormal{MV}(Q_{1},\dots,Q_{r})$ is non-zero.

###  7.2 Piecewise-Linear Functions

In this section we review some of notions from [Kaz14a] and [Kaz14], and use
them to derive the following result which will be used later on.

######  Theorem 7.4.

Let $L\times W$ be an additively free, rotund variety, with $L$ linear not
defined over $\mathbb{R}$. Then there is $\tau\in\textnormal{Trop}(W)$,
$\dim\tau>0$, such that:

  * 1.

$L\times W_{\tau}$ is rotund;

  * 2.

$(\textnormal{Supp}(\textnormal{Re}(L)\cap_{st}\textnormal{Trop}(W))\cap\textnormal{relint}\tau\neq\varnothing$.

This result will be given a geometric interpretation in the next section.

In what follows, let $\langle z,w\rangle$ denote the usual Hermitian product
on $\mathbb{C}^{n}$, so that if $z=(z_{1},\dots,z_{n})$ and
$w=(w_{1},\dots,w_{n})$ then $\langle
z,w\rangle=z_{1}\overline{w}_{1}+\dots+z_{n}\overline{w}_{n}$.

######  Remark 7.5.

Identify $z=(z_{1},\dots,z_{n})=(x_{1}+iy_{1},\dots,x_{n}+iy_{n})$ with the
real vector $(x_{1},y_{1},\dots,x_{n},y_{n})\in\mathbb{R}^{2n}$. Then we have
that $\textnormal{Re}\left(\langle z,z^{\prime}\rangle\right)$ coincides with
the real scalar product in $\mathbb{R}^{2n}$, as

$\textnormal{Re}(\langle(x_{1}+iy_{1},\dots,x_{n}+iy_{n}),(x_{1}^{\prime}+iy_{1}^{\prime},\dots,x_{n}^{\prime}+iy_{n}^{\prime})\rangle)=$
$=x_{1}x_{1}^{\prime}+y_{1}y_{1}^{\prime}+\dots+x_{n}x_{n}^{\prime}+y_{n}y_{n}^{\prime}.$

######  Definition 7.6.

A piecewise-linear function on $\mathbb{C}^{n}$ is a function
$h:\mathbb{C}^{n}\rightarrow\mathbb{R}$ for which there are polyhedra
$P_{1},\dots,P_{k}\subseteq\mathbb{C}^{n}\cong\mathbb{R}^{2n}$ and vectors
$a_{1},\dots,a_{k}\in\mathbb{C}^{n}$ such that
$\bigcup_{j=1}^{k}P_{j}=\mathbb{C}^{n}$ and $h$ coincides with the function
$z\mapsto\textnormal{Re}\langle z_{j},a_{j}\rangle$ on each polyhedron $P_{j}$
.

Any $\mathbb{R}$-linear function $\mathbb{C}^{n}\rightarrow\mathbb{R}$ is
clearly piecewise-linear.

######  Definition 7.7.

Let $A$ be a non-empty closed convex subset of $\mathbb{C}^{n}$. The support
function of $A$ is the function $h_{A}:\mathbb{C}^{n}\rightarrow\mathbb{R}$
defined by

$h_{A}(z)=\sup\\{\textnormal{Re}\langle z,a\rangle\mid a\in A\\}.$

The support function of a convex polyhedron is then clearly a piecewise-linear
function, whose domains of linearity form a polyhedral complex which coincides
with the normal fan to the polyhedron (and whose locus of non-
differentiability coincides with the $(n-1)-$skeleton of such a fan).

In [Kaz14] Kazarnovskii introduced a numerical invariant for collections of
polytopes in $\mathbb{C}^{n}$, the pseudo-mixed volume, and used it to study
tropical properties of systems of exponential sums with complex exponents,
with a particular attention to the behaviour of stable intersections. He also
proved that non-vanishing of the pseudo-mixed volume is equivalent to a
certain occurrence of the Rado property - we will show in a bit how to tie
this to rotundity using Lemma 5.5.

Given a Laurent polynomial $f\in\mathbb{C}[w_{1}^{\pm 1},\dots,w_{n}^{\pm
n}]$, consider its Newton polytope $P$. We have seen how this is a subset of
the dual space $(\mathbb{C}^{n})^{\vee}$: we can therefore interpret the
support function $h_{P}:(\mathbb{C}^{n})^{\vee}\rightarrow\mathbb{R}$ as

$h_{P}(z)=\\{\max\textnormal{Re}(\varphi(z))\mid\varphi\in P\\}.$

As the exponents of $f$ are integer (and therefore real) numbers for any
function in $P$ we will have $\varphi(z+iv)=\varphi(z)$ for all
$v\in\mathbb{R}^{n}$: therefore, the domains of linearity of $h_{P}$ are
subsets of the form $D+i\mathbb{R}^{n}$ for some polyhedra
$D\subseteq\mathbb{R}^{n}$.

On the other hand, consider the function $z\mapsto
l_{1}z_{1}+\dots+l_{n}z_{n}$ on $\mathbb{C}^{n}$: this is a point in
$(\mathbb{C}^{n})^{\vee}$. Consider the convex hull of
$\\{0,(l_{1},\dots,l_{n})\\}$: this is a segment in $(\mathbb{C}^{n})^{\vee}$,
orthogonal to the real hyperplane

$H:=\\{z\in\mathbb{C}^{n}\mid\textnormal{Re}\langle
z,\overline{l}\rangle=0\\}.$

If $(l_{1},\dots,l_{n})\in\mathbb{R}^{n}$, then $H$ itself is invariant under
translation by elements in $i\mathbb{R}^{n}$.

######  Definition 7.8.

Let $\tau\subseteq\mathbb{R}^{n}$ be a polyhedron. We denote by
$\tau_{\mathbb{C}}$ the complexification
$\textnormal{aff}(\tau)\otimes\mathbb{C}$ of its span.

Consider a linear space $L\leq\mathbb{C}^{n}$. Every space $L$ may be written
as $L=\textnormal{Re}(L)_{\mathbb{C}}\cap H$ for some
$H\subseteq\mathbb{C}^{n}$ with $\textnormal{Re}(H)=\mathbb{R}^{n}$. Thus, the
equations defining $L$ may be assumed to take the form

$\begin{cases}r_{1,1}z_{1}+\dots+r_{1,n}z_{n}=0\\\ \vdots\\\
r_{k,1}z_{1}+\dots+r_{k,n}z_{n}=0\\\
\lambda_{1,1}z_{1}+\dots+\lambda_{1,n}z_{n}=0\\\ \vdots\\\
\lambda_{n-d-k,1}z_{1}+\dots+\lambda_{n-d-k,n}z_{n}=0\\\ \end{cases}$

where the vectors $(r_{j,1},\dots,r_{j,n})$ are in $\mathbb{R}^{n}$ and the
corresponding equations define $\textnormal{Re}(L)_{\mathbb{C}}$ and the
vectors $(\lambda_{j,1},\dots,\lambda_{j,n})$ are in
$\mathbb{C}^{n}\setminus\mathbb{R}^{n}$. We denote by $h_{r_{j}}$ the function
$z\mapsto\langle z,r_{j}\rangle$ and by $h_{\lambda_{j}}$ the function
$z\mapsto\langle z,\overline{\lambda_{j}}\rangle$.

Consider the system of exponential sums associated to the variety $L\times W$,
in the sense of Definition 5.2. We know how to attach to this system a set of
convex polytopes. For each of these polytopes, the locus of non-
differentiability of the support function is a polyhedral complex. In
particular, the corner locus of $h_{P_{j}}$ will be
$\Sigma_{j}+i\mathbb{R}^{n}$ where $\Sigma_{j}$ denotes the tropicalization of
the hypersurface defined by the polynomial $f_{j}$; the corner locus of
$h_{r_{j}}$ (resp. $h_{\lambda_{j}}$) will be the hyperplane defined by
$\textnormal{Re}\langle z,r_{j}\rangle=0$ (resp. $\textnormal{Re}\langle
z,\overline{\lambda}_{j}\rangle$). We refer to these as the collection of
polyhedral complexes associated to the variety $L\times W$.

Non-emptiness of the stable intersection of these complexes can be
characterized using the mixed Monge-Ampère operator. This is an operator which
associates to a tuple of $k$ piecewise-linear functions on $\mathbb{C}^{n}$ a
current, i.e. a linear functional on the space of smooth compactly supported
differential forms. We do not get into the details, but rather refer the
reader to [Kaz14, Section 3]; the main idea, which will be used in the
upcoming proof, is that non-vanishing of the mixed Monge-Ampère operator for
the support functions of $n$ convex polytopes in $\mathbb{C}^{n}$ is
equivalent to non-emptiness of the stable intersections of the corresponding
polyhedral complexes. This is essentially [Kaz14, Theorem 3.1], which
establishes an isomorphism of rings between a ring of currents and a ring of
equivalence classes of polyhedral complexes, in which the product is given by
stable intersections.

######  Lemma 7.9.

The stable intersection of the collection of polyhedral complexes associated
to the variety $L\times W$ is non-empty if and only if the variety $L\times W$
is rotund.

###### Proof.

By Lemma 5.5, rotundity is equivalent to the Rado property for the collection
of convex polytopes associated to $L\times W$. By [Kaz14, Corollary 3.3], this
collection has the Rado property if and only if the mixed pseudo-volume of the
polytopes is non-zero; by Theorem 3.5 in the same paper this is equivalent to
non-vanishing of the mixed Monge-Ampère operator on the collection of support
functions of the polytopes, and finally by [Kaz14, Proposition 3.1] this is
equivalent to non-emptiness of the stable intersection. ∎

As an aside, we note that going back to the case in which $L$ is defined over
the reals this says that $L\times W$ is rotund if and only if
$\textnormal{Re}(L)\cap_{st}\textnormal{Trop}(W)\neq\varnothing$: therefore a
version of Fact 7.2 holds for this kind of varieties, establishing that the
stable intersection between $\textnormal{Trop}(W)$ and $\textnormal{Re}(L)$
(which acts as a sort of “$\textnormal{Trop}(\textnormal{exp}(L))$”) can be
lifted to an actual intersection between $W$ and $\textnormal{exp}(L)$.

This allows us to prove the main result of this section, Theorem 7.4.

###### Proof of Theorem 7.4.

As $L\times W$ is rotund, Lemma 7.9 guarantees that the collection of
polyhedral complexes associated to the variety has non-empty stable
intersection. This stable intersection is obviously a subset of the stable
intersection of any subset of the collection of polyhedral complexes:
therefore, it needs to be contained in the stable intersection of those
complexes which are loci of non-differentiability of $\mathbb{R}$-linear
functions, that is the stable intersections of the tropical hypersurfaces
$\Sigma_{j}+i\mathbb{R}^{n}$ and the hyperplanes defined by
$\textnormal{Re}\langle r_{j},z\rangle$ for $r_{j}\in\mathbb{R}^{n}$. (Note
that if $L$ were defined over the reals then these would be all the complexes,
and the stable intersection would turn out to be exactly $i\mathbb{R}^{n}$.)

Since $L$ is not defined over the reals, there is at least one complex linear
equation (and so at least one hyperplane not defined over $\mathbb{R}$ in the
collection of polyhedral complexes associated to $L\times W$). This does not
contain $i\mathbb{R}^{n}$, and therefore the stable intersection of the
complexes cannot be of the form $iT$ with $T$ an $n$-dimensional polyhedral
complex in $\mathbb{R}^{n}$. So let $\tau_{0}$ be any maximal cell of the
stable intersection: this contains, in its relative interior, a point of the
form $x+iy\in\mathbb{C}^{n}$ with $x\neq 0$. Therefore $x$ is in the support
of $\textnormal{Re}(L)\cap_{st}\textnormal{Trop}(W)$; let
$\tau\in\textnormal{Trop}(W)$ be the cell such that
$x\in\textnormal{relint}(\tau)$.

Consider the variety $L\times W_{\tau}$. As $W_{\tau}$ is an initial variety,
its tropicalization is equal to
$\textnormal{star}_{\tau}(\textnormal{Trop}(W))$ (by Lemma 4.22); in other
words it contains cells with the same affine spans as the cells of
$\textnormal{Trop}(W)$ which contain $\tau$. Therefore, if the collection of
complexes associated to $L\times W$ has non-empty stable intersection, then so
must the collection associated to $L\times W_{\tau}$: by the converse
implication of Lemma 7.9, then, $L\times W_{\tau}$ is rotund.

So $L\times W_{\tau}$ satisfies the statement. ∎

##  8 Raising to Complex Powers

In this section we establish our main result, the existence of solutions to
all systems of equations associated to additively free rotund varieties of the
form $L\times W$ with $L$ linear.

To do so we are going to use Theorem 7.4 to show that in a suitable
compactification of $(\mathbb{C}^{\times})^{n}$, $W$ and $\textnormal{exp}(L)$
have got sequences of points with the same limit, and that these sequences can
be taken to avoid phenomena of asymptoticity. To make this clear, the first
step is to give a geometric interpretation of Theorem 7.4, explaining the
second clause of the statement (the consequences of
$\textnormal{Re}(L)\cap_{st}\textnormal{Trop}(W)$ intersecting the relative
interior of $\tau$).

We will study compactifications of $W$ which live in toric varieties.

######  Definition 8.1.

A complex toric variety is a complex algebraic variety $Y$ for which there
exists an embedding $\iota:(\mathbb{C}^{\times})^{n}\hookrightarrow Y$ for
some $n$, such that:

  * 1.

The image of $\iota$ is an open dense subset of $Y$;

  * 2.

There is a continuous action of $(\mathbb{C}^{\times})^{n}$ on $Y$ which
coincides with the usual multiplication on the image of $\iota$.

The three easiest examples of complex toric varieties are the torus
$\mathbb{C}^{\times}$ itself, with just one orbit of the multiplicative
action, the affine line $\mathbb{A}^{1}(\mathbb{C})$ with the two orbits
$\\{0\\}$ and $\mathbb{A}^{1}(\mathbb{C})\setminus\\{0\\}$ and the projective
line $\mathbb{P}^{1}(\mathbb{C})$ with orbits $\\{0\\},\\{\infty\\}$ and
$\mathbb{P}^{1}(\mathbb{C})\setminus\\{0,\infty\\}$. The theory of toric
varieties is classical and several books on the subject exist, such as [Ful93]
and [CLS11].

A standard construction associates a toric variety to a polyhedral fan. We
refer the reader to [MS15, Section 6.1] or [NS13, Section 4] for details on
this construction and just recall that given the polyhedral fan
$\Sigma\subseteq\mathbb{R}^{n}$ there is a toric variety $Y_{\Sigma}$ such
that the action of $(\mathbb{C}^{\times})^{n}$ on $Y_{\Sigma}$ has precisely
one orbit $\mathcal{O}_{\tau}$ for each polyhedron $\tau\in\Sigma$. Denoting
by $\mathbb{T}_{\tau}$ the algebraic subgroup of $(\mathbb{C}^{\times})^{n}$
obtained as $\textnormal{exp}(\tau_{\mathbb{C}})$, we have that the orbit
$\mathcal{O}_{\tau}$ is itself a torus, isomorphic to
$(\mathbb{C}^{\times})^{n}/\mathbb{T}_{\tau}$: each point of
$\mathcal{O}_{\tau}$ is a point at infinity of a translate
$w\cdot\mathbb{T}_{\tau}$, and different translates have different points at
infinity.

######  Example 8.2.

Consider once again the curve
$W=\\{(w_{1},w_{2})\in(\mathbb{C}^{\times})^{2}\mid w_{1}+w_{2}+1=0\\}$: we
know that $\Sigma=\textnormal{Trop}(W)$ is a polyhedral complex consisting of
three half-lines and the origin. In this case, thus $Y_{\Sigma}$ needs to have
four orbits. The orbit $\mathcal{O}_{0}$ coincides with
$(\mathbb{C}^{\times})^{2}$. The other three are attached to the algebraic
subgroups of $(\mathbb{C}^{\times})^{2}$ defined by $w_{1}w_{2}^{-1}=1$,
$w_{1}=1$ and $w_{2}=1$ respectively. Thus, for example, $Y_{\Sigma}$ contains
a point at infinity for each translate of the group defined by $w_{1}=1$, of
the form $(c,\infty)$ for some $c\in\mathbb{C}^{\times}$.

We are particularly interested, of course, in the case in which $\Sigma$ is
the tropicalization of a variety $W$. In that case Tevelev has defined and
studied tropical compactifications of varieties: the tropical compactification
of $W$ is a compact subset of $Y_{\textnormal{Trop}(W)}$. Again, we refer the
reader to [Tev07], [MS15, Section 6.4] or [NS13, Section 4] for details and
just recall the following properties of tropical compactifications.

######  Theorem 8.3 (Tevelev, see [NS13, Lemma 12 and Corollary 13]).

Let $W\subseteq(\mathbb{C}^{\times})^{n}$ be an algebraic variety, and let
$\Sigma=\textnormal{Trop}(W)$. Then there is a subvariety
$\overline{W}\subseteq Y_{\Sigma}$ such that:

  * 1.

$\overline{W}$ is complete;

  * 2.

$\overline{W}\cap\mathcal{O}_{0}=W$;

  * 3.

For every $\tau\in\Sigma$, $\mathbb{T}_{\tau}$ acts by translation on the
initial variety $W_{\tau}$, and
$W_{\tau}/\mathbb{T}_{\tau}=\overline{W}\cap\mathcal{O}_{\tau}$ under the
isomorphism
$(\mathbb{C}^{\times})^{n}/\mathbb{T}_{\tau}\cong\mathcal{O}_{\tau}$.

In other words, limits of sequences in $W$ belong to the orbits
$\mathcal{O}_{\tau}$, and any such sequence can be approximated by a sequence
on the initial variety $W_{\tau}$ that lies in just one translate of
$\mathbb{T}_{\tau}$ and has the same limit in $Y_{\Sigma}$.

As we mentioned above, we are going to use these objects to give a geometric
interpretation of Theorem 7.4.

######  Lemma 8.4.

Let $L\times W$ be an additively free rotund variety, $L$ linear not defined
over the reals. Let $\Sigma=\textnormal{Trop}(W)$, and $Y_{\Sigma}$ be the
toric variety associated to the fan. Then there is $\tau\in\Sigma$,
$\dim(\tau)>0$, such that:

  * 1.

$L\times W_{\tau}$ is rotund;

  * 2.

There exist $s\in\mathbb{S}_{1}^{n}$ and a sequence
$\\{s\cdot\textnormal{exp}(l_{j})\\}_{j\in\omega}\subseteq(s\cdot\textnormal{exp}(L))\cap
W_{\tau}$ such that
$\lim_{j}s\cdot\textnormal{exp}(l_{j})\in\mathcal{O}_{\tau}$ and the
$\delta$-map of $L\times W_{\tau}$ is open at
$(0,s\cdot\textnormal{exp}(l_{j}))$ for each $j\in\omega$.

###### Proof.

Let $L\times W$ be an additively free rotund variety, and take
$\tau\in\textnormal{Trop}(W)$ to be the polyhedron whose existence is granted
by Theorem 7.4, so that $L\times W_{\tau}$ is rotund and
$\textnormal{Supp}(\textnormal{Re}(L)\cap_{st}\textnormal{Trop}(W))\cap\textnormal{relint}(\tau)\neq\varnothing$.

Consider the initial variety $W_{\tau}$. By Proposition 4.25, it is invariant
under translation by elements of $\textnormal{exp}(\tau_{\mathbb{C}})$.

By Proposition 3.7, there is a Zariski-open dense subset $(W_{\tau})^{\circ}$
of $W_{\tau}$ such that the $\delta$-map of $L\times W_{\tau}$ is open on
$L\times(W_{\tau})^{\circ}$. The variety $L\times(W_{\tau})$ is also
additively free, because $L\times W$ is.

By Corollary 6.17,
$\overline{\textnormal{exp}(\textnormal{Re}(L)_{\mathbb{C}})}=\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$
intersects $W_{\tau}$ in a Zariski-dense subset, and therefore it must in
particular intersect $(W_{\tau})^{\circ}$; then, let $w\in(W_{\tau})^{\circ}$
be a point in this intersection, so that $w\in s_{0}\cdot\textnormal{exp}(L)$
for some $s_{0}\in\mathbb{S}_{1}^{n}$. The $\delta$-map, as we said, is open
at $(0,w)$, and therefore there are neighbourhoods $V$ of $0$ in $L$ and $U$
of $w$ in $W$ such that
$\frac{U}{\textnormal{exp}(V)}\subseteq\textnormal{im}(\delta)$ contains an
open ball around $w$.

Since $\textnormal{Re}(L)\cap\textnormal{relint}(\tau)\neq\varnothing$, we may
consider a sequence
$\\{t_{j}\\}_{j\in\omega}\subseteq\textnormal{Re}(L)_{\mathbb{C}}\cap\tau_{\mathbb{C}}$
such that $\lim_{j\in\omega}\textnormal{exp}(t_{j})\in\mathcal{O}_{\tau}$
(this follows from Theorem 8.3, since
$\textnormal{exp}(t_{j})\in\mathbb{T}_{\tau}$ for each $j$). Hence,
$\lim_{j\in\omega}w\cdot\textnormal{exp}(t_{j})=w\cdot\lim_{j\in\omega}\textnormal{exp}(t_{j})\in\mathcal{O}_{\tau}$
as well, since the multiplicative action of $(\mathbb{C}^{\times})^{n}$ on
$Y_{\Sigma}$ is continuous and $\mathcal{O}_{\tau}$ is an orbit. Also,
$\textnormal{exp}(t_{j})\cdot U\subseteq W_{\tau}$ for each $j$, given that
$W_{\tau}$ is invariant under translation by
$\textnormal{exp}(\tau_{\mathbb{C}})$.

Moreover, as
$\\{\textnormal{exp}(t_{j})\\}_{j\in\omega}\subseteq\textnormal{exp}(L)\cdot\mathbb{S}_{1}^{n}$,
for each $j$ there is $l_{j}^{0}\in L$ such that
$\frac{\textnormal{exp}(t_{j})\cdot
w}{\textnormal{exp}\left(l_{j}^{0}\right)}\in\mathbb{S}_{1}^{n}.$ By
compactness of $\mathbb{S}_{1}^{n}$, we may assume that the sequence
$\left\\{\frac{\textnormal{exp}(t_{j})\cdot
w}{\textnormal{exp}\left(l_{j}^{0}\right)}\right\\}_{j\in\omega}$ has a limit,
call it $s\in\mathbb{S}_{1}^{n}$.

Consider for each $j$ the open set $\frac{\textnormal{exp}(t_{j})\cdot
U}{\textnormal{exp}\left(l_{j}^{0}\right)\cdot\textnormal{exp}(V)}=\frac{\textnormal{exp}(t_{j})}{\textnormal{exp}\left(l_{j}^{0}\right)}\cdot\frac{U}{\textnormal{exp}(V)}$.
As the set $\frac{U}{\textnormal{exp}(V)}$ was built to contain an open ball
around $w$, for $j$ sufficiently large we must have
$s\in\frac{\textnormal{exp}(t_{j})}{\textnormal{exp}\left(l_{j}^{0}\right)}\cdot\frac{U}{\textnormal{exp}(V)}$.

Therefore we can take a sequence
$\\{s\cdot\textnormal{exp}(l_{j})\\}_{j\in\omega}$, where each
$\textnormal{exp}(l_{j})$ lies in
$\textnormal{exp}\left(l_{j}^{0}\right)\cdot\textnormal{exp}(V)\subseteq\textnormal{exp}(L)$
and each $s\cdot\textnormal{exp}(l_{j})$ belongs to
$\textnormal{exp}(t_{j})\cdot U\subseteq W_{\tau}$; in other words, so that
$\\{s\cdot\textnormal{exp}(l_{j})\\}_{j\in\omega}\subseteq
s\cdot\textnormal{exp}(L)\cap W_{\tau}$, as required. ∎

######  Corollary 8.5.

Let $L\times W$ be an additively free, rotund variety, $L$ linear not defined
over the reals, $\tau\in\textnormal{Trop}(W)$ as given by Lemma 8.4, $U$ a
neighbourhood of the identity in $(\mathbb{C}^{\times})^{n}$. Then there is a
sequence $\\{t_{j}\\}_{j\in\omega}\subseteq\tau_{\mathbb{C}}$ such that
$\lim_{j}\textnormal{exp}(t_{j})\in\mathcal{O}_{\tau}$ and
$\textnormal{exp}(t_{j})\cdot U\cap\textnormal{exp}(L)\neq\varnothing$ for all
$j\in\omega.$

###### Proof.

By replacing $U$ by $U\cap U^{-1}$ if necessary we assume that $U$ is
symmetric.

Consider the sequence $\\{s\cdot\textnormal{exp}(l_{j})\\}_{j\in\omega}$ given
by Lemma 8.4.

As
$\lim_{j}s\cdot\textnormal{exp}(l_{j})\in\mathcal{O}_{\tau}\cong(\mathbb{C}^{\times})^{n}/\mathbb{T}_{\tau}$,
there is a translate $w\cdot\mathbb{T}_{\tau}$ such that
$\lim_{j}s\cdot\textnormal{exp}(l_{j})\cdot\mathbb{T}_{\tau}=w\cdot\mathbb{T}_{\tau}$.
Then we may assume that $s\cdot\textnormal{exp}(l_{j})\cdot U\cap
w\cdot\mathbb{T}_{\tau}\neq\varnothing$ for all $j\in\omega.$ Therefore, we
automatically get that for all sufficiently large $j$’s,

$\frac{s\cdot\textnormal{exp}(l_{j})}{w}\cdot
U\cap\mathbb{T}_{\tau}\neq\varnothing$
$\frac{s}{w}\cdot\textnormal{exp}(l_{j})\cdot
U\cap\mathbb{T}_{\tau}\neq\varnothing$

and this implies that taking a subsequence if necessary we may assume

$\textnormal{exp}(l_{j})\cdot U\cap\mathbb{T}_{\tau}\neq\varnothing.$

Hence, we can extract from each $\textnormal{exp}(l_{j})\cdot
U\cap\mathbb{T}_{\tau}$ a point $a_{j}\in\mathbb{T}_{\tau}$. Taking a sequence
$\\{t_{j}\\}_{j\in\omega}$, where for each $j$ we have
$\textnormal{exp}(t_{j})=a_{j}$, we prove the corollary. ∎

Next we show that if we can take $s$ in the statement of Lemma 8.4 to be the
identity $(1,\dots,1)$, then we can use the points in the sequence in
$\textnormal{exp}(L)\cap W_{\tau}$ as approximations for points in
$\textnormal{exp}(L)\cap W$.

######  Lemma 8.6.

Let $L\times W$ be an additively free rotund variety, $L$ linear not defined
over the reals. Let $\Sigma=\textnormal{Trop}(W)$, and $Y_{\Sigma}$ be the
toric variety associated to the fan. Suppose there is $\tau\in\Sigma$,
$\dim(\tau)>0$, such that:

  * 1.

$L\times W_{\tau}$ is rotund;

  * 2.

$\textnormal{exp}(L)\cap W_{\tau}$ contains a point $w$ such that the
$\delta$-map of $L\times W_{\tau}$ is open at $(0,w)$.

Then $\textnormal{exp}(L)\cap W\neq\varnothing$.

###### Proof.

Let $w$ be the point given by Assumption 2. Take
$z\in\textnormal{exp}^{-1}(w)$.

Let $L^{*}$ be any subspace of $\mathbb{C}^{n}$ such that $L\oplus
L^{*}=\mathbb{C}^{n}$; let
$\pi_{L}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}/L\cong L^{*}$ denote the
composition of the projection on the quotient with the isomorphism between
$\mathbb{C}^{n}/L$ and $L^{*}$. Since the $\delta$-map of $L\times W_{\tau}$
is open at $(0,w)$, the point $z$ is isolated in $z+L\cap\log W_{\tau}$:
hence, there is a ball $B_{1}$ around 0 in $L$ such that $z+\partial
B_{1}\cap\log W_{\tau}=\varnothing$. Given $B_{1}$, there must be a ball
$B_{2}$ around 0 in $L^{*}$ such that for all $x\in z+B_{1}+B_{2}\cap\log
W_{\tau}$, $x+\partial B_{1}\cap\log W_{\tau}=\varnothing$: this is because
$z+\partial B_{1}$ is compact and $\log W_{\tau}$ is closed, therefore if
their intersection is empty then it must be empty for all translates of
$z+\partial B_{1}$ that are sufficiently close to $z$. Let $U$ denote the
bounded open neighbourhood of $0$ in $\mathbb{C}^{n}$ given by $B_{1}+B_{2}$.

By Proposition 2.2, then, the projection $\pi_{L}$ is proper as a map from
$z+U\cap\log W_{\tau}$ to $\pi_{L}(z)+B_{2}$: this is because $z+U\cap\log
W_{\tau}$ has no limit points on $z+\partial B_{1}+B_{2}\cap\log W_{\tau}$. We
already know the projection is open on this domain, and therefore it has to be
surjective. In other words, $(z+U\cap\log W_{\tau})+L=z+B_{2}+L$.

By Corollary 8.5, there is a sequence
$\\{t_{j}\\}_{j\in\omega}\subseteq\tau_{\mathbb{C}}$ such that $z+t_{j}+U\cap
L\neq\varnothing$ for all $j\in\omega$.

It is easy to see that, since $\log W_{\tau}$ is invariant under translation
by $\tau_{\mathbb{C}}$, $(t_{j}+z+U)\cap\log W_{\tau}=t_{j}+(z+U\cap\log
W_{\tau})$ for each $j$. Hence, all the sets $U\cap-(z+t_{j})+\log W_{\tau}$
are equal to a set $K$. Note that, by the choice of the sets $B_{1}$ and
$B_{2}$, $\overline{K}\cap(\partial B_{1}+B_{2})=\varnothing$.

On the other hand, as $\log W$ is not invariant under translation by
$\tau_{\mathbb{C}}$, the sets $K_{j}:=U\cap-(z+t_{j})+\log W$ may not be all
equal, and their closures $\overline{K_{j}}$ converge in the Hausdorff
distance to $\overline{K}$. This means that for sufficiently large $j$’s,
$\overline{K_{j}}\cap(\partial B_{1}+B_{2})=\varnothing$: if this were not the
case, there would be a point in $K\cap(\partial B_{1}\cap B_{2})$. Therefore,
for sufficiently large $j$’s, the map $\pi_{L}$ is open and proper (again by
Proposition 2.2) as a map from $K_{j}$ into $B_{2}$, and therefore
$K_{j}+L=B_{2}+L$: we assume this holds for all $j$.

Since $K_{j}\subseteq-(z+t_{j})+\log W$, $z+t_{j}+K_{j}\subseteq\log W$. Also,
$z+t_{j}+K_{j}+L=z+t_{j}+B_{2}+L$ for all $j$’s. As $(z+t_{j}+B_{2}+L)\cap
L\neq\varnothing$ (it contains $z+t_{j}+U$, and thus a point which lies in
$\log W_{\tau}\cap L$), we may thus conclude: there are $z^{\prime}\in
z+t_{j}+K_{j}\subseteq\log W$ and $l\in L$ such that $z^{\prime}+l\in L$, and
thus $z^{\prime}\in L$. ∎

Finally we set out to prove that the intersection with the original variety
exists. This is done by induction, exploiting first the simple geometric
structure of initial varieties of curves and then the fact that initial
varieties are invariant under translation by subgroups.

######  Proposition 8.7.

Let $L\times W$ be an additively free rotund variety with $L$ linear not
defined over the reals and $\textnormal{codim}L=\dim W=1$. Then there is
$\tau\in\textnormal{Trop}(W)$, $\dim(\tau)>0$, such that:

  * 1.

$L\times W_{\tau}$ is rotund;

  * 2.

There is $w\in\textnormal{exp}(L)\cap W_{\tau}$ such that the $\delta$-map of
$L\times W_{\tau}$ is open at $(0,w)$.

###### Proof.

Let $\tau\in\textnormal{Trop}(W)$ be the polyhedron given by Lemma 8.4. The
initial variety $W_{\tau}$ is then a finite union of cosets of the subgroup
$\mathbb{T}_{\tau}$. By rotundity of $L\times W_{\tau}$, the $\delta$-map of
this variety is open (as $W_{\tau}$ has dimension 1, if it is open on a
Zariski-open dense subset then it is open everywhere) and therefore its image
is a subgroup of $(\mathbb{C}^{\times})^{n}$ with non-empty interior - that
is, it is $(\mathbb{C}^{\times})^{n}$. Then $\textnormal{exp}(L)\cap
W_{\tau}\neq\varnothing$, and the $\delta$-map is open everywhere. ∎

This allows us to establish our main result.

######  Theorem 8.8.

Let $L\times W$ be an additively free rotund variety, $L$ a linear space. Then
$\textnormal{exp}(L)\cap W\neq\varnothing$.

###### Proof.

We prove this by induction on the dimension of $W$.

Assume $\dim W=1$. If $L$ is defined over the reals, then this follows from
Theorem 6.16. Otherwise, by Proposition 8.7 there is a point
$w\in\textnormal{exp}(L)\cap W_{\tau}$ with $L\times W_{\tau}$ rotund, such
that the $\delta$-map is open around $(0,w)$. Hence by Lemma 8.6,
$\textnormal{exp}(L)\cap W\neq\varnothing$.

Now assume $\dim W>1$. Again, if $L$ is defined over the reals then we apply
Theorem 6.16. Otherwise, consider the polyhedron $\tau\in\textnormal{Trop}(W)$
given by Lemma 8.4: the initial variety $W_{\tau}$ is then invariant under
translation by the algebraic group $\mathbb{T}_{\tau}$. Consider then the
quotients $\pi_{\tau_{\mathbb{C}}}(L)$ and
$\pi_{\mathbb{T}_{\tau}}(W_{\tau})$: their product
$\pi_{\tau_{\mathbb{C}}}(L)\times\pi_{\mathbb{T}_{\tau}}(W_{\tau})$ is
additively free because $\pi_{\tau_{\mathbb{C}}}(L)$ is a quotient of $L$
(here we need additive freeness of $L$), it is rotund because it is a quotient
of a rotund variety, and $\dim(\pi_{\mathbb{T}_{\tau}}(W_{\tau}))<\dim W$
because $\dim W_{\tau}=\dim W$ and $W_{\tau}$ is invariant under translation
by $\mathbb{T}_{\tau}$. By the induction hypothesis,
$\textnormal{exp}(\pi_{\tau_{\mathbb{C}}}(L))\cap\pi_{\mathbb{T}_{\tau}}(W)\neq\varnothing$.
By applying the Rabinowitsch trick if necessary (as that does not change the
dimension of $\pi_{\mathbb{T}_{\tau}}(W)$) we may further assume that there is
a point $w$ in this intersection such that the relevant $\delta$-map is open
at $(0,w)$.

Therefore, there is $(l,w)\in L\times W_{\tau}$ such that the $\delta$-map of
$L\times W_{\tau}$ is open at $(l,w)$, and
$\frac{w}{\textnormal{exp}(l)}\in\mathbb{T}_{\tau}$. Since $W_{\tau}$ is
invariant under translation by $\mathbb{T}_{\tau}$, this implies that actually
$\frac{w}{t}\in\textnormal{exp}(L)\cap W_{\tau}$ for some
$t\in\mathbb{T}_{\tau}$, and the $\delta$-map is open at
$\left(0,\frac{w}{t}\right)$ \- this is again implied by invariance of
$W_{\tau}$ under $\mathbb{T}_{\tau}$, as if the $\delta$-map is open at
$(0,w)$ it must be open at every point of
$\\{0\\}\times(w\cdot\mathbb{T}_{\tau})$ by definition. Hence, by Lemma 8.6,
$\textnormal{exp}(L)\cap W\neq\varnothing$. ∎

######  Example 8.9.

We conclude by testing this argument in the example

$\begin{cases}iz_{1}-z_{2}=0\\\ w_{1}+w_{2}+1=0\\\ \end{cases}$

In this case obviously $\textnormal{Re}(L)=\mathbb{R}^{2}$, and for every
initial variety $W_{\tau}$ we have that $L\times W_{\tau}$ is rotund (the fact
that the dimension is low makes it very easy to check: rotundity of $L\times
W_{\tau}$ just means that $\log W_{\tau}$ and $L$ are not parallel as complex
lines).

So let us fix an initial variety, say $W_{\tau}$ for $\tau$ the positive half-
line spanned by $(1,1)$. The initial variety $W_{\tau}$ is then defined by
$w_{1}+w_{2}=0$, and

$\log W_{\tau}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}\mid
z_{1}-z_{2}\in(2\mathbb{Z}+1)\pi i\ \\}.$

The points of $L\cap\log W_{\tau}$ are easy to find: solving

$\begin{cases}iz_{1}-z_{2}=0\\\ z_{1}-z_{2}=ik\pi\\\ \end{cases}$

for all odd $k\in\mathbb{Z}$, one gets the points

$\frac{2h+1}{2}(\pi-i\pi,\pi+i\pi)$

for all $h\in\mathbb{Z}$. Considering those with $h\gg 0$, the points will be
very close to points in $\log W$. Hence there are sequences
$\\{a_{j}\\}_{j\in\omega}\subseteq W$ and
$\\{b_{j}\\}_{j\in\omega}\subseteq(W_{\tau})\cap\textnormal{exp}(L)$ with the
following properties:

  * 1.

The two sequences converge to the same limit in the tropical compactification
of $W$.

  * 2.

A sequence of neighbourhoods of the $b_{j}$’s can be approximated by a
sequence of neighbourhoods of the $a_{j}$’s.

  * 3.

The $\delta$-map ot $L\times W_{\tau}$ is open at each $(0,b_{j})$.

Combining these three properties, we get that there must be a sequence
$\\{U_{j}\\}_{j\in\omega}$ such that each $U_{j}$ is a neighbourhood of
$a_{j}$, and a neighbourhood $U$ of 0 in $L$ such that the $\delta$-map of
$L\times W$ is open on each $U\times U_{j}$. Hence for very large $j$ we will
have that $\frac{U_{j}}{\textnormal{exp}(U)}$ is an open set intersecting
$\textnormal{exp}(L)$, thus giving a point in the intersection
$W\cap\textnormal{exp}(L)$.

Finally, let us note that the intersection is again Zariski-dense.

######  Corollary 8.10.

Let $L\times W$ be an additively free rotund variety, $L$ a linear space. Then
$\textnormal{exp}(L)\cap W$ is Zariski-dense in $W$.

###### Proof.

By the same proof as Corollary 6.17. ∎

## References

  * [AKM22] Vahagn Aslanyan, Jonathan Kirby and Vincenzo Mantova  “A Geometric Approach to Some Systems of Exponential Equations”  In _International Mathematics Research Notices_ , 2022, pp. 1–36  DOI: 10.1093/imrn/rnab340
  * [BK18] Martin Bays and Jonathan Kirby  “Pseudo-exponential maps, variants, and quasiminimality”  In _Algebra And Number Theory_ 12.3 Mathematical Sciences Publishers, 2018, pp. 493–549  DOI: 10.2140/ant.2018.12.493
  * [BKW10] Martin Bays, Jonathan Kirby and Alex J. Wilkie  “A Schanuel property for exponentially transcendental powers”  In _Bulletin of the London Mathematical Society_ 42.5 Oxford University Press, 2010, pp. 917–922  DOI: 10.1112/blms/bdq054
  * [Bay+14] Martin Bays et al.  “Quasiminimal structures and excellence”  In _Bulletin of the London Mathematical Society_ 46.1 Wiley Online Library, 2014, pp. 155–163  DOI: 10.1112/blms/bdt076
  * [BM17] Woodrow Brownawell and David Masser  “Zero estimates with moving targets”  In _Journal of the London Mathematical Society_ 95, 2017  DOI: 10.1112/jlms.12014
  * [Chi12] Evgeni Mikhailovich Chirka  “Complex analytic sets” 46, Mathematics and its Applications  Springer ScienceBusiness Media, 2012
  * [CLS11] David A. Cox, John B. Little and Henry K. Schenck  “Toric varieties” 124, Graduate Studies in Mathematics  American Mathematical Society, 2011
  * [DFT21] Paola D’Aquino, Antongiulio Fornasiero and Giuseppina Terzo  “A weak version of the Strong Exponential Closure”  In _Israel Journal of Mathematics_ 242.2 Springer, 2021, pp. 697–705  DOI: doi.org/10.1090/tran/7206
  * [Ful93] William Fulton  “Introduction to Toric Varieties”  Princeton University Press, 1993
  * [GKZ94] Israel M. Gelfand, Mikhail M. Kapranov and Andrei V. Zelevinsky  “Discriminants, resultants, and multidimensional determinants”, Mathematics: Theory & Applications  Birkhäuser Boston, 1994
  * [Jon16] Mattias Jonsson  “Degenerations of amoebae and Berkovich spaces”  In _Mathematische Annalen_ 364.1 Springer, 2016, pp. 293–311  DOI: 10.1007/s00208-015-1210-3
  * [Kaz14] Boris Ya. Kazarnovskii  “Action of the complex Monge-Ampère operator on piecewise-linear functions and exponential tropical varieties”  In _Izvestiya: Mathematics_ 78.5 IOP Publishing, 2014, pp. 902–921  DOI: 10.1070/IM2014v078n05ABEH002712
  * [Kaz14a] Boris Ya. Kazarnovskii  “On the action of the complex Monge-Ampère operator on piecewise linear functions”  In _Functional Analysis and Its Applications_ 48.1 Springer, 2014, pp. 15–23  DOI: 10.1007/s10688-014-0042-3
  * [Kho91] Askold G. Khovanskii  “Fewnomials” 88, Translations of Mathematical Monographs  American Mathematical Society, 1991
  * [Kir19] Jonathan Kirby  “Blurred Complex Exponentiation”  In _Selecta Mathematica_ 25.5 Birkhäuser Verlag Basel, 2019  DOI: 10.1007/s00029-019-0517-4
  * [Lan66] Serge Lang  “Introduction to transcendental numbers”  Addison-Wesley Publishing Company, 1966
  * [MS15] Diane Maclagan and Bernd Sturmfels  “Introduction to Tropical Geometry” 161, Graduate Studies in Mathematics  American Mathematical Society, Providence, RI, 2015
  * [MZ16] Vincenzo Mantova and Umberto Zannier  “Polynomial–exponential equations and Zilber’s conjecture”  In _Bulletin of the London Mathematical Society_ 48.2 Wiley, 2016, pp. 309–320  DOI: 10.1112/blms/bdv096
  * [Mar06] David Marker  “A Remark on Zilber’s Pseudoexponentiation”  In _The Journal of Symbolic Logic_ 71.3 Cambridge University Press, 2006, pp. 791–798  DOI: 10.2178/jsl/1154698577
  * [Mik04] Grigory Mikhalkin  “Amoebas of algebraic varieties and tropical geometry”  In _Different faces of geometry_ Springer, 2004, pp. 257–300  DOI: 10.1007/0-306-48658-X˙6
  * [Mik04a] Grigory Mikhalkin  “Decomposition into pairs-of-pants for complex algebraic hypersurfaces”  In _Topology_ 43.5 Elsevier, 2004, pp. 1035–1065  DOI: 10.1016/j.top.2003.11.006
  * [Mil08] James S. Milne  “Algebraic Geometry (v5.10)” Available at www.jmilne.org/math/, 2008
  * [NS13] Mounir Nisse and Frank Sottile  “The phase limit set of a variety”  In _Algebra and Number Theory_ 7.2 Mathematical Sciences Publishers, 2013, pp. 339–352  DOI: 10.2140/ant.2013.7.339
  * [Pur08] Kevin Purbhoo  “A Nullstellensatz for amoebas”  In _Duke Mathematical Journal_ 141.3 Duke University Press, 2008, pp. 407–445  DOI: 10.1215/00127094-2007-001
  * [Rad42] Richard Rado  “A theorem on independence relations”  In _The Quarterly Journal of Mathematics_ Oxford Academic, 1942, pp. 83–89  DOI: 10.1093/qmath/os-13.1.83
  * [Tev07] Jenia Tevelev  “Compactifications of subvarieties of tori”  In _American Journal of Mathematics_ 129.4 Johns Hopkins University Press, 2007, pp. 1087–1104  DOI: 10.1353/AJM.2007.0029
  * [Zil97] Boris Zilber  “Generalized analytic sets”  In _Algebra and Logic_ 36, 1997, pp. 226–235  DOI: 10.1007/S10469-997-0064-5
  * [Zil02] Boris Zilber  “Exponential Sums Equations and the Schanuel Conjecture”  In _Journal of the London Mathematical Society_ 65.1 Cambridge University Press, 2002, pp. 27–44  DOI: 10.1112/S0024610701002861
  * [Zil05] Boris Zilber  “Pseudo-exponentiation on algebraically closed fields of characteristic zero”  In _Annals of Pure and Applied Logic_ 132.1 Elsevier, 2005, pp. 67–95  DOI: 10.1016/j.apal.2004.07.001
  * [Zil15] Boris Zilber  “The theory of exponential sums”, 2015  arXiv:1501.03297