# The tropical non-properness set of a polynomial map

Boulos El Hilany

###### Abstract.

We provide a polyhedral characterization for the Jelonek set of a polynomial
map $f:K^{n}\to K^{n}$ over an algebraically closed field of characteristic
zero. This is the set of points in $K^{n}$ at which $f$ is not finite. We
study the tropical analogue for the Jelonek set and prove a correspondence
theorem relating classical and tropical polynomial maps. Our description
allows to re-prove $K$-uniruleness of the Jelonek set and to compute its
tropical hypersurface via polyhedral means. Furthermore, we show how to use
our results to recover the face-lattice of the Newton polytope of the Jelonek
set for a large family of complex polynomial maps.

For this work, the author was supported by the DFG Walter Benjamin Programme
EL 1092/1-1

MSC: Primary 14D06, Secondary: 14T20, 14M25, 58K15

Key words: Dominant polynomial maps, Jelonek set, Tropical Geometry, Newton
Polytopes, Polyhedral Subdivisions

##  1\. Introduction

A polynomial map $f:=(f_{1},\ldots,f_{n}):\mathbb{C}^{n}\to\mathbb{C}^{n}$ is
_not proper_ at $y\in\mathbb{C}^{n}$ if there is a sequence
$\\{x_{k}\\}_{k}\subset\mathbb{C}^{n}$ satisfying $\|x_{k}\|\rightarrow\infty$
and $f(x_{k})\rightarrow y$. Investigating the set ${\mathcal{S}}(f)$ of
points for which $f$ is not proper was initiated by Jelonek in [18], where he
described its topological properties and provided computational results. He
showed that ${\mathcal{S}}(f)$ is a _$\mathbb{C}$ -uniruled_ algebraic
hypersurface (i.e. for every $y\in{\mathcal{S}}(f)$, there exists a polynomial
map $\varphi:\mathbb{C}\to{\mathcal{S}}(f)$ with $\varphi(0)=y$), and used
this study to solve classical problems on the topology of polynomial maps [20,
19, 23, 24]. These results and others were later extended to polynomial maps
over arbitrary algebraically closed fields, as well as over the real numbers
[22, 43, 40, 42, 27, 1]. We call ${\mathcal{S}}(f)$ the _Jelonek set_ of $f$.

Our main result is Theorem 1.1. We establish in it a correspondence relating
the Jelonek set to the so-called “tropical non-properness” of certain
piecewise-affine maps, called _tropical polynomial maps_. Tropical non-
properness occurs whenever a preimage under a piecewise-affine map satisfies a
particular degeneracy condition (see definition below). In Theorem 1.3, we
introduce a recipe for recovering the tropical non-properness set.
Additionally, we apply our results to complex polynomial maps to recover the
$\mathbb{C}$-uniruledness of the Jelonek set and to characterize its Newton
polytope (see Theorem 1.5).

###  1.1. Motivation

The study of the Jelonek set ushered new methods to tackle long-standing
problems such as the Jacobian conjecture [28], the classification of
topological types of polynomial maps [25], and estimating minima in polynomial
optimization [26, 29]. Polynomial maps also appear as ubiquitously in
applications such as robotics [6, 38], computer vision [33], algebraic
statistics [35] and chemical reaction networks [13]. In each of these applied
contexts, a map represents the model of the problem, for which the Jelonek set
describes a subset of points where the input of the problem has a “critical”
output. Depending on the application, it can describe so-called “paradoxically
moving linkages”, distorted reconstructions of objects, or reaction networks
with lower number of steady states.

Computing ${\mathcal{S}}(f)$ can be done using standard elimination methods
[18, Proposition 7]: if $\mathbb{C}^{n}$ has coordinates $z_{1},\ldots,z_{n}$,
then ${\mathcal{S}}(f)$ is the zero locus of the product of polynomials
$P_{1},\ldots,P_{n}\in\mathbb{C}[w_{1},\ldots,w_{n}]$, where each $P_{i}$
multiplies the highest degree in $z_{i}$ of the polynomial generating the
elimination ideal

(1.1) $\langle
f_{1}-w_{1},\ldots,f_{n}-w_{n}\rangle\cap\mathbb{C}[z_{i},w_{1},\ldots,w_{n}].$

Similar computations can be done for more general polynomial maps and over
arbitrary algebraically closed fields [43], where elimination techniques
remain to be key for a thorough description of the Jelonek set. However, given
their inefficiency, these can often slacken the study of objects related to
${\mathcal{S}}(f)$, especially when the degrees of the polynomials involved
are high (c.f. [17, Theorem 2.4]).

###  1.2. Theoretical framework and main results

A polynomial map $f:X\to Y$ between two affine varieties over an algebraically
closed field $K$ of characteristic zero, is said to be _finite_ if $K[X]$ is
integral over $K[Y]$. In particular, the number of preimages (counted with
multiplicities) in $X$ remains constant over any point in $Y$ [39, Ch. I, §3].
We say that $f$ is _not finite at_ $y\in Y$ if there is no Zariski open
neighborhood $U$ of $y$ for which $f_{|f^{-1}(U)}:f^{-1}(U)\to U$ is finite.
As it coincides with the Jelonek set whenever $K=\mathbb{C}$ [21, 43], we will
keep the terminology and notation from the complex case when referring to the
set of points at which $f$ is not finite.

In this paper, we consider locally convergent one-parametric series
$c(t):=c_{0}t^{r_{0}}+c_{1}t^{r_{1}}+\cdots$ with complex coefficients and
real exponents $r_{0}<r_{1}<\cdots$. These form an algebraically closed field
$\mathbb{K}$ of characteristic zero [32]. It is endowed with a function
$\operatorname{val}:\mathbb{K}\to\mathbb{R}\cup\\{-\infty\\}$ sending zero to
$-\infty$ and a non-zero element $c(t)$ above to the real value $-r_{0}$. The
image $\operatorname{Val}(Z)$ under
$\operatorname{Val}:=(\operatorname{val}(\cdot),\ldots,\operatorname{val}(\cdot))$,
also called _tropicalization_ , of an algebraic hypersurface $Z$ over
$\mathbb{K}$, is a finite rational polyhedral complex of pure dimension $n-1$
[16]. It is known (see e.g. [4, 31]) that these objects, called _tropical
hypersurfaces_ , preserve important topological criteria and other geometric
invariants of their classical counterparts in the form of combinatorial and
polyhedral data.

The first correspondence theorem in tropical geometry was shown in [8],
stating that the tropicalization of an algebraic hypersurface
$\\{P=0\\}\subset(\mathbb{K}\setminus 0)^{n}$ is the locus in $\mathbb{R}^{n}$
at which the _tropical polynomial_ of $P$ is not differentiable. A tropical
polynomial is a function $\mathbb{R}^{n}\to\mathbb{R}$ of the form
$\max(L_{1},\ldots,L_{r})$, where each $L_{1},\ldots,L_{r}$ is affine with
positive integer coefficients. Tropical polynomials can be constructed from
polynomials $\mathbb{K}^{n}\to\mathbb{K}$ by replacing the coefficients with
their valuations, and the addition and multiplication operations with their
respective tropical counterparts, defined for any values $a,b\in\mathbb{R}$,
as $\max(a,b)$ and $a+b$ respectively. For example, the corner locus of the
two tropical polynomials,

(1.2) $\begin{array}[]{rl}F_{1}:=&\max(b,\leavevmode\nobreak\
-5+2b,\leavevmode\nobreak\ -3+a+2b,\leavevmode\nobreak\
2a+b,\leavevmode\nobreak\ 4a+2b),\\\ F_{2}:=&\max(b,\leavevmode\nobreak\
-2+2a+b,\leavevmode\nobreak\ -2+3a+2b,\leavevmode\nobreak\
-4+4a+2b),\end{array}$

is represented as the left-most graph in Figure 2, where (1.2) are
tropicalizations of the two polynomials

(1.3) $\begin{array}[]{ll}f_{1}:=&v+t^{5}\leavevmode\nobreak\
v^{2}+t^{3}\leavevmode\nobreak\ uv^{2}+3u^{2}v-4u^{4}v^{2},\\\\[4.0pt]
f_{2}:=&-v+2t^{2}\leavevmode\nobreak\ u^{2}v-3t^{2}\leavevmode\nobreak\
u^{3}v^{2}+5t^{4}\leavevmode\nobreak\ u^{4}v^{2}.\end{array}$

In this fashion, any tropical polynomial map $\mathbb{R}^{n}\to\mathbb{R}^{m}$
can be obtained from a polynomial map $\mathbb{K}^{n}\to\mathbb{K}^{m}$ by
replacing the polynomials with their tropical counterparts.

####  1.2.1. Correspondence for polynomial maps

The _virtual preimage_ of a value $b\in\mathbb{R}$ under a tropical polynomial
function $\Lambda:\mathbb{R}^{n}\to\mathbb{R}$,
$x\mapsto\max(L_{1}(x),\ldots,L_{r}(x))$, is the set of all points in
$\mathbb{R}^{n}$ at which the function $\max(L_{1},\ldots,L_{r},b)$ is not
differentiable. The virtual preimage contains the (standard) preimage
$\Lambda^{-1}(b)$ (equality holds only whenever $r=1$), and can be viewed as
the tropicalization of the preimage of a polynomial function. The set
${\mathcal{T}}_{y}(F)$ of a tropical polynomial map
$F:=(F_{1},\ldots,F_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$, is defined as the
intersection in $\mathbb{R}^{n}$ of all virtual preimages of
$y_{1},\ldots,y_{n}$, under the respective functions $F_{1},\ldots,F_{n}$. In
Figure 1, the virtual preimage of $a$ under $F$ from (1.2), for example,
contains the union of the orange dot with the half-line (locally represented
in the window labeled by $a$), whereas $F^{-1}(a)$ is the half-line. A half-
line $H\subset\mathbb{R}^{n}$ is said to be _dictritical_ if no translation of
$H$ is contained in the orthant $(\mathbb{R}_{\leq 0})^{n}$. In other words,
if a vector $\alpha$ positively spans $H$, the $\alpha$ has a positive
coordinate.

######  Definition (Tropical non-properness).

The _tropical non-properness set_ ${\mathcal{S}}_{\mathbb{T}}(F)$ of a
tropical polynomial map $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the set of all
$y\in\mathbb{R}^{n}$ at which the virtual preimage contains a dicritical half-
line.

Our paper concerns large families of polynomial maps
$\mathbb{K}^{n}\to\bm{T}$, where $\bm{T}:=(\mathbb{K}\setminus 0)^{n}$. The
_support_ of a polynomial is the set of exponent vectors of its monomials in
$\mathbb{N}^{n}$ appearing with non-zero coefficients. Given a tuple of
polynomials $(f_{1},\ldots,f_{n})$, its _support_ is the tuple
$A:=(A_{1},\ldots,A_{n})$ formed by the respective supports of the
corresponding polynomials. All polynomials sharing a support $\sigma$ form a
configuration space $\mathbb{K}^{|\sigma|}$ of points, whose coordinates
represent the coefficients. In this way, we use $\mathbb{K}[A]$ to denote the
space of polynomial maps $\mathbb{K}^{n}\to\bm{T}$ supported on $A$, where

$\mathbb{K}[A]\cong\mathbb{K}^{|A_{1}|}\times\cdots\times\mathbb{K}^{|A_{n}|}.$

######  Theorem 1.1.

Let $A$ be any tuple of $n$ sets in
$\mathbb{N}^{n}\setminus\\{(0,\ldots,0)\\}$. Then, there exists a Zariski open
subset $\Omega\subset\mathbb{K}[A]$ in which any $f\in\Omega$, representing a
polynomial map $\mathbb{K}^{n}\to\bm{T}$ supported on $A$ satisfies

$\operatorname{Val}({\mathcal{S}}(f))={\mathcal{S}}_{\mathbb{T}}(F),$

where $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the tropical polynomial map
corresponding to $f$ (see e.g., (1.3) and (1.2)).

$a$$\alpha$$b$$c$$\beta$$\operatorname{Val}({\mathcal{S}}(f))$$\beta$$\alpha$$a$$b$$c$
Figure 1. Local illustrations of the virtual preimage under $F$ of some points
in the set $\operatorname{Val}({\mathcal{S}}(f))$. The maps $f$ and $F$ are
taken from Example 1.2.

######  Example 1.2.

Consider the polynomial map (1.3), whose corresponding tropical polynomial map
is (1.2). The set ${\mathcal{S}}(f)$ is given by the polynomial

$25t^{8}\leavevmode\nobreak\ r^{2}+30t^{6}\leavevmode\nobreak\
r+40t^{4}\leavevmode\nobreak\ rs+16t^{4}\leavevmode\nobreak\
r-45t^{4}\leavevmode\nobreak\ s-24t^{2}\leavevmode\nobreak\
s+16\leavevmode\nobreak\ s^{2}.$

It was computed using the Algorithm SparseJelonek-2 of [17]. Its tropical
curve $\operatorname{Val}({\mathcal{S}}(f))$ is shown in purple in Figure 1.
For each
$y\in\\{a,b,c,\alpha,\beta\\}\subset\operatorname{Val}({\mathcal{S}}(f))$, the
virtual preimage (in orange) ${\mathcal{T}}_{y}(F)$ contains a half-line
spanned by the vector $(1,-2)$. ∎

####  1.2.2. Tropical non-properness from cell-decompositions

We also provide a description of the tropical non-properness set of tropical
maps arising from polynomial maps in the set $\Omega$ of Theorem 1.1. These
will be called _face-generic_. A tropical polynomial map
$F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ induces a polyhedral cell-decomposition

$\mathbb{R}^{n}=\bigsqcup_{\xi\in\Xi}\xi,$

where every element in $\Xi$ is the relative interior of a convex polytope in
$\mathbb{R}^{n}$, and the restriction $F_{|\xi}$ at each
$\xi\subset\mathbb{R}^{n}$ is an affine map such that if
$F_{|\xi}=F_{|\xi^{\prime}}$, then either $\xi=\xi^{\prime}$ or its closure
(in the Euclidean topology) $\overline{\xi}$ is a face of
$\overline{\xi^{\prime}}$ (c.f [3, 15]). For example, the map (1.2), gives
rise to the decompsition of $\mathbb{R}^{n}$ represented in Figure 2 on the
left. The following result is detailed in §5, and provides an effective
description of ${\mathcal{S}}_{\mathbb{T}}(F)$ using finitely-many polyhedra.

######  Theorem 1.3.

Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a tropical polynomial map
corresponding to a face-generic polynomial map $\mathbb{K}^{n}\to\bm{T}$ and
let $\Xi$ denote the polyhedral cell-decomposition of $\mathbb{R}^{n}$ induced
by $F$. Then, for any $\xi\in\Xi$ the set of all $y\in\mathbb{R}^{n}$, for
which ${\mathcal{T}}_{y}(F)$ has a dicritical half-line intersecting $\xi$, is
a convex polytope in ${\mathcal{S}}_{\mathbb{T}}(F)$.

$a$$b$$c$$\alpha$$\beta$$\beta$$\alpha$$a$$b$$c$ |
---|---
Figure 2. From left to right: The decomposition induced by the map from
Example 1.2, its tropical non-properness set, and its tropical discriminant
(see [10]).

######  Example 1.4.

Figure 2 illustrates the cell-decomposition $\Xi$ corresponding to the map
$\mathbb{R}^{2}\to\mathbb{R}^{2}$ with the pair of tropical polynomials from
Equation (1.2). The virtual preimage ${\mathcal{T}}_{y}(F)$ of any $y$ in one
of the edges of ${\mathcal{S}}_{\mathbb{T}}(F)$ with the labeling $a$, $b$ and
$c$ has a dicritical half-line in the cell in $\Xi$ with the identical
labeling. The same goes for the one-dimensional cells $\alpha$ and $\beta$. ∎

Thanks to the well-known duality between Newton polytopes and tropical
hypersurfaces and (see e.g., Proposition 3.6), computing
${\mathcal{S}}_{\mathbb{T}}(F)$ becomes useful for describing the Newton
polytope of the Jelonek set of complex polynomial maps. The _face-vector_ of a
convex polytope $\Pi\subset\mathbb{R}^{n}$ is the finite integer sequence
$({\mathcal{F}}_{0},\ldots,{\mathcal{F}}_{n-1})$, where the component in
position $i$ is the number of $i$-dimensional faces of $\Pi$.

######  Theorem 1.5.

Let $A$ be any tuple of $n$ sets in
$\mathbb{N}^{n}\setminus\\{(0,\ldots,0)\\}$. Then, there exists a Zariski open
subset $\widetilde{\Omega}\subset\mathbb{C}[A]$ of complex polynomial maps, in
which for any $g\in\widetilde{\Omega}$, the face-vector of the Newton polytope
in $\mathbb{R}^{n}$ of ${\mathcal{S}}(g)$, together with the slopes of its
faces can be recovered by means of Theorems 1.1 and 1.3.

###  1.3. Organization

§2 is devoted to introducing some of the necessary conditions for a polynomial
map to be face-generic, called _almost face-generic_ (Definition 2.6). These
conditions apply for any polynomial map over an algebraically-closed field $K$
of characteristic zero. We show in Proposition 2.7 that almost face-genericity
is an open condition in $K[A]$. This will be useful for describing the Jelonek
set as a union of _face-resultants_ in $\bm{T}$ (Definition 2.8 and
Proposition 2.16). Face-resultants are obtained from square polynomial systems
by restricting individual polynomials of the map onto faces of their
respective Newton polytopes. Thanks to this description, we re-prove Jelonek’s
result on $K$-uniruledness for almost face-generic maps over $K$ (Corollary
2.17). In §3 we introduce notations and some classical results from basic
tropical algebraic geometry. These will be necessary for proving Theorems 1.1
and 1.3 in §4 and §5 respectively. Theorem 1.5 is proven at the end of §4.

##  2\. Jelonek set from faces of polytopes

The first task in this paper is to relate the Jelonek set of a polynomial map
to its face-resultants. We start this section with some notations and
definitions. In §2.1 we define almost face-generic polynomial maps, and show
that they form a Zariski open subset in the space of all maps with the same
support. In §2.2, we describe some relevant properties of face-resultants for
almost face-generic maps. We recall in §2.3 Bernstein’s theorem for polynomial
tuples and apply it to determine the topological degree of a plynomial map.
The correspondence between face-resultants and the Jelonek set culminates in
§2.4. We finish this section by using this description to re-prove
$\mathbb{K}$-uniruledness for the Jelonek set.

###  2.1. Preliminaries on Newton polytopes

A _polytope_ in $\mathbb{R}^{n}$ is the intersection of finitely-many closed
half-spaces. Recall that a face $\Phi$ of a polytope
$\Pi\subset\mathbb{R}^{n}$ is given by the intersection of $\Pi$ with a
hyperplane for which $\Pi$ is contained in one of the two halfspaces
determined by the hyperplane. We call this a _supporting_ hyperplane of
$\Phi$. We say that $\Phi$ is _dicritical_ if a normal vector
$\alpha:=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{R}^{n}$ of a supporting
hyperplane of $\Phi$ is directed outwards of $\Pi$ and has one positive
coordinate $\alpha_{i}$.

Given a tuple of polytopes $\Delta:=(\Delta_{1},\ldots,\Delta_{n})$, the
Minkowski sum of their members is a polytope in $\mathbb{R}^{n}$, denoted by
$\Sigma\Delta$. Here, the _Minkowski sum_ of any two subsets
$A,B\subset\mathbb{R}^{n}$ is the coordinate-wise sum
$A+B:=\\{a+b\leavevmode\nobreak\ |\leavevmode\nobreak\ a\in
A,\leavevmode\nobreak\ b\in B\\}$. The sub-tuple $(\Delta_{i})_{i\in I}$ is
denoted by $I\Delta$ for any $I\subset[n]$. A tuple
$\Gamma:=(\Gamma_{1},\ldots,\Gamma_{n})$ of polytopes in $\mathbb{R}^{n}$ is
said to be a _tuple-face_ of $\Delta$ if for any $i=1,\ldots,n$, the subset
$\Gamma_{i}$ is a face of $\Delta_{i}$ and $\Sigma\Gamma$ is a face of
$\Sigma\Delta$. We use the notation $\Gamma\prec\Delta$. When it is clear from
the context, we will use the term _face_ of $\Delta$ instead of the term
“tuple-face”.

######  Notation 2.1.

We use $\underline{0}$ to denote the point $(0,\ldots,0)$. ∎

######  Definition 2.2.

A face $\Gamma$ of a tuple of polytopes in $\mathbb{R}^{n}$ is said to be
_dicritical_ if $\Sigma\Gamma$ is dicritical, and for any $i=1,\ldots,n$, no
member $\Gamma_{i}$, coincides with $\underline{0}$. ∎

######  Definition 2.3.

The cone of $\mathbb{R}^{n}$, formed by all points with non-negative
coordinates is denoted by $\mathbb{R}^{n}_{\geq 0}$. Since this cone emanates
from $\underline{0}$, for any polytope $\Pi\subset\mathbb{R}^{n}_{\geq 0}$,
either $\underline{0}$ is a vertex of $\Pi$, or $\underline{0}\not\in\Pi$. A
face, $\Gamma$, of a tuple of polytopes $\Delta$ is called _origin_ if
$\underline{0}\in\Sigma\Gamma$. This is equivalent to $\Gamma_{i}$ containing
$\underline{0}$ for all $i\in[n]$. We say that $\Gamma$ is _pre-origin_ if
$\underline{0}\in\Gamma_{i}$ for some $i\in[n]$, and _strictly pre-origin_ if
it is pre-origin, but not origin. ∎

$(1,1,1)$$(0,1,1)$$(2,2,2)$$(1,1,0)$$(0,1,2)$$(0,2,4)$$(1,1,2)$$(2,2,4)$$(1,1,1)$$(1,0,0)$$(2,1,1)$
Figure 3. A triple of polytopes
$\Delta^{0}=(\Delta^{0}_{1},\Delta^{0}_{2},\Delta^{0}_{3})$ in
$\mathbb{R}^{3}$ obtained from Example 2.9.

######  Example 2.4.

Figure 3 represents a triple $\Delta^{0}$ of polytopes in $\mathbb{R}^{3}$.
The face $(\Gamma_{1},\Gamma_{2},\Gamma_{3})$ traced in red is strictly pre-
origin dicritical. So is the face traced in green. The face obtained by
intersecting $\Sigma\Delta^{0}$ with the horizontal coordinate plane is origin
and not dicritical. ∎

Let $K$ be an algebraically closed field of characteristic zero, and let $f$
be a polynomial in $K[z_{1},\ldots,z_{n}]$. Then $f$ is expressed as a linear
combination

$\sum_{a\in A}c_{a}z^{a}$

of monomials $z^{a}:=z_{1}^{a_{1}}\cdots z_{n}^{a_{n}}$, where $A$ is a finite
subset of $\mathbb{N}^{n}$, and $c_{a}\in{\\!K}\setminus 0$. We call $A$ the
_support_ of $f$, and its _Newton polytope_ is the convex hull of $A$ in
$\mathbb{R}^{n}$. If $\sigma$ is a subset of $A$, the _restriction of $f$ to
$\sigma$_, denoted by $f_{|\sigma}$, is the polynomial
$\sum_{a\in\sigma}c_{a}z^{a}$. The notation $\mathbb{V}^{*}_{\\!\\!K}(f)$ will
refer to the zero locus $\\{z\in\bm{T}\leavevmode\nobreak\
|\leavevmode\nobreak\ f(z)=0\\}$. Now, if $f$ is a tuple of polynomials
$f_{1},\ldots,f_{k}\in K[z_{1},\ldots,z_{n}]$, we use the same notation
$\mathbb{V}^{*}_{\\!\\!K}(f)$ for the intersection
$\mathbb{V}^{*}_{\\!\\!K}(f_{1})\cap\cdots\cap\mathbb{V}^{*}_{\\!\\!K}(f_{k})\subset\bm{T}$.

Let $A:=(A_{1},\ldots,A_{n})$ be a tuple of subsets in
$\mathbb{N}^{n}\setminus\\{\underline{0}\\}$, and let
$\Delta:=\big{(}\Delta_{1},\ldots,\Delta_{n}\big{)}$ be the tuple where
$\Delta_{i}$ is the convex hull of $A_{i}$ ($i=1,\ldots,n$). Recall that
$K[A]$ refers to the space of all tuples of poloynomials supported on $A$, and
let $f:=(f_{1},\ldots,f_{n})$ be a tuple in $K[A]$.

######  Notation 2.5.

For any $\Gamma\prec\Delta$, we use $f_{i\Gamma}$ ($i=1,\ldots,n$) to denote
the restricted polynomial $f_{i|\Gamma_{i}\cap A_{i}}$. The tuple
$(f_{1\Gamma},\ldots,f_{n\Gamma})$ is denoted by $f_{\Gamma}$, and for any
subset $I\subset[n]$, we use the notation

$f_{I\Gamma}:=(f_{i\Gamma})_{i\in I}.$

∎

###  2.2. Face-resultants for almost face-generic maps

Let $f:=(f_{1},\ldots,f_{n}):K^{n}\to K^{n}$ be a dominant polynomial map. Let
$\Delta^{0}(f)$ denote the tuple formed by the Newton polytopes of
$(f_{1},\ldots,f_{n})$ after adding a generic constant to each polynomial.
That is, if $A:=(A_{1},\ldots,A_{n})$ is the support of $f$, then
$\Delta^{0}\\!:=\Delta^{0}(f)$ coincides with the tuple
$(\Delta^{0}_{1},\ldots,\Delta^{0}_{n})$ of polytopes in $\mathbb{R}^{n}_{\geq
0}$ defined as

(2.1)
$\Delta_{i}^{0}:=\operatorname{conv}\big{(}\underline{0}\cup\\!A_{i}\big{)}\leavevmode\nobreak\
\text{for }i=1,\ldots,n.$

######  Definition 2.6.

The tuple $f$ is said to be _almost face-generic_ if for any
$\Gamma\prec\Delta^{0}$ the following three conditions hold:

  1. (a)

for any $\Gamma^{\prime}\prec\Delta^{0}$ with $\Gamma^{\prime}\neq\Gamma$ and
$\mathbb{V}^{*}_{\\!\\!K}(f_{\Gamma^{\prime}})\neq\emptyset$, it holds
$\mathbb{V}^{*}_{\\!\\!K}(f_{\Gamma})\neq\mathbb{V}^{*}_{\\!\\!K}(f_{\Gamma^{\prime}})$
,

  2. (b)

for any $I\subset[n]$, the set $\mathbb{V}^{*}_{\\!\\!K}(f_{I\Gamma})$ is a
smooth intersection that satisfies

(2.2)
$\mathbb{V}^{*}_{\\!\\!K}(f_{I\Gamma})\neq\emptyset\Leftrightarrow|I|-\dim\sum_{i\in
I}\Gamma_{i}\leq 0,$

  3. (c)

for any coordinate hyperplane $H$ of $K^{n}$, it holds
$\dim{\mathcal{S}}(f)\cap f(H)\leq n-2$.

∎

######  Proposition 2.7.

Almost face-generic tuples form a Zariski open subset of $K[A]$.

###### Proof.

Let $\Omega_{a}$, $\Omega_{b}$, and $\Omega_{c}$ be the three subsets
consisting of tuples in $K[A]$ satisfying conditions (a), (b) and (c)
respectively. It is enough to show that each of these sets is Zariski open in
$K[A]$.

Concerning (a): Up to re-enumeration of indexes, we have
$\Gamma_{1}\neq\Gamma^{\prime}_{1}$. For any $a$ in
$\Gamma_{1}\setminus\Gamma_{1}^{\prime}$ (the case
$a\in\Gamma_{1}^{\prime}\setminus\Gamma_{1}$ is analogous), consider the
monomial term $cz^{a}$ appearing in a tuple $f_{\Gamma}$, for some $f\in
K[A]$. Clearly, there are at most finitely-many values of $c\in K$ satisfying
$\mathbb{V}^{*}_{\\!\\!K}(f_{\Gamma})=\mathbb{V}^{*}_{\\!\\!K}(f_{\Gamma^{\prime}})$.
The latter are given as roots of univariate polynomials. Since this holds true
for any choice of $f$, we conclude that $\Omega_{a}$ is Zariski open in
$K[A]$.

Concerning (b): Consider the projection
$P_{I\Gamma}:Z_{I\Gamma}\subset\bm{T}\times K[A]\to K[A]$, where
$Z_{I\Gamma}:=\\{(z,f)\leavevmode\nobreak\ |\leavevmode\nobreak\
f_{I\Gamma}=\underline{0}\\}$. Here, $\operatorname{Crit}P_{I\Gamma}$ forms
the collection of all $(z,f)$ for which
$\mathbb{V}^{*}_{\\!\\!K}(f_{I\Gamma})$ is not smooth at $z\in\bm{T}$.
Bertini-Sard Theorem applied to each $I\subset[n]$ implies that $K[A]\setminus
P_{I\Gamma}(\operatorname{Crit}P_{I\Gamma})$ is Zariski open in $K[A]$.

Concerning Equation (2.2): The system $f_{I\Gamma}=\underline{0}$ having a
solution in $\bm{T}$, implies that $f$ lies in the _multivariate resultant_
$\operatorname{Res}_{I\Gamma}\subset K[A]$ corresponding to $\Gamma$ (see
[14]). This is the set of all tuples $f$ such that $f_{I\Gamma}=\underline{0}$
has a solution in $\bm{T}$. The resultant is an algebraic set in $K[A]$,
satisfying the following well-known equality (see also [41, 11])

(2.3) $\operatorname{cdim}\operatorname{Res}_{I\Gamma}=\max_{J\subset
I}\big{(}|J|-\dim\sum_{i\in J}\Gamma_{i}\big{)}.$

Since $\Gamma$ is a proper face of $\Delta^{0}$, the corresponding resultant
belongs to a hypersurface in $K[A]$. Hence $\Omega_{b}$ being Zariski open
follows from $\Delta^{0}$ having finitely-many proper faces.

Concerning (c): For this proof, we consider the space
$M:=K[A_{1}\cup\\{0\\},\ldots,A_{n}\cup\\{0\\}]$. Similarly to the above item,
let $Z$ be the set $\\{(z,f)\leavevmode\nobreak\ |\leavevmode\nobreak\
f=\underline{0}\\}$, and consider the projection $\bm{T}\times M\to M$. Let
$\mathsf{T}$ be the toric compactification of $\bm{T}$, compatible with $A$
(e.g. the one considered in [12, §4.1]), we denote by $\overline{Z}$ the
closure of $Z$ inside $\mathsf{T}\times M$, and by $\overline{P}$ the
extension of $P$ to $\overline{Z}$ so that $\overline{P}_{|Z}=P$. The boundary
$\partial\overline{Z}$ is a normal crossing divisor on $\overline{Z}$ formed
by irreducible components $D_{1},\ldots,D_{r}$.

For any tuple $f\in M$, consider the embedding $\eta_{f}:\bm{T}\hookrightarrow
M$ sending $f(K^{n})$ onto the linear subspace
$L_{f}:=\\{f+c\leavevmode\nobreak\ |\leavevmode\nobreak\ c\in\bm{T}\\}$. Then,
the Bertini-Sard Theorem shows that there is a Zariski open subset
$\mathcal{O}\subset M$ in which any $f\in\mathcal{O}$, the set $L_{f}$ is
transversal to the image of each stratum
$\overline{P}(D_{1}),\ldots,\overline{P}(D_{r})$ of
$\overline{P}(\partial\overline{Z})$. Furthermore, since $\mathsf{T}$ is a
compactification of $\bm{T}$, for any $f\in\Omega_{c}$, the images
$\eta_{f}({\mathcal{S}}(f)),\eta_{f}(f(\mathcal{H}))\subset
L_{f}\cap\overline{P}(\partial\overline{Z})$.

To finish the proof, recall that the compactification is toric. Then, each
component in the divisor $D\subset\partial\overline{Z}$ corresponds to a face
$\Gamma\prec\Delta^{0}$. Hence, since $f\in\Omega_{c}$, any top-dimensional
stratum $S$ of $\overline{P}(\partial\overline{Z})\cap L_{f}$ cannot be shared
by both $\eta_{f}({\mathcal{S}}(f))$ and $\eta_{f}(f(\mathcal{H}))$.
Therefore, ${\mathcal{S}}(f)$ and $f(\mathcal{H})$ do not share maximal-
dimensional components. Therefore, we choose $\Omega_{c}$ to be the image of
$\mathcal{O}$ under the projection $M\to K[A]$ by removing the constant terms
of any tuple $f\in M$. ∎

######  Definition 2.8.

Let $\Gamma$ be a proper face of $\Delta^{0}$, and let $X_{\Gamma}$ denote the
set

$\left\\{(z,w)\in\bm{T}\times\bm{T}\leavevmode\nobreak\
\left|\leavevmode\nobreak\ (f(z)-w)_{\Gamma}=0\right\\}\right..$

The _face-resultant_ of the map $f$ at the face $\Gamma$, denoted by
${\mathcal{R}}_{\Gamma}(f)$, is the image in $\bm{T}$ of $X_{\Gamma}$ under
the projection $\pi:\leavevmode\nobreak\ (z,w)\mapsto w$. ∎

######  Example 2.9.

Let $f$ be the below triple of polynomials in three variables $a,b,c$, defined
over the field $\mathbb{K}$ of generalized Puiseux series in $t$:

(2.4) $\begin{array}[]{ll}f_{1}:=&abc+\leavevmode\nobreak\
bc+\leavevmode\nobreak\ a^{2}b^{2}c^{2},\\\\[4.0pt]
f_{2}:=&ab+\leavevmode\nobreak\ abc^{2}+\leavevmode\nobreak\
t^{7}\leavevmode\nobreak\ bc^{2}+\leavevmode\nobreak\
t^{3}\leavevmode\nobreak\ b^{2}c^{4}+\leavevmode\nobreak\
t^{2}\leavevmode\nobreak\ ab^{2}c^{4}+\leavevmode\nobreak\
t^{5}\leavevmode\nobreak\ a^{2}b^{2}c^{4},\\\\[4.0pt]
f_{3}:=&t\leavevmode\nobreak\ a+\leavevmode\nobreak\ ab+\leavevmode\nobreak\
abc+\leavevmode\nobreak\ a^{2}bc.\end{array}$

The triple of polytopes $\Delta^{0}$ in Figure 3 correspond to (2.4). If
$\Gamma\prec\Delta^{0}$ is the triple in Figure 3, so that
$\Gamma_{1}=\Delta_{1}^{0}$ (highlighted in green), then
${\mathcal{R}}_{\Gamma}(f)$ is the surface defined by the below polynomial in
$\mathbb{K}[u,v,w]$:

$t^{4}(-2+4t+t^{3})+\leavevmode\nobreak\
t(1-2t-4t^{3}+6t^{4}+2t^{6})\leavevmode\nobreak\ u+\leavevmode\nobreak\
t^{3}(-2+3t+4t^{2}+t^{3})\leavevmode\nobreak\ w\\\ +\leavevmode\nobreak\
t(1-2t+t^{2}-2t^{3}+2t^{4}+t^{6})\leavevmode\nobreak\
u^{2}+(1-2t-t^{2}-2t^{3}+8t^{5}+t^{6})\leavevmode\nobreak\ uw\\\
+\leavevmode\nobreak\
t^{3}(-1+4t+t^{2}-2t^{3}+4t^{4}+t^{6})\leavevmode\nobreak\
w^{2}+\leavevmode\nobreak\ 2t(-1+t^{3}+t^{4}-t^{6})\leavevmode\nobreak\
uw^{2}+\leavevmode\nobreak\ t^{4}(1-t^{2}+4t^{3})\leavevmode\nobreak\
w^{3}+\leavevmode\nobreak\ t^{7}\leavevmode\nobreak\ w^{4}.$

This equation is obtained by introducing new variables
$(r,s):=(bc,\leavevmode\nobreak\ a)$, and computing

$\langle rs+r+s^{2}r^{2}-\leavevmode\nobreak\ u,\leavevmode\nobreak\
t+s+t^{3}\leavevmode\nobreak\ s^{2},\leavevmode\nobreak\
s+rs+s^{2}r-w,\rangle\cap\Bbbk[u,v,w].$

If $\Gamma$ is the triple of faces highlighted in red (i.e. $\Gamma_{2}$ is
spanned by the vectors $(1,1,2)$ and $(1,1,0)$), for example, then
${\mathcal{R}}_{\Gamma}(f)$ is the hyperplane defined by the polynomial
$t^{2}-t-u$, whose equation is obtained by introducing new variables
$(r,s):=(ab,\leavevmode\nobreak\ c)$, and computing the elimination ideal
$\langle rs+r^{2}s^{2}-\leavevmode\nobreak\ u,\leavevmode\nobreak\
r+rs^{2}+t^{5}r^{2}s^{4}-\leavevmode\nobreak\ v,\leavevmode\nobreak\
t+rs-\leavevmode\nobreak\ w\rangle\cap\Bbbk[u,v,w].$ ∎

We finish this part by proving two technical properties for the face-
resultants.

######  Lemma 2.10.

Assume that $f$ is almost face-generic, and let $\Gamma$ be a proper face of
$\Delta^{0}$. Then, the face-resultant ${\mathcal{R}}_{\Gamma}(f)$ is empty if
$\Gamma$ is not pre-origin, and $\dim{\mathcal{R}}_{\Gamma}(f)\leq n-1$
otherwise. Furthermore, for any other proper face $\Gamma^{\prime}$ of
$\Delta^{0}$, it holds
$\dim{\mathcal{R}}_{\Gamma}(f)\cap{\mathcal{R}}_{\Gamma^{\prime}}(f)\leq n-2$.

###### Proof.

If $\Gamma$ is not pre-origin, then for any $w\in\bm{T}$, it holds
$(f-w)_{\Gamma}=f_{\Gamma}$. Equation (2.2) of Definition 2.6 shows that
$X_{\Gamma}=\emptyset$, and thus so its projection under $\pi$.

Let $\Gamma$ be a pre-origin face. Now, we show that
$\dim{\mathcal{R}}_{\Gamma}(f)\leq n-1$. Note that if $\dim X_{\Gamma}\leq
n-1$, then $\dim\pi(X_{\Gamma})\leq n-1$ and we are done. Assume in what
follows that $\Gamma$ is pre-origin, and that $\dim X_{\Gamma}\geq n$. Then,
Definition 2.6:(b) shows that $\dim X_{\Gamma}=n$. Since $\Gamma$ is a face of
$\Delta^{0}$, up to multiplying each polynomial by an appropriate monomial,
for any $w\in\bm{T}$, the system $(f-w)_{\Gamma}=\underline{0}$ is formed by
quasi-homogeneous polynomials in $z$. Consequently, the fibers under the map
$\pi_{|X_{\Gamma}}:X_{\Gamma}\to\pi(X_{\Gamma})$ have positive dimension $d$.
Since $\dim X_{\Gamma}\geq\dim\pi(X_{\Gamma})+d$, we deduce that
$\dim{\mathcal{R}}_{\Gamma}(f)=\dim\pi(X_{\Gamma})\leq n-1$. This yields the
proof of the first part of the Lemma.

Now we show that
$\dim{\mathcal{R}}_{\Gamma}(f)\cap{\mathcal{R}}_{\Gamma^{\prime}}(f)\leq n-2$.
By contradiction, assume that ${\mathcal{R}}_{\Gamma}(f)$ and
${\mathcal{R}}_{\Gamma^{\prime}}(f)$ share an $(n-1)$-dimensional component.
Since $f$ is almost face-generic, both $X_{\Gamma}$ and $X_{\Gamma^{\prime}}$
are smooth, and thus irreducible. Consequently we obtain
${\mathcal{R}}_{\Gamma}(f)={\mathcal{R}}_{\Gamma^{\prime}}(f)$. Recall that
$\pi$ has fibers of positive dimension when restricted to either $X_{\Gamma}$
or $X_{\Gamma^{\prime}}$. Therefore, the equality

$\dim{\mathcal{R}}_{\Gamma}(f)=\dim{\mathcal{R}}_{\Gamma^{\prime}}(f)=n-1$

implies that $X_{\Gamma}=X_{\Gamma^{\prime}}$. This contradicts Definition 2.6
(a) as $f$ is almost face-generic and $\Gamma\neq\Gamma^{\prime}$. ∎

######  Remark 2.11.

Lemma 2.10 can be proven using known results on multivariate resultants and
the set $L_{f}\subset K[A]$ introduced in the proof of Proposition 2.7: Up to
perturbations, the set $L_{f}\cap\operatorname{Res}_{\Gamma}$ is a complete
intersection. Its codimension inside $L_{f}$ is equal to the right-hand-side
of (2.3). The first statement of Lemma 2.10 can be deduced from the fact that
the canonical embedding $f(\bm{T})\hookrightarrow L_{f}$ sends
${\mathcal{R}}_{\Gamma}(f)$ onto $L_{f}\cap\operatorname{Res}_{\Gamma}$. Its
second statement can be deduced from other known properties of resultants (see
e.g. [12, Lemma 2.31]).

Let $\mathcal{H}$ denote the union of coordinate hyperplanes in $K^{n}$.

######  Lemma 2.12.

Let $\Gamma$ be a dicritical face of $\Delta^{0}$, and let $f\in K[A]$ be
almost face-generic. Then, it holds

$\dim{\mathcal{R}}_{\Gamma}(f)\cap f(\mathcal{H})\leq n-2.$

###### Proof.

Let $H$ be a coordinate hyperplane of $K^{n}$. Then, we have
$f(H)={\mathcal{R}}_{\Gamma^{\prime}}(f)$ for the face $\Gamma^{\prime}$ of
$\Delta^{0}$ satisfying $\Sigma\Gamma^{\prime}\subset H$. If
$\Gamma^{\prime}\neq\Gamma$, then the result follows from Lemma 2.10.

Assume that $\Gamma=\Gamma^{\prime}$. We show that
$\dim{\mathcal{R}}_{\Gamma}(\Gamma)\leq n-2$. Since $\Gamma$ is dicritical
satisfying $\Sigma\Gamma\subset H$, we deduce that $\dim\Sigma\Gamma\leq n-2$.
Indeed, a dicritical face cannot be spanned by a vector $-e_{i}$ from the
canonical basis $(e_{1},\ldots,e_{n})$ of $\mathbb{R}^{n}$. Then, one can
perform a change of coordinates $z\to x$ so that $f_{\Gamma}$ is written as a
tuple $g$ of polynomials $g_{1},\ldots,g_{n}\in K[x_{1},\ldots,x_{k}]$, where
$k:=\dim\Sigma\Gamma$ (c.f. [2] or [30, §4]). This implies that the set of all
$w\in\bm{T}$ for which $f-w=\underline{0}$ has a solution in $\bm{T}$
coincides with the set of all $w$ for which $g-w=\underline{0}$ has a solution
in $(K\setminus 0)^{k}$. The latter is the image under the map $g:(K\setminus
0)^{k}\to\bm{T}$. This image has dimension at most $k\leq n-2$, and coincides
with $f(H)$. ∎

###  2.3. Topological degree of a map

If $\Delta$ is a convex body in $\mathbb{R}^{n}$, we use
$\operatorname{Vol}_{n}(\Delta)$ to denote its Euclidean volume. Let
$\Delta_{1},\ldots,\Delta_{n}$ be $n$ convex bodies in $\mathbb{R}^{n}$. Then
the classical notion of _mixed volume_ of $\Delta_{1},\ldots,\Delta_{n}$ is
defined as

$V(\Delta_{1},\ldots,\Delta_{n}):=\frac{1}{n!}\sum_{k=1}^{n}(-1)^{k+n}\sum_{i_{1}<\cdots<i_{k}}\operatorname{Vol}_{n}(\Delta_{i_{1}}+\cdots+\Delta_{i_{k}}),$

where the summation between polytopes is the Minkowski sum. For example, if
$n=2$, we have

$V(\Delta_{1},\Delta_{2})=\operatorname{Vol}_{2}(\Delta_{1}+\Delta_{2})-\operatorname{Vol}_{2}(\Delta_{1})-\operatorname{Vol}_{2}(\Delta_{2}).$

We have the following well-known version of Bernstein’s Theorem [2] (see e.g.
[37]).

######  Theorem 2.13.

Let $g_{1},\ldots,g_{n}$ be polynomials in $n$ variables
$K[z_{1},\ldots,z_{n}]$, such that $V(\Delta_{1},\ldots,\Delta_{n})\neq 0$,
where $\Delta_{1},\ldots,\Delta_{n}$ denote their respective Newton polytopes
in $\mathbb{R}^{n}$, and let $I^{*}(g)$ denote the number of isolated
solutions in $\mathbb{V}^{*}_{\\!\\!K}(g)$, counted with multiplicities. Then,
it holds $I^{*}(g)\leq V(\Delta_{1},\ldots,\Delta_{n})$, with equality if and
only if $g_{\Gamma}=\underline{0}$ does not have solutions in $\bm{T}$ for any
face $\Gamma$ of $\Delta$.

Bernstein’s theorem determines the topological degree of almost face-generic
polynomial maps.

######  Lemma 2.14.

Let $f$ be a dominant almost face-generic map. Then, there exists a Zariski
open subset $W\subset\bm{T}$ containing all $w\in W$ for which the preimage
$f^{-1}(w)$ is contained in $\bm{T}$, and has exactly $V(\Delta^{0})$ isolated
points counted with multiplicities.

###### Proof.

Take $W=f(\bm{T})\cap({\mathcal{R}}\cup{\mathcal{S}}(f))$, where
${\mathcal{R}}$ is the union of all face-resultants of $f$. Recall that $f$ is
dominant, and the Jelonek set is a hypersurface [18]. These facts, together
with Proposition 2.7, imply that $W$ is Zariski open in $\bm{T}$. Since $w\in
f(\bm{T})$, we get $f^{-1}(w)\subset\bm{T}$. By definition, the map is finite
outside ${\mathcal{S}}(f)$, and thus has a constant number of preimages
(counted with multiplicities) $\mu(f)$ that is independent of
$w\not\in{\mathcal{S}}(f)$ (see e.g. §1.3). In the notation of Theorem 2.13,
this shows that $\mu(f)=I^{*}(f-w)$. Finally, Theorem 2.13 shows that for any
$w\in W$, it holds $I^{*}(f-w)=V(\Delta_{1},\ldots,\Delta_{n})$, and we are
done. ∎

###  2.4. The Jelonek set and face-resultants

We finish this section by proving that the Jelonek set of a map is the union
of its face-resultants.

######  Notation 2.15.

For any tuple $\Delta$ of polytopes in $\mathbb{R}^{n}$, we use
${\mathscr{F}}_{\\!D}(\Delta)$ to denote the set of all its dicritical faces.
∎

######  Proposition 2.16.

Let $f:K^{n}\to\bm{T}$ be an almost face-generic map, and let
$\Delta^{0}:=\Delta^{0}(f)$. Then, it holds

(2.5)
${\mathcal{S}}(f)=\bigcup_{\Gamma\in\mathscr{F}_{\\!D}(\Delta^{0})}{\mathcal{R}}_{\Gamma}(f).$

###### Proof.

Recall that $\dim{\mathcal{S}}(f)=n-1$ (c.f. [18]) and $f(K^{n})=f(\bm{T})\cup
f(\mathcal{H})$. Then it is enough to show

(2.6) ${\mathcal{S}}(f)\cap
f(\bm{T})=\bigcup_{\Gamma\in\mathscr{F}_{\\!D}(\Delta^{0})}{\mathcal{R}}_{\Gamma}(f)\cap
f(\bm{T}).$

This is so thanks to Lemma 2.12 and Definition 2.6 (c). Indeed, the Zariski
closures of the left hand side and the right hand side in the equation (2.6)
coincide with their respective analogues in (2.5). We only show the inclusion
“$\subset$” in (2.6) as the other one follows the same steps in reverse.

Let $w\in{\mathcal{S}}(f)\cap f(\bm{T})$, and let $\mu(f)$ denote the
topological degree of $f$. Then, the preimage $f^{-1}(w)$ has less than
$\mu(f)$ isolated points in $\bm{T}$ counted with multiplicity. Lemma 2.14
shows that the number of these preimages is less than $V(\Delta^{0}(f))$.
Then, Theorem 2.13 shows that $(f-w)_{\Gamma}=\underline{0}$ has a solution in
$\bm{T}$ for some face $\Gamma$ of $\Delta^{0}$. This implies that
$w\in{\mathcal{R}}_{\Gamma}(f)$. If $\Gamma$ is not dicritical, then
${\mathcal{R}}_{\Gamma}(f)\subset f(\mathcal{H})$, where $\mathcal{H}$ is the
union of coordinate hyperplanes in $K^{n}$. Since $w\not\in f(\mathcal{H})$,
we conclude that $\Gamma$ is dicritical. ∎

###  2.5. Uniruledness

We recall that a set $S\subset\bm{T}$ is _$K$ -uniruled_ if for any $w\in S$,
there exists a polynomial map $\varphi:\mathbb{K}\to\bm{T}$ with
$\varphi(0)=w$, and $\varphi(\mathbb{K})\subset{\mathcal{S}}(f)$.

######  Corollary 2.17.

The Jelonek set of any almost face-generic polynomial map $f:K^{n}\to\bm{T}$
is $K$-uniruled.

###### Proof.

We retain the notation for $f$ of Proposition 2.16. Thanks to (2.5), it is
enough to show $K$-uniruledness of ${\mathcal{R}}_{\Gamma}(f)$ for each pre-
origin face $\Gamma$ of $\Delta^{0}$. If $\Gamma$ is strictly pre-origin, then
${\mathcal{R}}_{\Gamma}(f)$ is the intersection of $\bm{T}$ with a set of the
form $K^{r}\times X$, where $X$ is an algebraic subset of $K^{n-r}$ (see
Definition 2.8). Hence, the sets ${\mathcal{R}}_{\Gamma}(f)$ are $K$-uniruled
whenever $\Gamma$ is strictly pre-origin.

Assume now that $\Gamma$ is an origin-face of $\delta^{0}$. Then, Definition
2.8 shows that ${\mathcal{R}}_{\Gamma}(f)=f_{\Gamma}(\bm{T})$. For every
$w\in{\mathcal{S}}(f)$, consider the line $L$ given by the parametrization
$z_{i}(t)=t-p_{i}$ ($i=1,\ldots,n$), where $f_{\Gamma}(p_{1},\ldots,p_{n})=w$.
Then, the polynomial map
$\varphi:=f_{\Gamma}(z_{1}(\cdot),\ldots,z_{n}(\cdot)):K\to{\mathcal{S}}(f)$
is well-defined and satisfies $\varphi(0)=w$. ∎

##  3\. Preliminaries on tropical geometry

We state in this section some well-known facts about tropical geometry (see
e.g. [4, 36, 31], and the references therein). Some of the exposition and
notations here are taken from [5, 3, 9]. We will introduce in §3.1 some
tropical objects such as tropical hypersurfaces, and illustrate their relation
to their classical analogues. A key result in this section is Propositon 3.6
which relates the common intersection of tropical hypersurfaces to
decompositions of the Newton polytopes and their faces.

###  3.1. The base field

A _locally convergent generalized Puiseux series_ is a formal series of the
form

$c(t)=\underset{r\in R}{\sum}c_{r}t^{r},$

where $R\subset\mathbb{R}$ is a well-ordered set that is bounded from below,
for which all $c_{r}\in\mathbb{C}$. We denote by $\mathbb{K}$ the set of all
locally convergent generalized Puiseux series. It is an algebraically-closed
field of characteristic 0 [32]. The field $\mathbb{K}$ can be equipped with
the function

$\begin{array}[]{lccc}\displaystyle\operatorname{val}:&\mathbb{K}&\longrightarrow&\mathbb{R}\cup\\{-\infty\\}\\\
\displaystyle&0&\longmapsto&-\infty\\\ \displaystyle&\underset{r\in
R}{\sum}c_{r}t^{r}\neq 0&\longmapsto&-\min_{R}\\{r\ |\ c_{r}\neq
0\\}.\end{array}$

We will call $\operatorname{val}$ the _valuation function_ or simply,
_valuation_. This extends to a map
$\operatorname{Val}:\mathbb{K}^{n}\rightarrow(\mathbb{R}\cup\\{-\infty\\})^{n}$
by evaluating $\operatorname{val}$ coordinate-wise, i.e.
$\operatorname{Val}(z_{1},\ldots,z_{n})=(\operatorname{val}(z_{1}),\ldots,\operatorname{val}(z_{n}))$.

######  Remark 3.1.

In the standard literature (c.f. [7]), the valuation of an element in
$\mathbb{K}$ is defined as $-\operatorname{val}$. Our choice for it to be the
opposite was motivated by some of the conventions in tropical geometry.

Given a a polynomial $f\in\mathbb{K}[z_{1},\ldots,z_{n}]$, its _tropical
hypersurface_ , denoted by $\mathbb{V}_{\\!\mathbb{T}}(f)$, is the subset in
$\mathbb{R}^{n}$ defined as
$\operatorname{Val}(\mathbb{V}^{*}_{\\!\mathbb{K}}(f))$. Tropical
hypersurfaces can be determined combinatorially. Consider the function

(3.1)
$\begin{array}[]{lccc}\displaystyle\nu_{f}:&\mathbb{Z}^{n}&\longrightarrow&\mathbb{R}\cup\\{-\infty\\}\\\
\displaystyle&a&\longmapsto&\begin{cases}\operatorname{val}(c_{a}),&\text{if
$a\in A$,}\\\ -\infty,&\text{otherwise.}\end{cases}\\\ \end{array}$

Its _Legendre transform_ is a piecewise-linear convex function
$\mathcal{L}(\nu_{f}):\mathbb{R}^{n}\to\mathbb{R}$

$x\mapsto\max_{a\in A}\\{\langle x,a\rangle+\nu_{f}(a)\\},$

where $\langle\leavevmode\nobreak\ ,\leavevmode\nobreak\
\rangle:\leavevmode\nobreak\
\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ is the standard
Eucledian product. The set of points $p\in\mathbb{R}^{n}$ at which
$\mathcal{L}(\nu_{f})$ is not differentiable is called the _corner locus_ of
$\mathcal{L}(\nu_{f})$. We have the famous fundamental theorem of tropical
geometry [8], [31, Theorem 3.13] that was first announced by Kapranov.

######  Theorem 3.2 (Kapranov).

The tropical hypersurface $\mathbb{V}_{\\!\mathbb{T}}(f)$ of a polynomial $f$
defined over $\mathbb{K}$ is the corner locus of its Legendre transform
$\mathcal{L}(\nu_{f})$.

###  3.2. Subdivisions from tropical polynomials

For any two values $a,b\in\mathbb{T}:=\mathbb{R}\cup\\{-\infty\\}$, their
_tropical summation_ $a\oplus b$ is defined as their maximum $\max(a,b)$, and
their _tropical multiplication_ $a\otimes b$ is their usual sum $a+b$. This
gives rise to a _tropical semi-field_ $(\mathbb{T},\oplus,\otimes)$, where
$\max(a,-\infty)=a$, and $-\infty+a=-\infty$. A _tropical polynomial_ $F$ is
defined over the semifield $(\mathbb{T},\oplus,\otimes)$, which gives rise to
a function

$\begin{array}[]{lccc}F:&\mathbb{T}^{n}&\longrightarrow&\mathbb{T}\\\
&x&\longmapsto&\displaystyle\max_{a\in A}\\{\langle
x,a\rangle+\gamma_{a}\\},\\\ \end{array}$

where $A$ is a finite set containing all $a\in\mathbb{N}^{n}$ for which
$\gamma_{a}\in\mathbb{R}$. The set $A$ is called the _support_ of the tropical
polynomial $F$, and the linear terms appearing in $F$ are called the _tropical
monomials_. The _tropicalization_ of the polynomial $f$ above is the tropical
polynomial

$f^{\operatorname{trop}}(x):=\max_{a\in A}\\{\langle
x,a\rangle+\operatorname{val}(c_{a})\\}.$

This coincides with the piecewise-linear convex function
$\mathcal{L}(\nu_{f})$ defined above, and thus Theorem 3.2 asserts that
$\mathbb{V}_{\\!\mathbb{T}}(f)$ is the corner locus of
$f^{\operatorname{trop}}$. Conversely, the corner locus of any tropical
polynomial is a tropical hypersurface.

####  3.2.1. Regular polyhedral cell-decompositions

All polytopes in this paper are assumed to be convex.

######  Definition 3.3.

A _polyhedral subdivision_ of a polytope $\Delta\subset\mathbb{R}^{n}$ is a
set of polytopes $\\{\Delta_{i}\\}_{i\in I}$ satisfying $\cup_{i\in
I}\Delta_{i}=\Delta$, and if $i,j\in I$, then $\Delta_{i}\cap\Delta_{j}$ is
either empty or it is a common face of $\Delta_{i}$ and $\Delta_{j}$. ∎

######  Definition 3.4.

Let $\Delta$ be a polytope in $\mathbb{R}^{n}$ and let $\tau$ denote a
polyhedral subdivision of $\Delta$ consisting of polytopes. We say that $\tau$
is _regular_ if there exists a continuous, convex, piecewise-linear function
$\varphi:\Delta\rightarrow\mathbb{R}$ such that the polytopes of $\tau$ are
exactly the domains of linearity of $\varphi$. ∎

Let $\Delta$ be an integer polytope in $\mathbb{R}^{n}$ and let
$\varphi:\leavevmode\nobreak\ \Delta\leavevmode\nobreak\
\cap\leavevmode\nobreak\ \mathbb{Z}^{n}\rightarrow\mathbb{R}$ be a function.
We denote by $\widehat{\Delta}(\varphi)$ the convex hull of the set

$\\{(a,\leavevmode\nobreak\ \varphi(a))\in\mathbb{R}^{n+1}\leavevmode\nobreak\
|\leavevmode\nobreak\ a\in\Delta\cap\mathbb{Z}^{n}\\}.$

Then the polyhedral subdivision of $\Delta$ induced by projecting the union of
the lower faces of $\widehat{\Delta}(\varphi)$ onto the first $n$ coordinates,
is regular.

######  Example 3.5.

The blue part of the second polytope, $\Delta^{0}_{2}$, appearing in Figure 3
is also a convex polytope. It admits a regular subdivision induced by the
function $\varphi$ satisfying $\varphi(1,1,0)=\varphi(1,1,2)=0$,
$\varphi(0,1,2)=7$, $\varphi(0,2,4)=3$, $\varphi(1,2,4)=2$, and
$\varphi(2,2,4)=5$. ∎

####  3.2.2. Subdivisions and their duals

Keeping with the same notation as above, the tropical hypersurface
$\mathbb{V}_{\\!\mathbb{T}}(f)$ is an $(n-1)$-dimensional piecewise-linear
polyhedral complex which produces a _polyhedral cell-decomposition_ $\Xi$ of
$\mathbb{R}^{n}$. This is a finite collection of the relative interiors of
polyhedra in $\mathbb{R}^{n}$, whose closures satisfy Definition 3.3. An
element of $\Xi$ is called _cell_. The $n$-dimensional cells of $\Xi$, are the
connected components of the complement of $\mathbb{V}_{\\!\mathbb{T}}(f)$ in
$\mathbb{R}^{n}$. All together, cells of dimension less than $n$ form the
domains of linearity of $f^{\operatorname{trop}}$ at
$\mathbb{V}_{\\!\mathbb{T}}(f)$.

The cell-decomposition $\Xi$ induces a regular subdivision $\tau$ of the
Newton polytope $\Delta$ of $f$ in the following way (see also [3, Section
3]). Given a cell $\xi$ of $\mathbb{V}_{\\!\mathbb{T}}(f)$ and $x\in\xi$, the
set

$\mathcal{I}_{\xi}:=\\{a\in A\leavevmode\nobreak\ |\leavevmode\nobreak\
f^{\operatorname{trop}}(x)=\langle x,a\rangle+\operatorname{val}(c_{a})\\}$

does not depend on $x$. All together the polyhedra $\delta$, defined as the
convex hull of $\mathcal{I}_{\xi}$ form a subdivision $\tau$ of $\Delta$
called the _dual subdivision_ , and the polyhedron $\delta$ is called the
_dual_ of $\xi$. The subdivision $\tau$ is regular as the corresponding
function $\varphi$ in Definition 3.4 is the same as $\nu_{f}$ in 3.1 (c.f.
Example 3.5).

An analogous description holds true for any tuple $f$ of polynomials
$f_{1},\ldots,f_{k}\in\mathbb{K}[z_{1},\ldots,z_{n}]$. Let
$A_{1},\ldots,A_{k}\subset\mathbb{Z}^{n}$,
$\Delta_{1},\ldots,\Delta_{k}\subset\mathbb{R}^{n}$, and
$T_{1},\ldots,T_{k}\subset\mathbb{R}^{n}$ be their respective supports, Newton
polytopes, and tropical hypersurfaces respectively. The union of these latter
defines a polyhedral cell-decomposition $\Xi$ of $\mathbb{R}^{n}$. Any non-
empty cell of $\Xi$ can be written as

$\xi=\xi_{1}\cap\cdots\cap\xi_{k}$

with $\xi_{i}\in\Xi_{i}$ ($i=1,\ldots,k$), where $\Xi_{i}$ is the polyhedral
cell-decomposition of $\mathbb{R}^{n}$ produced by $T_{i}$. Any cell
$\xi\in\Xi$ can be uniquely written in this way. Similarly, the polyhedral
cell-decomposition induces a _mixed dual subdivision_ $\tau$ of the Minkowskii
sum $\Delta=\Delta_{1}+\cdots+\Delta_{k}$ in the following way. Any polytope
$\delta\in\tau$ is equipped with a unique representation
$\delta=\delta_{1}+\cdots+\delta_{k}$ with $\delta_{i}\in\tau_{i}$
($i=1,\ldots,k$), where each $\tau_{i}$ is the dual subdivision of
$\Delta_{i}$. The above duality-correspondence applied to the (tropical)
product of the tropical polynomials gives rise to the following well-known
fact (see e.g. [3, §3 & 4]).

######  Proposition 3.6.

There is a one-to-one duality correspondence between $\Xi$ and $\tau$, which
reverses the inclusion relations, and such that if $\delta\in\tau$ corresponds
to $\xi\in\Xi$, then

  1. (1)

$\xi=\xi_{1}\cap\cdots\cap\xi_{k}$ with $\xi_{i}\in\Xi_{i}$ for
$i=1,\ldots,n$, then $\delta=\delta_{1}+\cdots+\delta_{k}$ with
$\delta_{i}\in\tau_{i}$ is the polytope dual to $\xi_{i}$.

  2. (2)

$\dim\xi+\dim\delta=n$,

  3. (3)

the cell $\xi$ and the polytope $\delta$ span orthonogonal real affine spaces,

  4. (4)

$\delta$ lies on a proper face $F$ of $\Delta$ if and only if the cell $\xi$
is unbounded and contains a half-line spanned by the outward normal vector to
$F$.

###### Proof.

Item (1) is a consequence of [3, Proposition 4.1]. Items (2), and (3) are
obtained by applying item (1) and (2) of [3, §3] to the tropical hypersurface
$T_{1}\cup\cdots\cup T_{k}$. Item (4), is similarly obtained from Item (3) of
[3, §3], and diagram (3.1) in loc. cit. . ∎

We say that the above polyhedral cell-decomposition $\Xi$ is _induced by the
tuple $f$_.

Figure 4. To the left: The subdivisions $\tau_{1}$, $\tau_{2}$, and $\tau$ of
$\Delta^{0}_{1}$, $\Delta^{0}_{2}$, and $\Delta^{0}_{1}+\Delta^{0}_{2}$
respectively. To the right: the corresponding dual cell-decomposition of
$\mathbb{R}^{2}$. All obtained from Example 4.3

####  3.2.3. Transversal cell-decompositions

In what follows, we introduce a family of polynomial tuples having, whose
tropical hypersurfaces have non-degenerate intersections.

######  Definition 3.7.

The cell $\xi\in\Xi$ is _transversal_ if
$\dim(\delta)=\dim(\delta_{1})+\cdots+\dim(\delta_{k})$. A polyhedral cell-
decomposition is _transversal_ if so are all of its cells. ∎

######  Notation 3.8.

For any $\xi\in\Xi$ as in Proposition 3.6, we will use $\delta(\xi)$ to denote
the polytope $\delta$, dual to $\xi$ and we use $\delta(\xi_{i})$
($i=1,\ldots,n$) to denote the polytope $\delta_{i}$, dual to $\xi_{i}$. ∎

Applying [34, Theorem 1.1] recursively on transversal intersections of
tropical hypersurfaces in $\mathbb{R}^{n}$, we obtain the following
generalized version of Theorem 3.2 (see also [31, Ch. 4, §6, Theorem 4.6.18]).

######  Theorem 3.9.

Let $f$ be a tuple of polynomials
$f_{1},\ldots,f_{k}\in\mathbb{K}[z_{1},\ldots,z_{n}]$, and assume that the
polyhedral cell-decomposition of $\mathbb{R}^{n}$, induced by $f$ is
transversal. Then, it holds

(3.2)
$\operatorname{Val}(\mathbb{V}^{*}_{\\!\mathbb{K}}(f))=\mathbb{V}_{\\!\mathbb{T}}(f_{1})\cap\cdots\cap\mathbb{V}_{\\!\mathbb{T}}(f_{k}).$

######  Definition 3.10.

A polynomial tuple $f\in\mathbb{K}[A]$ is called _face-generic_ if it is
almost face-generic and if for any $\Gamma\prec\Delta$, and any $I\subset[n]$,
the intersection $\cap_{i\in I}\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma})$ is
transversal. ∎

######  Lemma 3.11.

Face-generic tuples form a Zariski open subset of $K[A]$.

###### Proof.

Let $m:=|A_{1}|+\cdots+|A_{k}|$, let $\mathbb{K}^{*}[A]$ be the largest torus
in $\mathbb{K}[A]$, let $\Gamma$ be a face of $\Delta$, and let $I\subset[n]$.
Consider the map
$\operatorname{\textbf{Val}}:\mathbb{K}^{*}[A]\to\mathbb{R}^{m}$,
$c\mapsto(\operatorname{val}(c_{1}),\ldots,\operatorname{val}(c_{m}))$ sending
a tuple $f$ onto the point in $\mathbb{R}^{m}$, whose coordinates form the
tuple of valuations of coefficients of $f$. Note that there are finitely-many
possible combinatorial types of cell-decompositions $\Xi$ of $\mathbb{R}^{n}$,
induced by $f_{\Gamma}$ obtained from points in
$\operatorname{\textbf{Val}}(c)\in\mathbb{R}^{m}$. Therefore, the set formed
by all $\operatorname{\textbf{Val}}(c)\in\mathbb{R}^{m}$ for which there
exists $f\in\mathbb{K}^{*}[A]$ satisfying $\operatorname{\textbf{Val}}(f)=c$
and $\cap_{i\in I}\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma})$ is not transversal,
form a finite union $\bm{H}_{I\Gamma}$ of hyperplanes
$H_{1},\ldots,H_{r}\subset\mathbb{R}^{m}$. Each such hyperplane is given by a
linear combination on $\operatorname{\textbf{Val}}c$, representing a necessary
condition for a collection of affine pieces
$L_{i}\subset\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma})$ ($i\in I$) to intersect
non-transversally (in the usual sense). The union $\bm{H}_{I\Gamma}$ of
hyperplanes is finite since there is a finite choice of above collections
$(L_{i})_{i\in I}$ and there are finitely-many combinatorial types for a non-
transversal intersection of such a collection.

Finally, notice that the map $\operatorname{\textbf{Val}}$ is continuous.
Therefore, the set
$\operatorname{\textbf{Val}}^{-1}(\mathbb{K}^{*}[A]\setminus\bm{H})$ is
Zariski open in $\mathbb{K}^{*}[A]$. The proof follows from Proposition 2.7. ∎

##  4\. Correspondence for the Jelonek set

In this section, we prove that the tropical non-properness set
${\mathcal{S}}_{\mathbb{T}}(F)$ of a tropical polynomial map
$F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the tropicalization of face resultants
of the corresponding polynomial map. Then, Theorem 1.1 would immediately
follow from Proposition 2.16. We finish this section by proving Theorem 1.5,
where we show how to describe the Newton polytope of the Jelonek set of
complex polynomial maps from its tropical counterpart.

######  Notation 4.1.

Let $F:\mathbb{R}^{n}\to\mathbb{R}$ be a tropical polynomial function
expressed as $x\mapsto\max_{a\in A}\\{\langle x,a\rangle+\gamma_{a}\\}$, where
$A$ is the support of $F$, and the values $\gamma_{a}$ are real. For any value
$y\in\mathbb{R}\cup\\{-\infty\\}$, we use the notation ${\mathcal{T}}_{y}(F)$
for the corner locus of the function

$x\mapsto\max_{a\in A}\\{\langle x,a\rangle+\gamma_{a},\leavevmode\nobreak\
y\\}.$

Similarly, if $F$ is a tropical polynomial map
$(F_{1},\ldots,F_{n}):\mathbb{R}^{m}\to\mathbb{R}^{n}$, then for any tuple
$y:=(y_{1},\ldots,y_{n})\in(\mathbb{R}\cup\\{-\infty\\})^{n}$ we use the
notation ${\mathcal{T}}_{y}(F)$ to refer to the intersection
${\mathcal{T}}_{y_{1}}(F_{1})\cap\cdots\cap{\mathcal{T}}_{y_{n}}(F_{n})$ in
$\mathbb{R}^{m}$. ∎

######  Definition 4.2 (virtual preimage).

The set ${\mathcal{T}}_{y}(F)$ corresponding to the tropical map $F$ in
Notation 4.1 will be called the _virtual preimage of $y$_ under $F$. ∎

######  Example 4.3.

Consider the polynomial map $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ defined by the
polynomials

$(F_{1},\leavevmode\nobreak\ F_{2})=\big{(}\max(a-1,\leavevmode\nobreak\
2a+b+2,\leavevmode\nobreak\ 3a+2b-5),\leavevmode\nobreak\
\max(a+b,\leavevmode\nobreak\ 2a+2b,\leavevmode\nobreak\ a+2b)\big{)}.$

The virtual preimage ${\mathcal{T}}_{(-2,-1)}(F)$ is represented in Figure 4
with the thick orange lines. ∎

Let $F:=(F_{1},\ldots,F_{n}):\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a tropical
polynomial map, let $A_{i}$ be the support of $F_{i}$ ($i=1,\ldots,n$). Using
this notation, we define $\Delta^{0}$ to be the same tuple of polytopes in
$\mathbb{R}^{n}$ defined as in (2.1) from §2.2. For the rest of the section we
make the following assumption:

For every $i=1,\ldots,n$, the tropical polynomial $F_{i}$ has no constant
tropical terms.

This implies that the support $A_{i}$ does not contain $\\{\underline{0}\\}$,
and thus $\Delta_{i}^{0}\setminus\Delta_{i}=\\{\underline{0}\\}$
($i=1,\ldots,n$). Since virtual preimages are intersections of tropical
hypersurfaces, for any $y\in\mathbb{R}^{n}$, the union

${\mathcal{T}}_{y_{1}}(F_{1})\cup\cdots\cup{\mathcal{T}}_{y_{n}}(F_{n})$

forms a polyhedral cell-decomposition of $\mathbb{R}^{n}$, denoted $\Xi(y)$,
and induces a mixed subdivision $\tau_{y}$ of $\Sigma\Delta^{0}$, dual to
$\Xi(y)$ in the sense of Proposition 3.6 in §3.2.2. Accordingly, the cell-
decomposition $\Xi$ of $\mathbb{R}^{n}$ induced by $F$ is obtained from
$\Xi(y)$ by taking $y_{i}=-\infty$ ($i=1,\ldots,n$). Figure 4 on the right
represents the cell-decomposition $\Xi(-2,-1)$ of $\mathbb{R}^{2}$ given by
the set ${\mathcal{T}}_{-2}(F_{1})\cup{\mathcal{T}}_{-1}(F_{2})$, where $F$ is
from Example 4.3.

######  Definition 4.4.

A cell $\xi\in\Xi(y)$ is said to _dicritical_ if it contains a dicritical
half-line. ∎

We deduce the following observation from the definitions and Proposition 3.6.

######  Lemma 4.5.

Let $\Gamma\prec\Delta^{0}$, and let $\xi\in\Xi(y)$, so that
$\delta(\xi_{i})\subset\Gamma_{i}$ ($i=1,\ldots,n$). Then, the following holds

  1. (1)

$\xi$ is dicritical if and only if $\Gamma$ is dicritical, and

  2. (2)

$\dim\delta(\xi_{i})>0$, for all $i=1,\ldots,n$ if and only if
$\xi\subset{\mathcal{T}}_{y}(F)$.

Similarly to Notation 2.5 of §2, for any $\Gamma\prec\Delta^{0}$, and any
$i\in[n]$, we define the function $F_{i\Gamma}:\mathbb{R}^{n}\to\mathbb{R}$,

$x\mapsto\max_{a\in\Gamma_{i}}(\langle x,a\rangle+\gamma_{a}).$

For any $I\subset[n]$, we define the _restriction of $F$ onto $I\Gamma$_ to be
the tropical polynomial map

$F_{I\Gamma}:=(F_{i\Gamma})_{i\in I}:\mathbb{R}^{n}\to\mathbb{R}^{|I|}.$

######  Example 4.6.

Consider the polytopes from Figure 4, and the map $F$ from Example 4.3. Let
$\Gamma=\big{(}\\{(1,0),\leavevmode\nobreak\ (2,1),\leavevmode\nobreak\
(3,1)\\},\leavevmode\nobreak\ \\{(0,0),\leavevmode\nobreak\
(1,1),\leavevmode\nobreak\ (2,2)\\},\big{)}$. Then, $F_{1\Gamma}=F_{1}$, and
$F_{2\Gamma}=\max(a+b,\leavevmode\nobreak\ 2a+2b)$. ∎

Recall that ${\mathcal{S}}_{\mathbb{T}}(F)$ is the set of points in
$\mathbb{R}^{n}$ for which ${\mathcal{T}}_{y}(F)$ has a dicritical half-line
in $\mathbb{R}^{n}$.

######  Proposition 4.7.

Let $f:=(f_{1},\ldots,f_{n}):\mathbb{K}^{n}\to\bm{T}$ be a face-generic
polynomial map, let $F$ be the tropical polynomial map
$(f_{1}^{\operatorname{trop}},\ldots,f_{n}^{\operatorname{trop}}):\mathbb{R}^{n}\to\mathbb{R}^{n}$,
and let $\Delta^{0}:=\Delta^{0}(f)$. Then, it holds

${\mathcal{S}}_{\mathbb{T}}(F)=\bigcup_{\Gamma\in{\mathscr{F}}_{D}(\Delta^{0})}\operatorname{Val}({\mathcal{R}}_{\Gamma}(f)).$

###### Proof.

The inclusion “$\supset$” does not require face-genericity. Hence, we prove it
first. Let $\Gamma$ be a dicritical face of $\Delta^{0}$ for which
${\mathcal{R}}_{\Gamma}(f)$ is not empty. Let $\bm{\circleddash}$ be the
subset of indexes $i\in[n]$ for which $\Gamma_{i}$ contains the origin
$\underline{0}$. Let $w$ be a point in ${\mathcal{R}}_{\Gamma}(f)$ and let $y$
be its valuation in $\mathbb{R}^{n}$. We use $\Sigma_{\Gamma}(y)$ to denote
the cell-decomposition of $\mathbb{R}^{n}$, induced by the collection of
tropical hypersurfaces $S_{1},\ldots,S_{n}$, where

$S_{i}:=\begin{cases}{\mathcal{T}}_{y_{i}}(F_{i\Gamma})&\text{if
}i\in\bm{\circleddash}\\\\[4.0pt]
{\mathcal{T}}_{-\infty}(F_{i\Gamma})&\text{otherwise}.\\\ \end{cases}$

The common locus $S_{1}\cap\cdots\cap S_{n}$ is non-empty as it contains the
set $\operatorname{Val}({\mathcal{R}}_{\Gamma}(f))$. Hence, there exists a
cell $\sigma\in\Sigma_{\Gamma}(y)$ satisfying the conditions of Lemma 4.5 (2).
Since $\Gamma$ is a face of $\Delta^{0}$, the cell-decomposition $\Xi(y)$
contains a cell $\xi\subset\sigma$ satisfying
$\delta(\xi_{i})=\delta(\sigma_{i})$ ($i=1,\ldots,n$) (e.g. if $\Gamma$ and
$F$ are as in Example 4.6, then $\Sigma_{\Gamma}(-2,-1)$ is represented in
Figure 5). Finally, the cell $\xi$ is dicritical follows from Lemma 4.5 (1).

Figure 5. The cell-decompositions $\Xi(-2,-1)$ (in transparent colors) and
$\Xi_{\Gamma}(-2,-1)$ (in opaque colors) corresponding to Example 4.6

Now, we prove the other direction “$\subset$”. Assume that
${\mathcal{T}}_{y}(F)$ has a dicritical half-line $L\subset\mathbb{R}^{n}$ for
some $y\in\mathbb{R}^{n}$. Then, Lemma 4.5 (2) shows that there exists a cell
$\xi\in\Xi(y)$, satisfying $\dim\delta(\xi_{i})>0$ ($i=1,\ldots,n$), and
$L\subset\xi$. Proposition 3.6 (4) shows that
$\delta(\xi_{i})\subset\Gamma_{i}$ ($i=1,\ldots,n$) for some
$\Gamma\prec\Delta^{0}$, and Lemma 4.5 (1) shows that $\Gamma$ is dicritical.
Then, for any $x\in\xi$, and any $i\in[n]$, the maximum

$\max_{a\in\Gamma_{i}}(\langle
x,a\rangle+\operatorname{val}(c_{a}(i)),\leavevmode\nobreak\ y_{i})$

is reached twice. Furthermore, if this maximum is reached by $y_{i}$, then
$\underline{0}\in\Gamma_{i}$. Theorem 3.2 shows that for any $w\in\bm{T}$ such
that $\operatorname{Val}(w)=y$, it holds

$x\in\mathbb{V}_{\\!\mathbb{T}}((f_{i}-w_{i})_{\Gamma})\leavevmode\nobreak\
\text{for }i=1,\ldots,n.$

In what follows, we show that $x$ is the valuation of a solution $z\in\bm{T}$
to $(f-w)_{\Gamma}=\underline{0}$. This would imply that
$w\in{\mathcal{R}}_{\Gamma}(f)$, which will yield
$y\in\operatorname{Val}({\mathcal{R}}_{\Gamma}(f))$.

We retain from the beginning of the proof the notation of $\bm{\circleddash}$
corresponding to the face $\Gamma$. Then, we have
$(f_{i}-w_{i})_{\Gamma}=f_{i\Gamma}-w_{i}$ if $i\in\bm{\circleddash}$, and
$(f_{i}-w_{i})_{\Gamma}=f_{i\Gamma}$ otherwise. Consequently, it holds

(4.1)
$\mathbb{V}_{\\!\mathbb{T}}((f_{i}-w_{i})_{\Gamma})=\begin{cases}\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma}-w_{i})&\text{if
}i\in\bm{\circleddash}\\\\[4.0pt]
\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma})&\text{otherwise}.\\\ \end{cases}$

Pick any of the largest subsets $\bm{\top}\subset[n]$ of indexes containing
$[n]\setminus\bm{\circleddash}$ so that the intersection
$\cap_{i\in{\bm{\top}}}\mathbb{V}_{\\!\mathbb{T}}((f_{i}-w_{i})_{\Gamma})$ is
transversal. Such subsets exist thanks to Definition 2.6 (b). Then, Theorem
3.9 shows that

(4.2) $(f_{i}-w_{i})_{\Gamma}=0,\leavevmode\nobreak\ i\in{\bm{\top}}$

has a solution $z\in\bm{T}$ satisfying $\operatorname{Val}(z)=x$. In fact,
since (4.2) is quasi-homogeneous, there are infinitely-many such solutions
$z$. Now, we show that $\bm{\top}$ is large enough.

Condition (b) of Definition 2.6 implies that one such solution $z$ can be
chosen so that for any $i\in\bm{\circleddash}\setminus{\bm{\top}}$, it holds
$\operatorname{val}(f_{i\Gamma}(z))=F_{i\Gamma}(x)=y_{i}$. This way, given any
$w\in\bm{T}$ satisfying $\operatorname{Val}(w)=y$, we constructed a solution
$z\in\bm{T}$ to the system $(f-w)_{\Gamma}=\underline{0}$. ∎

###  4.1. Proof of Theorem 1.5

We first show that there exists a polytope $\Pi\subset\mathbb{R}^{n}_{\geq
0}$, and a Zariski open subset $U\subset\mathbb{C}[A]$, such that for any
$g\in U$, it holds $\mathcal{N}(\mathsf{S}_{g})=\Pi$, where $\mathsf{S}_{g}$
is the polynomial realizing $\mathcal{S}(g)$. Let $K$ be an algebraically
closed field of characteristic zero. Then, any point in $K[A]$ is obtained
from a collection $h:=(h_{1},\ldots,h_{n})$ of polynomials
$h_{i}\in\mathbb{Z}[z_{1},\ldots,z_{n},c_{a}(i):a\in A_{i}]$ ($i=1,\ldots,n$),
where we substitute the parameters $c_{a}(i)$ with values in $K$. Accordingly,
(1.1) shows that the polynomial $\mathsf{S}_{h}$ is either a constant or is
expressed as a finite sum

$\sum\phi_{\alpha}(\bm{c})w^{\alpha},$

where $\bm{c}:=\big{(}(c_{a}(i))_{a\in A_{i}}\big{)}_{i\in[n]}$,
$\phi_{\alpha}\in\mathbb{Z}[\bm{c}]$, and
$w^{\alpha}:=w_{1}^{\alpha_{1}}\cdots w_{n}^{\alpha_{n}}$. If
$\mathsf{S}_{h}\in K$, then any map $K^{n}\to K^{n}$, obtained from $K[A]$, is
proper. Otherwise, if $P$ is defined as the polynomial $\prod\phi_{\alpha}$,
then for any $f,\widetilde{f}\in K[A]\setminus\mathbb{V}_{K}(P)$, it holds
$\phi_{\alpha}(f)=0$ $\Leftrightarrow$ $\phi_{\alpha}(\widetilde{f})=0$.
Therefore, the Newton polytopes of $\mathsf{S}_{f}$ and
$\mathsf{S}_{\widetilde{f}}$ coincide in $\mathbb{R}^{n}$. Thus, it is enough
to take $U:=\mathbb{C}[A]\setminus\mathbb{V}_{\\!\mathbb{C}}(P)$, and
$\mathcal{N}(\mathsf{S}_{g})=\Pi$ follows for every $g\in U$.

Next, we show that for any face-generic polynomial map $f\in\mathbb{K}[A]$, it
holds $\mathcal{N}({\mathcal{S}}_{f})=\Pi$. It suffices to show that
$\mathbb{K}[A]\setminus\mathbb{V}_{\\!\mathbb{K}}(P)$ contains all face-
generic polynomial maps. Let $f\in\mathbb{K}[A]$ be a face-generic polynomial
map, and let $g$ be a complex polynomial map in $U$. Then, the Newton-Puiseux
algorithm implies that one can make a perturbation $f\to\widetilde{f}$ only on
the coefficients of the first-order terms, so to get
$\phi_{\alpha}(\widetilde{f})\neq 0$ whenever $\phi_{\alpha}(g)\neq 0$. Then,
it holds $\mathcal{N}(\mathsf{S}_{\widetilde{f}})=\Pi$. As this perturbation
does not affect the valuations of the coefficients in $f$, Theorem 1.1 shows
that
$\operatorname{Val}({\mathcal{S}}(f))=\operatorname{Val}({\mathcal{S}}(\widetilde{f}))$.
Then, Proposition 3.6 shows that
$\mathcal{N}({\mathcal{S}}_{\widetilde{f}})=\mathcal{N}({\mathcal{S}}_{f})$.
Therefore, the Newton polytope of the Jelonek set of every face-generic
polynomial map in $\mathbb{K}[A]$ is equal to $\Pi$.

To finish the proof, note that the face-vector and the slopes of $\Pi$ can be
computed using the duality in Proposition 3.6 between the tropical
hypersurface ${\mathcal{S}}_{\mathbb{T}}(F)$ and the polytope $\Pi$.

##  5\. The Jelonek set via polyhedral geometry

Let $f:=(f_{1},\ldots,f_{n}):\mathbb{K}^{n}\to\bm{T}$ be a face-generic
polynomial map, and let $F$ be the tropical polynomial map
$(f_{1}^{\operatorname{trop}},\ldots,f_{n}^{\operatorname{trop}}):\mathbb{R}^{n}\to\mathbb{R}^{n}$.
Fix a face $\Gamma$ of $\Delta^{0}:=\Delta^{0}(f)$, and let $\Sigma_{\Gamma}$
denote the polyhedral cell-decomposition of $\mathbb{R}^{n}$ induced by the
polynomials $f_{\Gamma}$ (see e.g. Proposition 3.6). Since $f$ is face-
generic, the cell-decomposition $\Sigma_{\Gamma}$ is transversal. The proof of
Theorem 1.3 relies on an effective description of the tropical non-properness
set through the cell-decomposition of $\Sigma_{\Gamma}$ for each dicritical
face $\Gamma$.

###  5.1. Contributing cells

From the proof of Proposition 4.7, we retain the notation $\bm{\circleddash}$
for the subset of indexes $i\in[n]$ for which $\Gamma_{i}$ contains
$\underline{0}$. In what follows, we consider the intersection locus

$\operatorname{Trop}(\setminus\\!\bm{\circleddash}):=\bigcap_{i\in[n]\setminus\bm{\circleddash}}\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma}).$

Note that the cell-decomposition $\Sigma_{\Gamma}$ of $\mathbb{R}^{n}$ is a
refinement of the cell-decomposition induced by
$\cup_{i\in[n]\setminus\bm{\circleddash}}\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma})$.
Indeed, since $\Sigma_{\Gamma}$ is given by the union
$\cup_{i\in[n]}\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma})$. This justifies the
following notion.

######  Definition 5.1 (contributing cells).

For any cell $\sigma=\sigma_{1}\cap\cdots\cap\sigma_{n}\in\Sigma_{\Gamma}$,
either $\sigma\subset\operatorname{Trop}(\setminus\\!\bm{\circleddash})$, or
$\sigma\cap\operatorname{Trop}(\setminus\\!\bm{\circleddash})=\emptyset$. In
the former case, we say that $\sigma$ is _contributing_. ∎

Thanks to Theorem 3.9 applied to the transversal intersection
$\operatorname{Trop}(\setminus\\!\bm{\circleddash})$, we will see that for
each $\Gamma$, only contributing cells of $\Sigma_{\Gamma}$ are taken into
account for computing the tropical non-properness set. We will distinguish two
types of contributing cells: those who belong to the tropical hypersurface
$\mathbb{V}_{\\!\mathbb{T}}(f_{i\Gamma})$ for some $i\in\bm{\circleddash}$,
and those who don’t. This distinction takes into account the dimensions of the
collection of dual cells $\delta(\sigma_{1}),\ldots,\delta(\sigma_{n})$ of a
cell $\sigma\in\Sigma_{\Gamma}$. Namely, we define the subset

$I_{\sigma}^{+}:=\\{i\in[n]\leavevmode\nobreak\ |\leavevmode\nobreak\
\dim\delta(\sigma_{i})>0\\}.$

######  Remark 5.2.

If $\sigma$ is a contributing cell, then $[n]\setminus\bm{\circleddash}\subset
I_{\sigma}^{+}$. In other words, for all the members $\Gamma_{i}$ of $\Gamma$,
that do not contain the origin, the dual cell
$\delta(\sigma_{i})\subset\Gamma_{i}$ satisfies $\dim\delta(\sigma_{i})>0$.

For contributing cells $\sigma$, and for each $i\in\bm{\circleddash}$, it will
be useful later to distinguish whether $i\in
I_{\sigma}^{+}\cap\bm{\circleddash}$, or $i\in\bm{\circleddash}\setminus
I_{\sigma}^{+}$.

######  Notation 5.3.

If $\sigma$ is a contributing cell of $\Sigma_{\Gamma}$, we denote
$\operatorname{Trs}(\sigma)$ the subset of all $y\in\mathbb{R}^{n}$ for which

(5.1) $\bigcap_{i\in
I_{\sigma}^{+}\cap\bm{\circleddash}}{\mathcal{T}}_{y_{i}}(F_{i\Gamma})\cap\operatorname{Trop}(\setminus\\!\bm{\circleddash})$

is transversal at $x\in\sigma$. Thanks to Definition 3.10, we use the
convention $\operatorname{Trs}(\sigma)=\mathbb{R}^{n}$ if
$I_{\sigma}^{+}=[n]\setminus\bm{\circleddash}$. ∎

###  5.2. Tropical non-properness is a union of finitely-many polytopes

Several technical results are required in order to prove Theorem 1.3.

######  Lemma 5.4.

If $\sigma$ is contributing, then $\operatorname{Trs}(\sigma)$ equals to

(5.2) $\bigcup_{x\in\sigma}\left\\{y\in\mathbb{R}^{n}\leavevmode\nobreak\
\left|\leavevmode\nobreak\ y_{i}<F_{i\Gamma}(x),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in
I_{\sigma}^{+}\cap\bm{\circleddash}\right\\}\right.\\!,$

and is empty otherwise. Furthermore, if $y$ is outside the closure of the set
(5.2), then the set (5.1) is empty.

###### Proof.

The emptyness of $\operatorname{Trs}(\sigma)$ can be deduced from the
definitions if $\sigma$ is not contributing. Otherwise, let $i$ be an index in
$\bm{\circleddash}\cap I_{\sigma}^{+}$, and let $x$ be a point in
$\sigma_{i}$. Then, it holds $x\in{\mathcal{T}}_{y_{i}}(F_{i\Gamma})$ for any
$y_{i}<F_{i\Gamma}(x)$, where the maximum of $F_{i}$ is reached exactly twice.
Recall that $\Xi$ is transversal and that
$\operatorname{Trop}(\setminus\\!\bm{\circleddash})$ consists of transversal
intersection points. Therefore, the inclusion
$\operatorname{Trs}(\sigma)\subset$ (5.2) follows from the definitions.

Now we prove the second inclusion, and the rest of the Lemma. Since
$i\in\bm{\circleddash}\cap I_{\sigma}^{+}$, there are three tropical monomial
terms of $y_{i}\oplus F_{i}$ reaching the maximum at $x$ if $y_{i}=F_{i}(x)$.
Hence we get $y\not\in\operatorname{Trs}(\sigma)$. Otherwise, if $y$ is not in
the closure of (5.2) (i.e. $y_{i}>F_{i}(x)$ for some
$i\in\bm{\circleddash}\cap I_{\sigma}^{+}$), only one monomial terms of
$y_{i}\oplus F_{i}$ reaches the maximum. Hence, we get
$x\not\in{\mathcal{T}}_{y_{i}}(F_{i\Gamma})$, making the set (5.1) empty. In
particular, we get $y\not\in\operatorname{Trs}(\sigma)$. ∎

######  Notation 5.5.

For any cell $\sigma\in\Sigma_{\Gamma}$, we define
$Y(\sigma)\subset\mathbb{R}^{n}$ to be the set

$\bigcup_{x\in\sigma}\left\\{y\in\mathbb{R}^{n}\leavevmode\nobreak\
\left|\leavevmode\nobreak\ y_{i}=F_{i\Gamma}(x),\leavevmode\nobreak\
\forall\leavevmode\nobreak\ i\in\bm{\circleddash}\setminus
I_{\sigma}^{+}\right\\}\right.\\!.$

∎

######  Lemma 5.6.

For any contributing cell $\sigma\in\Sigma_{\Gamma}$, the set
$\overline{\operatorname{Trs}(\sigma)}\cap\overline{Y(\sigma)}$ is either
empty or is a polytope in $\mathbb{R}^{n}$ of dimension at most $n-1$.

###### Proof.

We start by showing that $\dim\operatorname{Trs}(\sigma)\cap Y(\sigma)\leq
n-1$. Assuming non-emptyness, we make the following re-enumeration:

$\begin{matrix}\bm{\circleddash}\setminus
I_{\sigma}^{+}=\\{1,\ldots,r\\},&\bm{\circleddash}\cap
I_{\sigma}^{+}=\\{r+1,\ldots,s\\},&\text{and}&[n]\setminus\bm{\circleddash}=\\{s+1,\ldots,n\\}.\end{matrix}$

Then, we have the following expressions thanks to Lemma 5.4

$\begin{matrix}\overline{\operatorname{Trs}(\sigma)}=&\mathbb{R}^{r}\times\prod_{i=r+1}^{s}\\{y_{i}\leq\varepsilon_{i}\\}\times\mathbb{R}^{n-s},&\text{and
}&Y(\sigma)=&F_{[r]\Gamma}(\sigma)\times\mathbb{R}^{s-r}\times\mathbb{R}^{n-s},\end{matrix}$

where $\varepsilon_{i}=\max_{x\in\sigma}F_{i\Gamma}(x)$ ($i=r+1,\ldots,s$),
and
$F_{[r]\Gamma}:=(F_{1\Gamma},\ldots,F_{r\Gamma}):\mathbb{R}^{n}\to\mathbb{R}^{r}$.
Then, we obtain

(5.3)
$\overline{\operatorname{Trs}(\sigma)}\cap\overline{Y(\sigma)}=\overline{F_{[r]\Gamma}(\sigma)}\times\prod_{i=r+1}^{s}\\{y_{i}\leq\varepsilon_{i}\\}\times\mathbb{R}^{n-s}.$

Since $F_{[r]\Gamma}$ is continuous, the right-hand-side of the set (5.3) has
dimension at most $n-1$ if

(5.4) $\dim F_{[r]\Gamma}(\sigma)\leq r-1.$

The transversality of $\Sigma_{\Gamma}$ implies that

$\operatorname{cdim}\sigma=\sum_{i=1}^{n}\operatorname{cdim}\sigma_{i}=\sum_{i\in
I_{\sigma}^{+}}\operatorname{cdim}\sigma_{i}\geq|I_{\sigma}^{+}|,$

where the second equality follows from Proposition 3.6 (2). Since
$[n]\setminus\bm{\circleddash}\subset I_{\sigma}^{+}$ (see Remark 5.2), it
holds $\operatorname{cdim}\sigma\leq n-r$. This yields inequality (5.4).

Now we show that
$\overline{\operatorname{Trs}(\sigma)}\cap\overline{Y(\sigma)}$ is a polytope.
On the one hand, up to re-enumeration, $\overline{\operatorname{Trs}(\sigma)}$
can be expressed as $M\times\mathbb{R}^{n+r-s}$, where $M$ is the Minkowski
sum

$\overline{F_{(\bm{\circleddash}\cap
I_{\sigma}^{+})\Gamma}(\sigma)}+(\mathbb{R}_{\leq 0})^{s-r},$

and $F_{(\bm{\circleddash}\cap I_{\sigma}^{+})\Gamma}$ is the map
$(F_{i\Gamma})_{i\in\bm{\circleddash}\cap
I_{\sigma}^{+}}:\mathbb{R}^{n}\to\mathbb{R}^{s-r}$. Notice that the map
$F_{(\bm{\circleddash}\cap I_{\sigma}^{+})\Gamma}$ is affine on $\sigma$ as
the latter is a cell of $\Sigma_{\Gamma}$. Since poltopes are closed under
product, Minkowski sums, and linear maps, we obtain that
$\overline{\operatorname{Trs}(\sigma)}$ is a polytope. On the other hand,
recall that
$Y(\sigma)=F_{[r]\Gamma}(\sigma)\times\mathbb{R}^{s-r}\times\mathbb{R}^{n-s}$.
Similarly here, the map $F_{[r]\Gamma}$ is affine on $\sigma$, and thus
$\overline{Y(\sigma)}$ is also a polytope in $\mathbb{R}^{n}$. Therefore,
since polytopes are closed under intersections, the set
$\overline{\operatorname{Trs}(\sigma)}\cap\overline{Y(\sigma)}$ is a polytope.
∎

Let $\Xi(\Gamma)$ be the subset of all cells in $\Xi$ for which
$\delta(\xi)\subset\Sigma\Gamma$.

######  Lemma 5.7.

There is a unique bijection $\varphi:\Xi(\Gamma)\to\Sigma_{\Gamma}$, sending a
cell $\xi$ onto $\varphi(\xi)$ so that $\xi\subset\varphi(\xi)$.

###### Proof.

The function $\varphi$ is well-defined since the cells $\Sigma_{\gamma}$ are
disjoint. Furthermore, as $\Xi$ is a refinement of the cell-decomposition
$\Sigma_{\Gamma}$, we get $\varphi$ is unique and injective. To show
surjectivity, recall that $\Xi$ is in bijection with the set of polytopes in
the dual subdivision $\tau$ of $\Sigma\Delta^{0}$ (Proposition 3.6). This
induces a bijection between $\Xi(\Gamma)$ and the polyhedra in the dual
subdivision of $\Sigma\Gamma$ induced by $\tau$. The latter is dual to
$\Sigma_{\Gamma}$. Hence, we get $|\Xi(\Gamma)|=|\Sigma_{\Gamma}|$, and
surjectivity of $\varphi$ follows. ∎

######  Definition 5.8.

Let $\Gamma$ be a tropical polynomial map $G:\mathbb{R}^{n}\to\mathbb{R}^{n}$.
The _virtual image_ of $x\in\mathbb{R}^{n}$ is the subset of all
$y\in\mathbb{R}^{n}$ for which ${\mathcal{T}}_{y}(G)\cap x\neq\emptyset$. We
will denote it by ${\mathcal{V}}_{\sigma}(G)$. ∎

######  Remark 5.9.

As, in general, it holds $G^{-1}(y)\neq{\mathcal{T}}_{y}(G)$ (c.f. Remark
LABEL:rem:preimages_tropical), we get ${\mathcal{V}}_{x}(G)\neq G(x)$.

######  Lemma 5.10.

If a face $\Gamma\prec\Delta^{0}$ is not pre-origin, then for any
$\sigma\in\Sigma_{\Gamma}$, it holds $\operatorname{Trs}(\sigma)\cap
Y_{\sigma}=\emptyset$.

###### Proof.

Recall that $F$ is the tropical map corresponding to a face-generic map
$f:\mathbb{K}^{n}\to\bm{T}$. Let $\Gamma\prec\Delta^{0}$, and let
$y\in\operatorname{Trs}(\sigma)\cap Y_{\sigma}$ for some
$\sigma\in\Sigma_{\Gamma}$. We first prove that
$\operatorname{Trs}(\sigma)\cap Y_{\sigma}\subset{\mathcal{R}}_{\Gamma}(f)$.

From the definition of $\operatorname{Trs}(\sigma)$ (see Notation 5.3), and
Theorem 3.9 for transversal tropical intersections, it holds that for any
point $w\in\bm{T}$ satisfying $\operatorname{Val}(w)=y$, there exists a point
$z\in\bm{T}$ satisfying $\operatorname{Val}(z)\in\sigma$ and is a solution to

(5.5)
$f_{i_{1}\Gamma}=\cdots=f_{i_{r}\Gamma}=f_{j_{1}\Gamma}-w_{j_{1}}=\cdots=f_{j_{s}\Gamma}-w_{j_{s}}=0,$

where $\\{i_{1},\ldots,i_{r}\\}=[n]\setminus\bm{\circleddash}$ and
$\\{j_{1},\ldots,j_{s}\\}=I_{\sigma}^{+}\cap\bm{\circleddash}$. Moreover,
since $y\in Y(\sigma)$, for any $i\in\bm{\circleddash}\setminus
I_{\sigma}^{+}$, we have $x\in F_{i\Gamma}^{-1}(y_{i})$. The condition
$\dim\delta(\sigma_{i})=0$ for $i\in\bm{\circleddash}\setminus I_{\sigma}^{+}$
implies the equality

(5.6)
$F_{i\Gamma}(\operatorname{Val}(z))=\operatorname{val}(f_{i\Gamma}(z)),\leavevmode\nobreak\
\text{for }i\in\bm{\circleddash}\setminus I_{\sigma}^{+}.$

Then, (5.5) and (5.6) show that
$y=\operatorname{Val}(w)\in\operatorname{Val}({\mathcal{R}}_{\Gamma}(f))$.

To conclude, recall that if $\Gamma$ is not pre-origin, then for any
$w\in\bm{T}$, it holds $(f-w)_{\Gamma}=f_{\Gamma}$, and thus
${\mathcal{R}}_{\Gamma}(f)=\emptyset$ whenever
$\mathbb{V}_{\\!\mathbb{K}}(f_{\Gamma})=\emptyset$ (see Definition 2.8).
Therefore, Definition 2.6 (b) (applied to $I=[n]$) shows that
${\mathcal{R}}_{\Gamma}(f)=\emptyset$ if $\Gamma$ is not pre-origin. ∎

######  Lemma 5.11.

Let $F$ be a tropical map $\mathbb{R}^{n}\to\mathbb{R}^{n}$ corresponding to a
face-generic polynomial map $f:\mathbb{K}^{n}\to\bm{T}$ with
$f(\underline{0})=\underline{0}$, and let $\Delta^{0}:=\Delta^{0}(f)$. Then,
for any $\Gamma\prec\Delta^{0}$, and any $\sigma\in\Sigma_{\Gamma}$, it holds

${\mathcal{V}}_{\sigma}(F_{\Gamma})=\overline{\operatorname{Trs}(\sigma)}\cap\overline{Y(\sigma)}.$

###### Proof.

If $\sigma$ is contributing, then the equality follows from the definitions
and Lemma 5.4. Otherwise, for any $x\in\sigma$, and any
$i\in[n]\setminus\bm{\circleddash}$, it holds
${\mathcal{T}}_{y_{i}}(F_{i\Gamma})\cap\sigma=\emptyset$. This shows that each
of ${\mathcal{V}}_{\sigma}(F_{\Gamma})$ and $\operatorname{Trs}(\sigma)$ is
empty whenever $\sigma$ is not contributing. ∎

###### Proof of Theorem 1.3.

Let $\Gamma$ be a face of $\Delta^{0}$, and let $\xi\in\Xi$ be a cell, so that
$\delta(\xi)\subset\Sigma\Gamma$. Let ${\mathcal{S}}(\xi)$ denote the set of
all points $y\in\mathbb{R}^{n}$, where ${\mathcal{T}}_{y}(F)$ has a dicritical
half-line intersecting $\xi$. Thanks to Lemma 4.5, one can easily check that
${\mathcal{S}}(\xi)={\mathcal{V}}_{\xi}(F)$ if $\Gamma$ is dicritical and
${\mathcal{S}}(\xi)=\emptyset$ otherwise. Similarly, and thanks to Lemma 5.7,
we furthermore have
${\mathcal{V}}_{\xi}(F)={\mathcal{V}}_{\varphi(\xi)}(F_{\Gamma})$. Therefore,
Lemma 5.11, together with Lemma 5.10 show that
${\mathcal{S}}(\xi)=\overline{\operatorname{Trs}(\sigma)\cap Y_{\sigma}}$ if
$\Gamma$ is a dicritical pre-origin face of $\Delta^{0}$, and
${\mathcal{S}}(\xi)$ is empty otherwise. Finally, Lemma 5.6 yields the proof.
∎

### Contact

Boulos El Hilany
Institut für Analysis und Algebra, TU Braunschweig,
Universitätsplatz 2, 38106 Braunschweig,
<EMAIL_ADDRESS>
boulos-elhilany.com

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