# Generic root counts and flatness
in tropical geometry

Paul Alexander Helminck  Department of Mathematics, Durham University, United
Kingdom<EMAIL_ADDRESS>https://paulhelminck.wordpress.com and
Yue Ren  Department of Mathematics, Durham University, United Kingdom.
<EMAIL_ADDRESS>https://yueren.de

###### Abstract.

We use tropical and non-archimedean geometry to study generic root counts of
families of polynomial equations. These families are given as morphisms of
schemes $X\to{Y}$ that factor through a closed embedding into a relative torus
over a parameter space $Y$. We prove a generalization of Bernstein’s theorem
for these morphisms, showing that the root count of a single well-behaved
tropical fiber spreads to an open dense subset of $Y$. We use this to express
the generic root count of a wide class of square systems in terms of the
matroidal degree of an explicit variety. This in particular gives tropical
formulas for the birational intersection indices and volumes of Newton-
Okounkov bodies from [KK10, KK12], and the generic root counts of the steady-
state equations of chemical reaction networks. An important role in these
theorems is played by the notion of tropical flatness, which allows us to
infer generic properties of $X\to{Y}$ from a single tropical fiber. We show
that the tropical analogue of the generic flatness theorem holds, in the sense
that $X\to{{Y}}$ is tropically flat over an open dense subset of the Berkovich
analytification of $Y$.

###### Key words and phrases:

tropical geometry, non-archimedean geometry, root counts, polynomial systems.

###### 2020 Mathematics Subject Classification:

14T90,14M25,13P15

##  1\. Introduction

In this paper we study the problem of determining the generic number of
solutions of parametrized polynomial systems using tropical and non-
archimedean geometry. These systems arise naturally in many areas, describing
for instance the $27$ lines on a smooth cubic surface, the dynamics of the Wnt
signaling pathway [GHRS16], the motion of the Gough-Steward platform [SW05],
and the central configurations in Newton’s $N$-body problem [HM06]. One of the
leading techniques in solving these systems numerically is homotopy
continuation, which is capable of computing a single root in average
polynomial time [Lai17]. If however all roots are required, then finding a
good upper bound on the total number of roots poses a major problem.

The archetypal example of a generic root count of a family of polynomial
systems is _Bézout’s theorem_ , which states that $n$ polynomials
$f_{1},\dots,f_{n}$ in $n$ variables of degrees $d_{1},\dots,d_{n}$
generically have $\prod_{i=1}^{n}d_{i}$ solutions. Here, the family consists
of all $n$-tuples $(f_{1},...,f_{n})$ of polynomials of degrees
$(d_{1},...,d_{n})$ and being generic means that the number of solutions is
attained on a Zariski open dense subset of the family. Another example is the
_Bernstein-Kushnirenko theorem_ [Kou76, Ber76], where the family consists of
systems with fixed monomial supports and the generic root count is the mixed
volume of their Newton polytopes. A third prominent example can be found in
the works of Kaveh and Khovanskii [KK10, KK12], who consider family of systems
with fixed polynomial supports. Their generic root count is the birational
intersection index, and in special cases it is the volume of the associated
_Newton-Okounkov body_.

The goal of this paper is to develop a new approach for determining generic
root counts using tropical geometry. We establish the notion of _tropical
flatness_ and use it to extend the Bernstein-Kushnirenko theorem to families
of properly intersecting schemes. Herein the mixed volume is naturally
replaced by the tropical intersection product, and the initial system
criterion [Ber76, Theorem B] is rephrased in terms of tropical intersections.
This allows us to introduce a method for constructing good re-embeddings of
square polynomial systems, and we show how it can be applied to derive a
formula for the number of steady states of chemical reaction networks [Dic16],
and the birational intersection indices and volumes of the aforementioned
Newton-Okounkov bodies from [KK10, KK12].

###  1.1. Outline and main results of this paper

In Section 2 we fix our notation and assumptions for the rest of the paper.
The polynomial families in this paper are given by morphisms of schemes
$X\to{Y}$, where $Y$ is the parameter space and $X$ is the base space. The
parameter space $Y=\mathrm{Spec}(A)$ will be integral of finite type over a
non-archimedean field $K$, and given a generically finite morphism $X\to Y$,
we define its _generic root count_ $\ell_{X/Y}$ to be the length of the fiber
of $X\to{Y}$ over the generic point $\eta$ of $Y$, see Section 2.2.

From Section 3 onwards,
$T_{Y}=\operatorname{Spec}(A[x_{1}^{\pm},\dots,x_{n}^{\pm}])$ will denote a
relative $n$-dimensional torus over $Y$, and we will assume that $X\to Y$
factors through a closed embedding $X\to T_{Y}$. This naturally gives rise to
the notion of a fiberwise tropicalization $\operatorname{trop}(X_{P})$, which
is the tropicalization of the fiber of $X$ over a point $P$ in the Berkovich
analytification $Y^{\mathrm{an}}$ of $Y$. As fiberwise tropicalizations need
not vary continuously, we introduce the notion of tropical flatness. We say
$X\to{T_{Y}}$ is tropically flat over $P\in{Y^{\mathrm{an}}}$ if, locally and
after a valued field extension, there exist polynomials $f_{i}$ such that:

  1. (1)

(Non-degeneracy) The coefficients of $f_{i}$ do not vanish around $P$.

  2. (2)

(Tropical basis) The $f_{i}$ form a tropical basis of $X$ around $P$.

We then prove the tropical analogue of the generic flatness theorem:

######  Theorem 3.22.

Let $Y=\mathrm{Spec}(A)$ be an integral scheme of finite type over a non-
archimedean field $K$. Then the tropically flat locus of a closed embedding of
$Y$-schemes $X\to{T_{Y}}$ contains an open dense subset of $Y^{\mathrm{an}}$.

Theorem 3.22 in particular implies that if $K$ is non-trivially valued and
algebraically closed, then $Y({K})$ contains a dense open subset over which
$X\to{T_{Y}}$ is tropically flat. To prove this theorem, we study tropical
bases of $X$ over points in $Y^{\mathrm{an}}$ that give valuations of the
function field $K(Y)$. For small enough neighborhoods of these points, we have
that tropical bases continue to give tropical bases. The density in Theorem
3.22 then follows from an approximation argument.

If $X\to{T_{Y}}$ is tropically flat over a point, then many properties of the
corresponding fiber of $X\to Y$ spread to an open dense subset of the
parameter space $Y$, thus becoming generic properties. In Section 4, we
investigate one instance of this phenomenon, and show that the tropical
intersection number over a tropically flat point is the generic root count:

######  Theorem 4.5.

Let $Y=\mathrm{Spec}(A)$ be an integral scheme that is of finite type over a
non-archimedean field $K$ and let $p\colon X\to{Y}$ be an affine morphism of
finite type. Suppose that the following hold:

  1. (1)

The morphism $p$ factors through a closed embedding $X\to{T_{Y}}$ of
$Y$-schemes, where $T_{Y}$ is an $n$-dimensional torus over $Y$.

  2. (2)

The closed subscheme $X\to{T_{Y}}$ is an intersection of closed subschemes
$X_{i}\to{T_{Y}}$, $i=1,\dots,r$, that are generically Cohen-Macaulay and pure
of relative dimension $k_{i}$ over $Y$ with

$\sum_{i=1}^{r}(n-k_{i})=n.$

  3. (3)

There is a $P\in{Y^{\mathrm{an}}}$ over which the $X_{i}\to T_{Y}$ are
tropically flat and the intersection $\bigcap_{i=1}^{k}\mathrm{trop}(X_{i,P})$
is bounded.

Then $X\to{Y}$ is generically finite and the generic root count $\ell_{X/Y}$
is the tropical intersection number $\prod_{i=1}^{r}\mathrm{trop}(X_{i,P})$.

Theorem 4.5 can be regarded as a generalization of Bernstein’s theorem to
families of properly intersecting schemes. Here the mixed volume in [Ber76,
Theorem A] is replaced by a tropical intersection number and the initial
system criterion of [Ber76, Theorem B] is phrased in terms of tropical
intersections. The concept of tropical flatness is crucial in Theorem 4.5, as
otherwise the tropical intersection numbers would be able to vary from fiber
to fiber.

To facilitate the use of Theorem 4.5 we introduce the notions of _torus-
equivariance_ and _algebraic independence_. We say that an embedding
$X\to{T_{Y}}$ is torus-equivariant if there is an action
$T_{Y}\times_{Y}{Y}\to{Y}$ such that $\lambda\cdot{X_{P}}=X_{\lambda\cdot{}P}$
for every point $(\lambda,P)$ of $T_{Y}\times_{Y}{Y}$, see Section 4.2 for
more details. We furthermore say that the families $X_{i}\to{Y}$ from Theorem
4.5 are algebraically independent if there exist $X^{\prime}_{i}\to Y_{i}$
with $\prod_{K}{Y_{i}}=Y$ such that $X^{\prime}_{i}\times_{Y_{i}}Y=X_{i}$. In
Proposition 4.19, we show that torus-equivariant and algebraically independent
families satisfy the conditions of Theorem 4.5. In a sense, if a family is
torus-equivariant then we can move the tropical fibers to obtain a stable
intersection. This concept furthermore allows us to eliminate intersections in
loci where the fibers $X_{i,P}$ are not Cohen-Macaulay.

From Section 5 on, we focus on square systems. The moduli of these square
systems are captured by a finite graph which we call the degeneracy graph of
$X\to{Y}$, see Section 5.1. In Section 5.2, we propose a re-embedding of $X\to
Y$ that splits $X$ into a linear part $\widehat{X}_{\mathrm{lin}}\to Y$ and a
nonlinear part $\widehat{X}_{\mathrm{nlin}}\to Y$. We say that $X\to{Y}$ is
_tropically modifiable_ if $\widehat{X}_{\mathrm{lin}}$ is torus-equivariant,
and $\widehat{X}_{\mathrm{lin}}\to{Y}$ and $\widehat{X}_{\mathrm{nlin}}\to{Y}$
are algebraically independent and of complementary codimension. For tropically
modifiable $X\to{Y}$, we obtain the following combinatorial formulas for their
generic root counts.

######  Theorem 5.11.

Let $X\to Y$ be tropically modifiable. Then there exists a dense open subset
$U\subset{T_{Y}}$ such that $X\cap{U}\to{Y}$ is generically finite. The
generic root count is given by

$\ell_{X\cap
U/Y}=\operatorname{trop}(\widehat{X}_{{\mathrm{lin}},P})\cdot\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$
(1)

for explicit schemes $\widehat{X}_{\mathrm{lin}}\to Y$ and
$\widehat{X}_{{\mathrm{nlin}}}\to Y$ and $P$ in an open dense subset of
$Y^{\mathrm{an}}$.

The open subset $U$ in Theorem 5.11 can moreover be chosen to be the maximal
one with $\ell_{X\cap
U/Y}=\operatorname{trop}(\widehat{X}_{{\mathrm{lin}},P})\cdot\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$.
We call this open subset the maximal compatible subset of the modification. It
arises from intrinsic obstructions similar to those in the works of Kaveh and
Khovanskii [KK10, KK12]. We describe the maximal compatible subset $U$ and
provide sufficient criteria for $X\cap U=X$.

As the right hand side of Equation (1) depends only on the matroid of
$\widehat{X}_{{\mathrm{lin}},P}$, we refer to it as a _matroidal degree_ of
$\widehat{X}_{{\mathrm{nlin}},P}$. If
$\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$ is a tropical complete
intersection, then the generic root count $\ell_{X\cap U/Y}$ is expressible as
a weighted sum of mixed volumes using cotransversal matroid decompositions.
More generally, we can use a result by Jensen and Yu [JY16, Corollary 5.2],
based on a result by McMullen [McM89, Lemma 20], to write
$\widehat{X}_{\mathrm{nlin}}$ and $\widehat{X}_{\mathrm{lin}}$ as sums of
powers of tropical hypersurfaces. This yields a formula for
$\ell_{X\cap{U}/Y}$ in terms of sums of mixed volumes.

In Section 6, we consider two classes of linearly parametrized square
polynomial systems: systems with _vertical dependencies_ inspired by steady
state equations of chemical reaction networks [Dic16], and systems with
_horizontal dependencies_ inspired by polynomials with fixed polynomial
support [KK10, KK12]. We show that the method from Section 5 is applicable to
these systems. This in particular gives a formula for the birational
intersection indices $[L_{1},...,L_{n}]$ from [KK10] for $L_{i}$ subspaces of
the function field of a torus in terms of tropical intersection numbers. Using
[KK12, Theorem 4.9], we then also obtain a formula for the volumes of Newton-
Okounkov bodies in terms of tropical intersection numbers.

These special cases provide simplifications of the tropical intersection
numbers from Section 5 in two ways. Namely, for systems with vertical
dependencies, we find that the
$\mathrm{trop}(\widehat{X}_{{\mathrm{nlin}},P})$ are tropical complete
intersections, whereas for systems with horzontal dependencies the
$\mathrm{trop}(\widehat{X}_{{\mathrm{lin}},P})$ are tropical complete
intersections. We demonstrate our technique in two examples: the stationary
equations of the Kuramoto model [CMMN19] and Duffing oscillators [BMMT22].

###  1.2. Connections to existing literature

A landmark result in the subject of counting roots of polynomial equations is
Bernstein’s theorem [Ber76, Kou76], which establishes the mixed volume as an
upper bound for the number of roots of a square polynomial system. An
algorithmic milestone in this area was the introduction of the technique of
polyhedral homotopy continuation [HS95]. More recently, Kaveh and Khovanskii
introduced the notion of Newton-Okounkov bodies [KK12], which in particular
gives the root count for specialized systems associated to a single linear
subspace. Our work is similar in purpose, but different in approach. To find
generic root counts, we use techniques from non-archimedean and tropical
geometry. Our approach allows us to treat overdetermined systems that come
with a suitable decomposition, as well as systems with more general
dependencies between coefficients. In particular, in Section 6.1 we discuss
systems which arise as steady state equations of chemical reaction networks,
and in Section 6.2 we discuss systems as considered by [KK12]. We show that
our techniques can be applied to both types of system to give generic root
counts as tropical intersection products.

Our work builds on two fundamental papers [OP13] and [OR13] by Osserman, Payne
and Rabinoff, which relate tropical intersection theory to algebraic
intersection theory. Our contribution with respect to these papers is that we
are able to identify a precise context in which these results can be
propagated to open dense subsets of arbitrary integral parameter spaces. The
key ingredient is tropical flatness, which gives very precise control over the
local variation of fiberwise tropicalizations. We furthermore obtain a generic
tropical flatness theorem, showing that the condition imposed here is
satisfied for most points in the parameter space.

### Acknowledgements

The authors would like to thank Mateusz Michałek and Tianran Chen for
providing valuable feedback and examples. The authors would like to thank Alex
Fink and Felipe Rincón for helpful discussions on tropical linear spaces. The
authors would further like to thank the participants of the Durham tropical
geometry seminar, Jeffrey Giansiracusa, James Maxwell, and Stefano Mereta, for
many productive discussions. Both authors are supported by UK Research and
Innovation under the Future Leaders Fellowship programme (MR/S034463/2).

##  2\. Preliminaries

In this section, we briefly review some basic concepts and fix our notation,
which is close to the one used in [Gub13]. We also define generic root counts,
decompositions and universal families of polynomial systems.

###  2.1. Berkovich spaces

All schemes in this paper are Noetherian unless mentioned otherwise. The
residue field of a point $P\in{X}$ is denoted by $k(P)$. For a commutative
unitary ring $A$, we denote a polynomial ring in $n$ variables over $A$ by
$A[x_{1},...,x_{n}]$, and a Laurent polynomial ring in $n$ variables over $A$
by $A[x_{1}^{\pm},...,x_{n}^{\pm}]$. Let $K$ be a non-archimedean field with
absolute value $|\cdot{}|$. We allow the absolute value to be trivial. The
valuation associated to $(K,|\cdot{}|)$ is $x\mapsto-\mathrm{log}|x|$, where
$\mathrm{log}(\cdot{})$ is the natural logarithm.

Let $X$ be a scheme that is locally of finite type over a non-archimedean
field $K$. The Berkovich analytification $X^{\mathrm{an}}$ of $X$ is the set
of pairs $(P,k(P)\to\mathbb{R}_{\geq{0}})$, where $P\in{X}$ and
$k(P)\to\mathbb{R}_{\geq{0}}$ is an absolute value extending the absolute
value on $K$. Equivalently, we can view points of $X^{\mathrm{an}}$ as
equivalence classes of $L$-valued points of $X$, where $L$ is a valued field
extension of $K$. If $p\colon X\to{Y}$ is a morphism of schemes, then we
denote its analytification by $p^{\mathrm{an}}\colon
X^{\mathrm{an}}\to{Y^{\mathrm{an}}}$. There is a natural forgetful map
$\pi\colon X^{\mathrm{an}}\to{X}$ that sends
$Q=(P,k(P)\to\mathbb{R}_{\geq{0}})$ to $P$. We say that $Q$ is rational if the
induced map $K\to{k(\pi(Q))}$ is an isomorphism. If $X=\mathrm{Spec}(B)$, then
$X^{\mathrm{an}}$ coincides with the usual definition of multiplicative
seminorms $B\to{\mathbb{R}_{\geq{0}}}$ that restrict to the given norm on $K$.
If $f\in{B}$ and $P\in X^{\mathrm{an}}$, then we write $f(P)$ for the image of
$f$ under $A\to{k(\pi(P))}$. Similarly, we write
$|f(P)|\in\mathbb{R}_{\geq{0}}$ for the image of $f$ under
$B\to{k(\pi(P))}\to\mathbb{R}_{\geq{0}}$. The topology on $X^{\mathrm{an}}$ is
generated by

$\mathbb{B}(f,r_{1},r_{2})=\\{P:r_{1}<|f(P)|<r_{2}\\}$

for $r_{1}\geq{0}$ and $r_{2}>0$. If $X$ is given the Zariski topology and
$X^{\mathrm{an}}$ the topology above, then $\pi\colon X^{\mathrm{an}}\to{X}$
is continuous. For more background information on Berkovich spaces and
tropicalizations, we recommend [BPR16], [Gub13] and [Tem15].

###### Remark 2.1

Let $X=\mathrm{Spec}(B)$ be of finite type over a non-archimedean field $K$
and let $P\in{X^{\mathrm{an}}}$ with field of definition $L=k(\pi(P))$. The
point $P$ induces a canonical point $P_{L}\in{X^{\mathrm{an}}_{L}}$ by the
following procedure. We represent $P$ as a ring homomorphism
$B=K[x_{1},\dots,x_{n}]/(g_{1},\dots,g_{m})\to{L}$. Then $P_{L}$ is given by
the natural ring homomorphism
$B_{L}=L[x_{1},\dots,x_{n}]/(g_{1},\dots,g_{m})\to{L}$. If $K\to{M}$ is a
valued field extension different from $K\to{L}$, then $P$ in general does not
induce a canonical point of $X^{\mathrm{an}}_{M}$. In this case we simply
choose a point $P_{M}$ lying over $P$. This exists by [Gub13, Lemma 2.3]
applied to
$X^{\mathrm{an}}_{\hat{M}}\to{X^{\mathrm{an}}_{M}}\to{X^{\mathrm{an}}}$. Here
$\hat{M}$ is the completion of $M$ with respect to the given valuation. If $P$
is rational, then there is only one point $P_{M}$ lying over $P$ for any
valued field extension $K\to{M}$.

###  2.2. Generic properties

In this section, we define generic root counts, generically Cohen-Macaulay
morphisms and generically pure morphisms. Determining generic root counts will
be main focus of the paper; the other two are important conditions required to
find generic root counts.

Let $p\colon X\to Y$ be locally of finite type with $Y$ integral. If $p$ is
generically finite, then there are finitely many points lying over the generic
point $\eta$ of $Y$. By [Stacks22, Lemma 02NG], we then have that the generic
fiber $X_{\eta}$ is a finite $k(\eta)$-scheme, where $k(\eta)$ is the residue
field of $\eta$. This is associated to a finite $k(\eta)$-algebra, whose
$k(\eta)$-dimension we call the length or degree of $X_{\eta}$.

###### Definition 2.2

The _generic root count_ of $p\colon X\to{Y}$ is the length of the
$k(\eta)$-scheme $X_{\eta}$. We denote it by $\ell_{X/Y}$. If $X\to{Y}$ is not
generically finite, then we set $\ell_{X/Y}=\infty$.

###### Example 2.3

Suppose $Y=\mathrm{Spec}(A)$ is integral and $p:X\to{Y}$ is affine with
$X=\mathrm{Spec}(B)$. Then

$\ell_{X/Y}=\mathrm{dim}_{K(A)}(B\otimes_{A}K(A)),$

where $K(A)$ is the function field of $A$.

###### Remark 2.4

By our assumptions, $X$ is Noetherian and thus $p:X\to{Y}$ is quasi-separated
and quasi-compact. If $p$ is generically finite, then there is an open affine
$U\subset{Y}$ such that $p^{-1}(U)\to{U}$ is finite [Stacks22, Lemma 02NW].
Using Grothendieck’s generic freeness theorem, we may shrink $U$ and assume
that $p_{U}:p^{-1}(U)\to{U}$ is finite and free. The rank of this morphism is
then exactly $\ell_{X/{Y}}$. For every $y\in{U}$, we then have that the length
of the fiber of $p_{U}$ over $y$ is $\ell_{X/Y}$.

We say that a morphism of schemes $p\colon X\to{Y}$ is generically Cohen-
Macaulay if there is an open dense $U\subset{Y}$ such that $p^{-1}(U)\to{U}$
is Cohen-Macaulay. We can rephrase this in terms of generic fibers as follows.

###### Lemma 2.5

Let $Y$ be integral and let $p\colon X\to{Y}$ a morphism of finite type.
Suppose that the generic fiber $X_{\eta}$ is Cohen-Macaulay. Then there is an
open dense subset $U\subset{Y}$ such that $p^{-1}(U)\to{U}$ is a Cohen-
Macaulay morphism.

###### Proof.

We can assume by generic flatness that $p$ is flat. Consider the open subset
$W$ from [Stacks22, Lemma 045U]. Its complement $Z=X\backslash{W}$ is closed
and the fiber of $Z\to{Y}$ over the generic point is empty by assumption. This
gives an open set $U_{\eta}$ containing $\eta$ for which
$p^{-1}(U_{\eta})\cap{Z}=\emptyset$ by [Stacks22, Lemma 02NE]. The induced
morphism $p^{-1}(U_{\eta})\to{U_{\eta}}$ is automatically Cohen-Macaulay. ∎

###### Definition 2.6

Let $p\colon X\to{Y}$ be a morphism of finite type. We say that $p$ is
_generically pure_ if there exists an open dense subset $U\subset{Y}$ such
that the irreducible components of the fibers of $f$ over $U$ are all of
constant dimension $k$.

###### Lemma 2.7

Let $p\colon X\rightarrow Y$ be a morphism of finite type. Suppose that the
fiber over some $y\in{Y}$ is pure of dimension $k$. Then there is an open
dense subset $V\subset\overline{\\{y\\}}=Z$ such that the base change
$p_{Z}\colon p^{-1}(Z)\to{Z}$ has pure fibers of dimension $k$ over $V$.

###### Proof.

We follow the proof of [Stacks22, Lemma 055A]. We first reduce to $Y=Z$
integral and find a surjective finite étale map $\pi:Y^{\prime}\to{V}$ for an
open dense $V\subset{Y}$ such that the generic fiber $X^{\prime}_{y}$ consists
of geometrically irreducible components $X^{\prime}_{i,y}$ (there is only one
point lying over $y$ by construction, so we denote it by $y$ again). These are
necessarily pure of dimension $k$ again, see [Stacks22, Lemma 04KX] and
[Stacks22, Section 07NB] for instance. Let $X^{\prime}_{i}$ be the closure of
$X^{\prime}_{i,y}$ in $X^{\prime}$. The proof of [Stacks22, Lemma 055A] then
shows that we can find an open $V^{\prime}\subset{Y^{\prime}}$ such that the
fibers of the $X^{\prime}_{i}$ are geometrically irreducible and
$X^{\prime}=\bigcup_{i}X^{\prime}_{i}$ over $V^{\prime}$. We can furthermore
assume by shrinking $V^{\prime}$ that all the fibers of the $X^{\prime}_{i}$
are of dimension $k$.

We now have a morphism $\pi\colon V^{\prime}\to\pi(V^{\prime})$ such that the
base change $p_{\pi}$ of $p$ is relatively pure of dimension $k$. Consider the
proof of [Stacks22, Lemma 0556], giving a bijection between the geometrically
irreducible components of $p$ over $\pi(V)$ and $p^{\prime}$ over $V$. Under
this bijection the Krull dimensions of the irreducible components are not
changed, since they are obtained by base change over a field extension. We
thus obtain the desired result. ∎

###### Corollary 2.8

If the generic fiber $X_{\eta}$ is pure of dimension $k$, then $f$ is
generically pure.

###### Proof.

This follows from Lemma 2.7. ∎

We note here that generically Cohen-Macaulay morphisms are almost generically
pure. Namely, if $p\colon X\to{Y}$ is Cohen-Macaulay, then there exist open
and closed subschemes $X_{k}$ with $\bigsqcup_{k}X_{k}=X$ such that the
$X_{k}$ are pure of relative dimension $k$ over $Y$. In our theorems, we
assume that there is only one closed and open subscheme $X=X_{k}$, so that $p$
is pure of relative dimension $k$.

###  2.3. Closed embeddings into relative tori

In this section, we introduce notation for morphisms $X\to{Y}$ that factor
through closed embeddings into tori. These provide a natural framework for
fiberwise tropicalizations, as we will see in Section 3. We furthermore define
decompositions of these closed embeddings.

###### Notation 2.9

For the remainder of the article, we write $A$ and $B$ for algebras of finite
type over a non-archimedean field $K$, and $p\colon X\rightarrow Y$ for a
morphism of finite type between their spectra $X=\mathrm{Spec}(B)$ and
$Y=\mathrm{Spec}(A)$. We assume that $Y$ is integral, so that the generic root
count $\ell_{X/Y}$ is defined. We write
$T_{Y,n}=\mathrm{Spec}(A[x_{1}^{\pm},\dots,x_{n}^{\pm}])$ for the relative
standard $n$-dimensional torus over $Y$. If $n$ is clear from context, then we
denote it by $T_{Y}$. We assume that $p$ factors through a closed embedding of
$Y$-schemes $\phi\colon X\rightarrow T_{Y}$, and we denote the ideal of the
embedding by $I\subseteq A[x_{1}^{\pm},\dots,x_{n}^{\pm}]$. For any Zariski
open $V$ in $Y$, we write $A_{V}$ for the corresponding coordinate ring and
$I_{V}$ for the corresponding ideal in $A_{V}[x_{1}^{\pm},...,x_{n}^{\pm}]$.

Let $Q\in{Y^{\mathrm{an}}}$. We write $I_{Q}$ for the image of
$I\otimes_{A}k(\pi(Q))$ in $k(\pi(Q))[x_{1}^{\pm},...,x_{n}^{\pm}]$. The
corresponding scheme gives the fiber of $V(I)$ over $\pi(Q)$. For
$f\in{A[x_{1}^{\pm},...,x_{n}^{\pm}]}$ and $\mathcal{G}\subset
A[x_{1}^{\pm},...,x_{n}^{\pm}]$, we similarly write $f_{Q}$ and
$\mathcal{G}_{Q}$ for the images of $f$ and $\mathcal{G}$ under
$A[x_{1}^{\pm},...,x_{n}^{\pm}]\to k(\pi(Q))[x_{1}^{\pm},...,x_{n}^{\pm}]$.

###### Definition 2.10

A _decomposition_ of the closed embedding $\phi$ is a finite set of ideals
$I_{1},\dots,I_{k}\subseteq A[x_{1}^{\pm},...,x_{n}^{\pm}]$ such that
$I=\sum_{i=1}^{k}I_{i}$. Writing
$X_{i}=\mathrm{Spec}(A[x_{1}^{\pm},...,x_{n}^{\pm}]/I_{i})$ for the
corresponding closed subschemes, we then have an isomorphism

$X\simeq{\prod_{T_{Y}}X_{i}},$

where the right-hand side is the fiber product over the embeddings
$\phi_{i}\colon X_{i}\to{}T_{Y}$.

We say a decomposition is _square_ if $k=n$ and $I_{i}=(f_{i})$ for $f_{i}\in
A[x_{1}^{\pm},...,x_{n}^{\pm}]$. We will assume that the $f_{i}$ are chosen so
that $V(f_{i})$ is relatively and generically pure of codimension one. We also
refer to these as _square polynomial systems_. The _mixed volume_ of the
decomposition is the normalized mixed volume of the Newton polytopes of the
$f_{1},\dots,f_{n}$. We denote it by $\mathrm{MV}(f_{1},...,f_{n})$.

###### Remark 2.11

For more general decompositions, we can view the tropical intersection number
of a general fiber as the analogue of the mixed volume, see Proposition 4.19.

###  2.4. Universal families

We now define universal families of polynomial systems with a fixed monomial
support. Square systems are studied in more detail in Sections 5 and 6.

###### Definition 2.12

Let $n,k\in\mathbb{N}$, and $[k]=\\{1,...,k\\}$. A _fixed monomial support_ is
a finite subset $S\subset\mathbb{Z}^{n}\times[k]$ such that
$S\cap(\mathbb{Z}^{n}\times\\{i\\})\neq\emptyset$ for every $i\in{S}$. We
write $S_{i}\subset\mathbb{Z}^{n}$ for the projection of
$S\cap(\mathbb{Z}^{n}\times{\\{i\\}})$ onto $\mathbb{Z}^{n}$, and we will
generally assume that $|S_{i}|>1$. The _parameter ring_ associated to a fixed
monomial support $S$ is the ring $A=K[c_{i,\alpha}|(\alpha,i)\in S]$ generated
by variables $c_{i,\alpha}$ with $(\alpha,i)\in S$.

Consider the (Laurent) polynomials

$f_{i}=\sum_{\alpha\in S_{i}}c_{i,\alpha}x^{\alpha}\in
A[x_{1}^{\pm},\dots,x_{n}^{\pm}].$

These are the universal polynomials with fixed monomial support $S$. We set
$X=\mathrm{Spec}(A[x_{1}^{\pm},\dots,x_{n}^{\pm}]/(f_{1},...,f_{k}))$ and
$Y=\mathrm{Spec}(A)$. If $k=n$, we call $X\to{Y}$ the _universal family_ of
square polynomial systems over $K$ with monomial support $S$.

We will see in Section 4 that the universal family
$\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ of square polynomial systems with a
fixed monomial support $S$ is generically finite and that its generic root
count is the mixed volume of the Newton polytopes of the $f_{i}$. Similar
statements hold for the universal families of square systems with fixed
polynomial supports, see Section 5.2.

##  3\. Fiberwise tropicalizations and generic flatness

In this section, we study fiberwise tropicalizations for morphisms $p\colon
X\to{{Y}}$ that factor through a closed embedding into a torus. We introduce
the notion of tropical flatness, as these fibers need not vary continuously.
For families that are tropically flat over a point, we show that the fiberwise
tropicalizations are locally constant after a non-archimedean base change. We
then show that closed embeddings $X\to{T_{Y}}$ are generically tropically
flat, see Section 3.2.

###  3.1. Fiberwise tropicalizations and flatness

Let $T_{K,n}$ be an $n$-dimensional torus over a non-archimedean field. Recall
that there is a natural tropicalization map
$T^{\mathrm{an}}_{K,n}\to\mathbb{R}^{n}$, see [BPR16] and [Gub13]. For a
scheme $Y=\mathrm{Spec}(A)$ of finite type over $K$, we similarly define a
_relative_ tropicalization map

$\mathrm{trop}\colon T_{Y}^{\mathrm{an}}\to\mathbb{R}^{n}$

as follows. For any $P\in{T_{Y}^{\mathrm{an}}}$ with ring homomorphism
$\psi\colon A[x_{1}^{\pm},...,x_{n}^{\pm}]\to{k(\pi(P))}$, we set

$\mathrm{trop}(P)=(-\log|\psi(x_{1})|,...,-\log|\psi(x_{n})|).$

We use this to define fiberwise tropicalizations of closed subschemes of
$T_{Y}$.

###### Definition 3.1

Let $P\in{Y^{\mathrm{an}}}$. We define the _fiberwise tropicalization_ of
$\phi\colon X\to{T_{Y}}$ over $P$ to be

$\mathrm{trop}(X_{P})\coloneqq\mathrm{trop}\big{(}\phi^{\mathrm{an}}((p^{\mathrm{an}})^{-1}(P))\big{)}.$

Given a decomposition $\phi_{i}\colon X_{i}\to{T_{Y}}$ and corresponding
$p_{i}\colon X_{i}\to Y$, the _fiberwise tropical prevariety_ over $P$ is
$\bigcap_{i}\mathrm{trop}(X_{i,P})$.

###### Remark 3.2

If we represent a point $P\in{Y^{\mathrm{an}}}$ as a morphism
$A\to{k(\pi(P))}$ together with a valuation on $k(P)=k(\pi(P))$, then the
fiberwise tropicalization of the torus over $P$ can be identified with the
tropicalization of the Berkovich spectrum of
$A[x_{1}^{\pm},...,x_{n}^{\pm}]\otimes_{A}{k(P)}=k(P)[x_{1}^{\pm},...,x_{n}^{\pm}]$.
The fiberwise tropicalization of a closed subscheme $X\to{T_{Y}}$ is then the
tropicalization of the corresponding ideal in
$k(P)[x_{1}^{\pm},...,x_{n}^{\pm}]$. In particular, we find that the fiberwise
tropicalization of a closed subscheme has the usual structure of a tropical
variety in $\mathbb{R}^{n}$.

General tropical fibers can behave quite erratically, so we impose the
following continuity conditions.

###### Definition 3.3

A $K$-rational _local tropical basis_ for $X\to{T_{Y}}$ around $P$ is an open
neighborhood $U$ of $P$, a Zariski open $V\subset{Y}$ with $\pi(U)\subset{V}$,
and generators $f_{1},\dots,f_{k}$ of $I_{V}$ satisfying:

  1. (1)

(Non-degeneracy) $c_{i,\alpha}(Q)\neq{0}$ for all $Q\in{U}$, where $f_{i}=\sum
c_{i,\alpha}x^{\alpha}$ and $c_{i,\alpha}\in{A_{V}}$.

  2. (2)

(Tropical basis) $f_{1,Q},\dots,f_{k,Q}$ form a tropical basis of $I_{Q}$ for
every $Q\in{U}$, i.e.,
$\operatorname{trop}(X_{Q})=\bigcap_{i=1}^{k}\operatorname{trop}(V(f_{i})_{Q})$.

A local tropical basis at $P$ is a valued field extension $K\to{L}$, a point
$P_{L}$ lying over $P$, and an $L$-rational local tropical basis at $P_{L}$
for $X_{L}\to{T_{Y,L}}$.

###### Lemma 3.4

Let $f_{1},\dots,f_{k}$ be a $K$-rational local tropical basis at $P$ and let
$K\to{L}$ be a valued field extension. Then $f_{1},\dots,f_{k}$ are an
$L$-rational local tropical basis at any $P_{L}$ mapping to $P$.

###### Proof.

The morphism $X_{L}^{\mathrm{an}}\to{X^{\mathrm{an}}}$ is continuous, so the
neighborhood $U$ of $P$ gives an open neighborhood $U_{L}$ of $P_{L}$. The
$f_{i}$ then still form a tropical basis on $U_{L}$ since tropicalizations are
invariant under field extensions by [MS15, Theorem 3.2.4]. ∎

###### Definition 3.5

We say a closed embedding $X\to{T_{Y}}$ is _tropically flat_ over
$P\in{Y^{\mathrm{an}}}$ if it admits a local tropical basis at $P$. The
tropically flat locus is the set of all $P\in{Y^{\mathrm{an}}}$ such that
$X\to{T_{Y}}$ is tropically flat at $P$.

###### Example 3.6

If $X=V(f)$ for some $f\in{A[x_{1}^{\pm},...,x_{n}^{\pm}]}$, then the
fiberwise tropicalizations of $f=\sum_{\alpha}c_{\alpha}x^{\alpha}$,
$c_{\alpha}\in A$, near any rational $P$ outside
$\bigcup_{\alpha}V(c_{\alpha})$ are locally constant, as we will see in Lemma
3.9. This in particular implies that $X\to{T_{Y}}$ is tropically flat outside
$\bigcup_{\alpha}V(c_{\alpha})$.

In general it is difficult to show that a given $X\to{T_{Y}}$ is not
tropically flat at a point $P$ if $P$ is outside the non-degenerate locus. We
refer the reader to Example 4.7 for an example of an embedding that is not
tropically flat over a non-degenerate $P\in{Y^{\mathrm{an}}}$.

We will see in Section 4 that if $X\to{T_{Y}}$ is tropically flat at $P$, then
many properties of the fiberwise tropicalization $\operatorname{trop}(X_{P})$
become generic properties of $p\colon X\to{Y}$. To prove this, we will need an
important fact about tropically flat embeddings, namely that they are locally
constant after a non-archimedean base change.

###### Lemma 3.7

Let $g\in{A}$ and $P\in{Y^{\mathrm{an}}}$ with $r=|g(P)|\neq{0}$. There is a
non-empty open neighborhood $U$ of $P_{L}$ in $Y^{\mathrm{an}}_{L}$ and an
element $c\in{L}$ with $|c|=r$ such that

$|g(Q)-c(Q)|<r$

for all $Q$ in $U$.

###### Proof.

Take any element $c\in{L}$ such that $g(P)=c$ and consider the open ball
$U_{L}=\mathbb{B}(g-c,r)=\\{Q\in{Y_{L}^{\mathrm{an}}}:|(g-c)(Q)|<r\\}$. For
any $P_{L}\in{Y_{L}^{\mathrm{an}}}$ mapping to $P$, we then have
$P_{L}\in{U_{L}}$ since $g(P_{L})=c$. The inequality in the lemma is then
satisfied by definition. Note that such a point $P_{L}$ exists by Remark 2.1.
∎

###### Remark 3.8

By applying the non-archimedean triangle inequality to $g$ as in Lemma 3.7, we
find that $g$ is locally constant after a base change to $L$. Without the base
change, this is false: tropicalizations will generally not be locally
constant. Take for instance a complete non-trivially valued algebraically
closed field $K$ and the affine line $Y=\mathrm{Spec}(K[x])$ over $K$. The
function $f=x$ is then not locally constant on $Y^{\mathrm{an}}$. Indeed,
$-\mathrm{log}|x|$ is linear with absolute slope $1$ on the line segment
emanating from $x=0$. It will however be locally constant after a base change.

###### Lemma 3.9

Let $P\in{Y^{\mathrm{an}}}$ be a rational point and let $f=\sum
c_{\alpha}x^{\alpha}$ be a non-constant polynomial in
$A[x_{1}^{\pm},...,x_{n}^{\pm}]$. If $c_{\alpha}(P)\neq{0}$ for all
$c_{\alpha}$, then the fiberwise tropicalizations of $V(f)$ are constant on a
non-empty open neighborhood of $P$.

###### Proof.

We apply Lemma 3.7 to every $c_{\alpha}$ to find a set of open neighborhoods
$U_{\alpha}$ of $P$. By the non-archimedean triangle inequality, we have that
$|c_{\alpha}(Q)|$ is constant on $U_{\alpha}$. We thus see that the
tropicalizations of $f$ are constant over the neighborhood
$U=\bigcap{U_{\alpha}}$ of $P$. ∎

###  3.2. Generic tropical flatness theorem

In this section, we prove that a closed embedding $X\to{T_{Y}}$ is generically
tropically flat if $Y$ is integral. The main idea is to use Gröbner bases over
valued fields to show that a tropical basis over a generic valuation
$P\in{Y^{\mathrm{an}}}$ gives a local tropical basis for an open subset of
$Y^{\mathrm{an}}$. By an approximation argument, we then show that the union
of these open subsets is dense in $Y^{\mathrm{an}}$.

###### Assumption 3.10

Throughout this section, except in Theorem 3.22, we assume that $K$ is
algebraically closed, non-trivially valued, and complete.

We start with two important results that allow us to approximate the behavior
of points in $Y^{\mathrm{an}}$ using rational points.

###### Lemma 3.11

Let $K$ be as in 3.10 and let $\Gamma^{\prime}$ denote the image of the
absolute value $|\cdot|\colon K\to\mathbb{R}_{\geq{0}}$. Let
$c_{1},\dots,c_{m}\in{A}$ and $P\in{Y^{\mathrm{an}}}$ be such that
$|c_{i}(P)|=r_{i}\in\Gamma^{\prime}$. Then there is a rational point $Q$ of
$Y$ such that $|c_{i}(Q)|=r_{i}$. If $V$ is a strictly $K$-affinoid subdomain
of $Y^{\mathrm{an}}$ and $P\in{V}$, then the same holds for a $K$-rational
point of $V$.

###### Proof.

We write $\Gamma\subset\mathbb{R}$ for the value group of $K$, so that
$\Gamma=\\{-\mathrm{log}(r):r\in\Gamma^{\prime}\setminus\\{0\\}\\}$. We extend
the $c_{i}$ to a generating set of $A$. This gives a closed embedding

$\phi\colon Y\to\mathbb{A}^{m}.$

Let $v\in(\Gamma\cup\\{\infty\\})^{m}$ be the tropicalization of
$\phi^{\mathrm{an}}(P)$. This lies in the tropicalization of $\phi(Y)$, so by
[Gub13, Proposition 3.8] we can find a $K$-rational point of $\phi(Y)$ that
tropicalizes to $v$. This also gives a $K$-rational point $Q$ of $Y$ and it
has the desired properties by construction.

If $V$ is a strictly $K$-affinoid subdomain of $Y^{\mathrm{an}}$, then [Ber90,
Proposition 2.1.15] shows that the $K$-rational points are dense in $V$. The
embedding above gives $V$ as a closed subset of $\mathbb{A}^{m,\mathrm{an}}$.
The arguments in [Gub13, Proposition 3.8] can now be repeated to obtain the
desired statement. ∎

###### Corollary 3.12

Let $K$ be as in 3.10 and let $P$, $c_{i}$, $r_{i}$, $Q$ be as in Lemma 3.11.
Suppose $r_{i}\neq{0}$. Then there is a non-empty open neighborhood $U$ of $Q$
with $|c_{i}(Q^{\prime})|=r_{i}$ for all $Q^{\prime}\in{U}$. The same holds
for a strictly $K$-affinoid subdomain of $Y^{\mathrm{an}}$.

###### Proof.

This follows from Lemma 3.7. ∎

###### Definition 3.13

A _(generic) valuation_ on $Y$ is a point $P\in Y^{\mathrm{an}}$ such that
$\pi(P)$ is the generic point of $Y$.

For points $Q\in{Y^{\mathrm{an}}}$ close to a valuation $P$ in the sense of
Corollary 3.12, we will show that the fiberwise tropicalizations
$\mathrm{trop}(V(I)_{P})$ and $\mathrm{trop}(V(I)_{Q})$ coincide. In
particular, this implies that a tropical basis of $\mathrm{trop}(V(I)_{P})$
gives a local tropical basis around $Q$. We first recall several concepts from
[MS15, Section 2.4].

###### Definition 3.14

Let $Q\in Y^{\mathrm{an}}$ and $w\in\mathbb{R}^{n}$. A finite set
$G_{Q}\subseteq I_{Q}$ is called a _Gröbner basis_ with respect to $w$, if
$G_{Q}$ generates $I_{Q}$ and $\\{\operatorname{in}_{w}(g):g\in G_{Q}\\}$
generates $\operatorname{in}_{w}(I_{Q})$. We define the _Gröbner polyhedron_
of $I_{Q}$ around $w$ to be
$C_{w}(I_{Q})=\mathrm{cl}(\\{w^{\prime}\in\mathbb{R}^{n}:\operatorname{in}_{w^{\prime}}(I_{Q})=\operatorname{in}_{w}(I_{Q})\\})$,
where $\mathrm{cl}(\cdot)$ denotes the closure with respect to the Euclidean
topology. We denote the _Gröbner complex_ of $I_{Q}$ by
$\Sigma(I_{Q})=\\{C_{w}(I_{Q}):w\in\mathbb{R}^{n}\\}$. If $I_{Q}$ is
homogeneous, then $\Sigma(I_{Q})$ is a finite polyhedral complex and
$\mathrm{trop}(V(I_{Q}))$ is the support of a subcomplex of $\Sigma(I_{Q})$.

###### Lemma 3.15

Let $K$ be as in 3.10 and let $I$ be a homogeneous ideal. Let $P\in
Y^{\mathrm{an}}$ be a valuation, and $w\in\mathbb{R}^{n}$ a weight vector.
Then there is a Zariski-open $V\subset{Y}$, a $\mathcal{G}\subseteq
A_{V}[x_{1},\dots,x_{n}]$, a rational point $Q\in V^{\mathrm{an}}$, and an
open neighborhood $U\subset{V^{\mathrm{an}}}$ of $Q$ such that

  1. (1)

$\mathcal{G}_{P}$ is a Gröbner basis of $I_{P}$ with respect to $w$,

  2. (2)

$\mathcal{G}_{Q^{\prime}}$ is a Gröbner basis of $I_{Q^{\prime}}$ with respect
to $w$ for all $Q^{\prime}\in U$,

  3. (3)

the monomial supports of $\mathcal{G}_{Q^{\prime}}$ and $\mathcal{G}_{P}$ are
equal for all $Q^{\prime}\in U$,

  4. (4)

the valuations of the coefficients of $G_{Q^{\prime}}$ and $\mathcal{G}_{P}$
coincide for all $Q^{\prime}\in U$.

###### Proof.

Fix a monomial ordering $>$ and let $\mathcal{G}_{P}$ be a Gröbner basis of
$I_{P}$ with respect to $w$ as computed by [CM19, Algorithm 2.9]. In
particular, $\mathcal{G}_{P}$ will satisfy a Buchberger-type criterion: if
$\mathrm{NF}$ and $S$ denote the normal form and $S$-polynomial from [CM19],
then $\mathrm{NF}_{\mathcal{G}_{P}}(S(g_{i},g_{j}))=0$ for every pair of
polynomials in $\mathcal{G}_{P}$. This implies that there exist $h_{i}$ such
that

$S(g_{i},g_{j})=\sum h_{k}g_{k},$

where $h_{k}g_{k}\geq{S(g_{i},g_{j})}$ in the sense of [CM19]. The polynomials
$g_{i}$ and $h_{i}$ are defined over a Zariski-open subset of $Y$. Using
Corollary 3.12, we find that there is a rational point $Q$ and an open
neighborhood $U$ of $Q$ such that $h_{i}g_{i}\geq{S(g_{i},g_{j})}$ still holds
for the order defined by $>,Q^{\prime}$ and $w$ for $Q^{\prime}\in{U}$. This
implies that the normal form of $S(g_{i},g_{j})$ with respect to this order is
zero. We then conclude using Buchberger’s criterion that
$\mathcal{G}_{Q^{\prime}}$ is a Gröbner basis with respect to $w$. ∎

###### Lemma 3.16

Let $K$ be as in 3.10 and let $I$ be a homogeneous ideal. Let
$P\in{Y^{\mathrm{an}}}$ be a valuation. Then there is an open neighborhood $U$
of a rational point $Q$ such that for all $Q^{\prime}\in{}U$, we have that

$\Sigma(I_{P})=\Sigma(I_{Q^{\prime}}).$

###### Proof.

We will show that $C_{w}(I_{P})=C_{w}(I_{Q})$ for all $w\in\mathbb{R}^{n}$.
Note that, by the proof of [MS15, Proposition 2.5.2], $C_{w}(I_{P})$ is
uniquely determined by the monomial support of a Gröbner basis with respect to
$w$ and the valuation of its coefficients. For every
$\sigma\in{\Sigma(I_{P})}$, fix a
$w_{\sigma}\in\operatorname{Relint}(\sigma)$, and let $\mathcal{G}_{\sigma}$
be a Gröbner basis of $I_{P}$ with respect to $w_{\sigma}$. By Lemma 3.15
there is an open neighborhood $U$ of a rational point $Q\in{Y^{\mathrm{an}}}$
such that every $\mathcal{G}_{\sigma}$ remains a Gröbner basis of $I_{Q}$ with
respect to $w_{\sigma}$ and the valuations of the coefficients of with respect
to $Q^{\prime}$ and $P$ coincide. Hence $\sigma=C_{w_{\sigma}}(I_{Q})$, and
thus $\Sigma(I_{P})=\Sigma(I_{Q^{\prime}})$. ∎

For the upcoming proofs, we will assume that $P$ admits an affinoid
neighborhood $V$ such that $P$ reduces to a generic point of the canonical
reduction $\tilde{V}$. We will see in Lemma 3.21 that these points form a
dense subset of $Y^{\mathrm{an}}$.

###### Lemma 3.17

Let $K$ be as in 3.10. Let $P\in{Y^{\mathrm{an}}}$ be a valuation that admits
a strictly $K$-affinoid neighborhood $V$ in $Y^{\mathrm{an}}$ such that $P$
reduces to a generic point of the canonical reduction $\tilde{V}$ of $V$ and
let $w\in\mathrm{trop}(V(I)_{P})$. Then there is an open neighborhood $U$ of a
rational point $Q$ such that $w\in\mathrm{trop}(V(I)_{Q})$.

###### Proof.

Since tropicalizations commute with homogenizations, we may assume that $I$ is
homogeneous. Consider the initial degeneration of $I_{P}$ at $w$. This scheme
over the residue field of $k(P)$ is cut out by the initial forms of a Gröbner
basis $\mathcal{G}_{P}$ of $I_{P}$ with respect to $w$. We assume without loss
of generality that $w=0$ and that the elements of $\mathcal{G}_{P}$ are
defined over $A$, so that $\mathcal{G}\subset{A[x_{1},\dots,x_{n}]}$. Consider
a strictly $K$-affinoid neighborhood $V$ of $P$ with canonical reduction
$\tilde{V}$ as in the statement of the lemma. By assumption, the intersection
of the initial degeneration at $w=0$ over $P$ with the torus is non-empty. We
interpret this in terms of models over $R$ as follows. By scaling the
generators of $\mathcal{G}_{P}$, we obtain a closed subscheme
$\mathcal{X}\to{T_{R_{P}}}$ over the valuation ring $R_{P}$ of $P$ such that
the toric initial degeneration of $I_{P}$ at $0$ is the special fiber of
$\mathcal{X}$, see [Gub13, Definition 5.1]. Consider the coefficients $c_{i}$
of the $\mathcal{G}_{P}$ and the polynomials used in Lemma 3.15. Note that for
every polynomial in $\mathcal{G}_{P}$, we have that the coefficients have
absolute value $\leq{1}$ with respect to $P$ by the scaling process mentioned
above. We denote their absolute values at $P$ by $r_{i}$. We then define the
Laurent domain

$W=\\{Q\in{V}:|c_{i}(Q)|=r_{i}\\},$

which contains $P$ by construction. We write $\mathcal{B}$ for the affinoid
algebra corresponding to $W$ and $\mathcal{B}^{0}$ for the ring of power-
bounded elements in $\mathcal{B}$. Note that the polynomials in $\mathcal{G}$
are defined over $\mathcal{B}^{0}$, so $\mathcal{X}$ can be defined over
$\mathrm{Spec}(\mathcal{B}^{0})$. For every $Q\in{W}$, we obtain a canonical
$R_{Q}$-valued point of $\mathrm{Spec}(\mathcal{B}^{0})$. By construction,
since $\mathcal{G}$ forms a Gröbner basis over $W$, the fiber of $\mathcal{X}$
over the closed point of $\mathrm{Spec}(R_{Q})$ is the toric initial
degeneration of $I_{Q}$ at $w$. Let $\tilde{P}$ denote the point of
$\mathrm{Spec}(B^{0})\times_{R}k$ induced by $P$. Since the fiber over
$\tilde{P}$ of $\mathcal{X}$ is non-empty, there is a Zariski-open
neighborhood of $\mathrm{Spec}(B^{0})\times_{R}k$ over which the fibers of
$\mathcal{X}$ are non-empty. We consider a basic open $D(\tilde{h})$ contained
in this Zariski-open neighborhood and we lift $\tilde{h}$ to $W$ to obtain
$h$. By restricting to an open neighborhood of $Y$, we can assume that
$h\in{A}$. Note that if $Q$ reduces to a point in $D(\tilde{h})$, then the
intersection of the initial degeneration of $I$ over $Q$ with the torus will
be non-empty. We now apply Corollary 3.12 to $W$ and $h$ to find a rational
$Q$ and an open neighborhood over which $w\in\mathrm{trop}(V(I)_{Q})$. ∎

###### Lemma 3.18

Let $K$ be as in 3.10. Let $P\in{Y^{\mathrm{an}}}$ be a valuation that admits
a strictly $K$-affinoid neighborhood $V$ such that $P$ reduces to a generic
point of the canonical reduction $\tilde{V}$ of $V$. Then there is an open
neighborhood $U$ of a rational point $Q\in{V}$ such that

$\mathrm{trop}(V(I)_{P})=\mathrm{trop}(V(I)_{Q}).$

###### Proof.

As in the proof of the previous lemma, we may assume that $I$ is homogeneous.
We will use Corollary 3.12 to create the desired rational point $Q$ and open
neighborhood $U$. To that end, we have to give a finite set of polynomials
$c_{i}$ in $A$ or a localization thereof. In the previous lemmas, we also
created neighborhoods using Corollary 3.12. To obtain the desired open
neighborhood in this lemma, we will expand our list of polynomials $c_{i}$
using polynomials from the previous lemmas.

We first note that there are finitely many $C_{w}(I_{P})\in\Sigma(I_{P})$.
Suppose that $C_{w}(I_{P})\subseteq\mathrm{trop}(V(I)_{P})$. By Lemma 3.17, we
can find a set of polynomials $c_{i}$ such that for $Q^{\prime}$ in the
induced $U$, we have that $w\in\mathrm{trop}(V(I)_{Q})$. Similarly, if
$C_{w}(I_{P})\nsubseteq\mathrm{trop}(V(I)_{P})$, then there is an
$f_{w,P}\in{I_{P}}$ whose initial with respect to any
$w^{\prime}\in\mathrm{Relint}(C_{w}(I_{P}))$ is monomial. Using Corollary
3.12, we can expand our list of polynomials $c_{i}$ so that $f_{w,Q}$ remains
monomial for $Q$ in the induced $U$. We now expand our list of polynomials one
more time using the polynomials from Lemma 3.16 to obtain the desired $U$. If
$C_{w}(I_{P})\nsubseteq\mathrm{trop}(V(I)_{P})$, then
$\mathrm{in}_{w}(f_{w,Q})$ is monomial. Hence,
$w\notin\mathrm{trop}(V(I)_{Q})$. But then
$C_{w}(I_{P})=C_{w}(I_{Q})\nsubseteq\mathrm{trop}(V(I)_{Q})$. If
$C_{w}(I_{P})\subseteq\mathrm{trop}(V(I)_{P})$, then by construction
$w\in\mathrm{trop}(V(I)_{Q})$. We then have
$C_{w}(I_{P})=C_{w}(I_{Q})\subset\mathrm{trop}(V(I)_{Q})$. We conclude that
$\mathrm{trop}(V(I)_{P})=\mathrm{trop}(V(I)_{Q})$. ∎

###### Proposition 3.19

Let $K$ be as in 3.10. Let $P\in{Y^{\mathrm{an}}}$ be a valuation that admits
a strictly $K$-affinoid neighborhood $V$ such that $P$ reduces to a generic
point of the canonical reduction $\tilde{V}$ of $V$. Suppose that
$f_{1},...,f_{m}$ give a tropical basis over $P$. Then there is a rational
point $Q$ and an open neighborhood $U$ of $Q$ such that $f_{1},...,f_{m}$ form
a tropical basis over $Q^{\prime}$ for all $Q^{\prime}\in U$.

###### Proof.

We may assume as before that $I$ and the $f_{1}$,…,$f_{m}$ are homogeneous. By
Lemma 3.9, the tropicalizations of the $f_{i}$ are locally constant near a
rational point. The result then follows from Lemma 3.18. ∎

###### Corollary 3.20

Let $K$ be as in 3.10, let $P$, $V$ and $f_{1},\dots,f_{m}$ be as in
Proposition 3.19 and let $h\in{A}$. Then the open neighborhood $U$ can be
chosen to satisfy $|h(Q^{\prime})|=|h(P)|$ for all $Q^{\prime}\in{U}$.

###### Proof.

By including $h$ in the proofs of Lemmas 3.15, 3.16, 3.17, 3.18 and
Proposition 3.19, one sees using Corollary 3.12 that a $Q$ and a $U$ can be
found with the desired properties. ∎

Finally, we show that the open sets given by the generic valuations of the
type considered in Lemma 3.18 give a dense open subset of $Y^{\mathrm{an}}$,
which means that $X\to T_{Y}$ is generically tropically flat.

###### Lemma 3.21

Let $Y$ be an integral scheme over a non-archimedean field $K$ as in 3.10. Let
$h\in{A}\backslash{K}$ and let $r=e^{-w}$ for $w$ in the value group of $K$.
Then there is a valuation $P\in Y^{\mathrm{an}}$ with $|h(P)|=r$ that admits a
strictly $K$-affinoid neighborhood $V$ such that $P$ reduces to a generic
point of the canonical reduction $\tilde{V}$.

###### Proof.

Consider the subalgebra $K[h]\subset{A}$, giving a non-constant morphism
$\mathrm{Spec}(A)\to\mathbb{A}^{1}_{K}$. We extend this to a transcendence
basis for $A$, so that $K(A)$ is finite over $K(h_{1},...,h_{n})$, where
$h_{1}=h$. This induces a rational map $\mathrm{Spec}(A)\to\mathbb{A}^{n}$
that is finite over an open subset $U$ of $\mathbb{A}^{n}$. Write $\Gamma$ for
the value group of $K$. For any $v=(v_{1},\dots,v_{n})\in\Gamma^{n}$, we can
now consider the strictly $K$-affinoid
$V_{v}=\\{P\in\mathbb{A}^{n,\mathrm{an}}:|h_{i}(P)|=e^{-v_{i}}\\}$. The
canonical reduction is
$\tilde{V}_{v}=\mathrm{Spec}(k[h_{1}^{\pm},...,h_{n}^{\pm}])$ and the generic
point of $\tilde{V}_{v}$ corresponds to a point
$P_{v}\in\mathbb{A}^{n,\mathrm{an}}$. The condition $\pi(P_{v})=(0)$ implies
that $P_{v}$ lies in the open subset $U^{\mathrm{an}}$. By restricting to a
subdomain of $V_{v}$, we can then assume that $V_{v}\subset{U^{\mathrm{an}}}$.
We take the first coordinate of $v$ to be $-\text{log}(r)$ and consider a
point $Q_{v}$ in the pre-image of $P_{v}$. Since the induced morphism over $U$
is finite, we find that this point is again generic, and the inverse image of
$V_{v}$ is a strictly $K$-affinoid neighborhood of $Q_{v}$ with the right
properties. ∎

###### Theorem 3.22

Let $Y$ be integral. Then the tropically flat locus of a closed embedding of
$Y$-schemes $X\to{T_{Y}}$ contains an open dense subset of $Y^{\mathrm{an}}$.

###### Proof.

Since we allow valued field extensions in our definition of tropically
flatness,we can assume that $K$ is algebraically closed, non-trivially valued
and complete. Consider an open neighborhood $\mathbb{B}(h,r_{0},r_{1})$ in
$Y^{\mathrm{an}}$, where $h\in{A}$ and $r_{1},r_{2}\in\mathbb{R}_{\geq 0}$. We
will show that $\mathbb{B}(h,r_{0},r_{1})$ intersects the tropically flat
locus.

Using Lemma 3.21, we can find a generic valuation $P$ with
$P\in\mathbb{B}(h,r_{0},r_{1})$ and a strictly $K$-affinoid neighborhood of
$P$ such that $P$ reduces to a generic point of $\tilde{V}$. We consider a
tropical basis of $I$ with respect to $P$. By Proposition 3.19, the tropical
basis over $P$ remains a tropical basis over an open neighborhood $U$ of a
rational point $Q$. Note that the open neighborhood $U$ lies outside the loci
of the coefficients of the monomials of the tropical basis. In particular,
$X\to T_{Y}$ is tropically flat over $Q$. Finally, by Corollary 3.20, we have
$|h(P)|=|h(Q)|$, so that $Q\in\mathbb{B}(h,r_{0},r_{1})$. ∎

###### Corollary 3.23

Let $Y$ be an integral scheme of finite type over a non-archimedean field $K$
as in 3.10. Then there is an open dense subset of
$Y({K})\subset{Y^{\mathrm{an}}}$ over which $X\to{T_{Y}}$ is tropically flat.

###### Proof.

We consider the set of rational $Q$ from the proof of Theorem 3.22 over which
we have a local tropical basis. By the arguments in the proof, this is a dense
open subset. Note that the basis is moreover defined over $K$ so we in fact
have a $K$-rational local tropical basis. ∎

##  4\. Tropical intersections and generic root counts

We now use the material from the previous two sections to show how generic
properties of morphisms of schemes can be detected using tropical geometry. We
will see that many properties of a single tropical fiber over a tropically
flat point propagate to an open dense subset of the parameter space. In
Section 4.1, we prove Theorem 4.5, which expresses the generic root count as a
tropical intersection product. This also gives a standalone proof of
Bernstein’s theorem, see Corollary 4.11. In Section 4.2 we study torus-
equivariant and algebraically independent systems, and we prove Proposition
4.19. Finally, we discuss extensions of the results given here to analytic
families of polynomial equations.

###  4.1. Generic root counts as tropical intersection numbers

In this section, we show that generic root counts can be expressed in terms of
tropical intersection numbers, provided that we have a tropically flat proper
intersection. This extends Bernstein’s theorem to possibly overdetermined
families of polynomial equations with non-trivial relations among the
coefficients. An important tool in proving this is the following lemma.

###### Lemma 4.1

Let $X$ be a scheme that is locally of finite type over a complete non-
archimedean field $K$ and let $U\subset{X}$ be a dense open set. Then
$U^{\mathrm{an}}$ is open and dense.

###### Proof.

This follows from [Ber93, Proposition 2.6.4]. ∎

We now show that the generic dimension is reflected in the fiber over a
tropically flat point.

###### Proposition 4.2

Let $Y$ be irreducible. Let $P\in{Y^{\mathrm{an}}}$ be a point over which
$X\to{T_{Y}}$ is tropically flat and suppose that the fiberwise
tropicalization of $X$ over $P$ is of dimension $k$. Then there is an open
dense $U\subset{Y}$ such that $X_{U}\to{U}$ has fibers of dimension $k$.

###### Proof.

We first note that Krull dimensions are unaffected by base changes $K\to{L}$,
so we can assume that $P$ is $K$-rational and that $K$ is complete. Let
$k^{\prime}$ be the dimension of the generic fiber of $X$. By [Stacks22, Lemma
05F7], there is a (non-empty) open subset $U$ over which the fiber dimension
is $k^{\prime}$. By Lemma 3.9, there is an open $V$ around
$P\in{Y^{\mathrm{an}}}$ such that the tropicalizations are all of dimension
$k$. By Lemma 4.1, $V$ intersects $U^{\mathrm{an}}$ so we conclude that
$k^{\prime}=k$ using the tropical structure theorem [MS15, Theorem 3.3.5]. ∎

###### Example 4.3

Let $X\to{T_{Y}}$ be a closed embedding with an empty generic fiber. If the
fiber of $X$ over some $P\in{Y^{\mathrm{an}}}$ is non-empty, then the
embedding is not tropically flat over $P$ by Proposition 4.2. More generally,
if the dimension of the fiberwise tropicalization $\mathrm{trop}(X_{P})$ is
higher than the generic dimension, then $X\to{T_{Y}}$ is not tropically flat
over $P$.

###### Lemma 4.4

Let $X_{1},\dots,X_{k}$ be closed subschemes of complementary codimension in a
torus $T_{K}$ over a field $K$, i.e.,
$\sum_{i=1}^{k}\operatorname{codim}(X_{i})=\dim(T_{K})$, and suppose that the
$X_{i}$ intersect properly in a zero-dimensional set. Suppose that
$\bigcap_{i=1}^{k}{X_{i}}$ lies in the Cohen-Macaulay-locus of every $X_{i}$.
Then the higher intersection multiplicities vanish.

###### Proof.

We will show that the higher intersection multiplicities for the $k$-fold
diagonal $\Delta^{k}\subset{T_{K}^{k}}$ and $\prod_{i=1}^{k}X_{i}$ vanish. The
support of the intersection corresponds to $\bigcap_{i=1}^{k}{X_{i}}$ embedded
in the product by the map $D_{T}\colon T\to{T^{k}}$. Note that
$\prod_{i=1}^{k}X_{i}$ is Cohen-Macaulay at every $D_{T}(z)$ for
$z\in{\bigcap_{i=1}^{k}{X_{i}}}$. Indeed, this follows from [Stacks22, Lemma
0C0W] and [Stacks22, Lemma 045T(1)]. The lemma now follows from [Stacks22,
Lemma 0B02]. ∎

###### Theorem 4.5

Let $Y=\operatorname{Spec}(A)$ be an integral scheme that is of finite type
over a non-archimedean field $K$, let $p\colon X\to{Y}$ be an affine morphism
of finite type with Berkovich analytification $p^{\mathrm{an}}\colon
X^{\mathrm{an}}\to{Y^{\mathrm{an}}}$. Suppose that the following hold:

  1. (1)

The morphism $p$ factors through a closed embedding $X\to{T_{Y}}$ of
$Y$-schemes, where $T_{Y}$ is an $n$-dimensional torus over $Y$.

  2. (2)

The closed subscheme $X\to{T_{Y}}$ is an intersection of closed subschemes
$X_{i}\to{T_{Y}}$, $i=1,\dots,k$, that are generically Cohen-Macaulay and pure
of relative dimension $k_{i}$ over $Y$ with

$\sum_{i=1}^{k}(n-k_{i})=n.$

  3. (3)

There is a $P\in{Y^{\mathrm{an}}}$ over which the $X_{i}$ are tropically flat
and the tropical prevariety $\bigcap_{i=1}^{k}\mathrm{trop}(X_{i,P})$ is
bounded.

Then $X\to{Y}$ is generically finite and the generic root count $\ell_{X/Y}$
is the tropical intersection number $\prod_{i=1}^{r}\mathrm{trop}(X_{i,P})$.

###### Proof.

It suffices to prove the theorem after a base change $K\to{L}$, where $L$ is a
valued field extension of $K$. Indeed, the local rank is stable under base
change since the rank of a free module is stable under base change and being
generically Cohen-Macaulay is stable under field extensions by [Stacks22,
Lemma 00RJ] or Lemma 2.5. We can thus suppose that $K$ is complete and that
$P$ is $K$-rational.

Since the $W_{i}$ are tropically flat over $P$, we can find a set of
generators that form a tropical basis on an open neighborhood $\mathbb{B}_{1}$
of $P$. By Lemma 3.9, there is an open neighborhood $\mathbb{B}_{2}$ of $P$
over which the tropicalizations of these generators are constant. Let
$U_{1}\subset{Y}$ be a dense open subset over which $p$ is locally free and
let $U_{2}$ be a dense open subset over which $p$ is Cohen-Macaulay. Since
$U^{\mathrm{an}}_{1}\cap{U^{\mathrm{an}}_{2}}$ is dense in $Y^{\mathrm{an}}$
by Lemma 4.1, it intersects $\mathbb{B}_{1}\cap\mathbb{B}_{2}$ in a point $Q$.
As the the tropical prevariety is bounded, the fibers $X_{i,Q}$ are pure and
their codimensions add up to $n$, we can apply [OR13, Corollary 6.13] to find
that the tropical intersection number is equal to the sum of the algebraic
intersection numbers. By Lemma 4.4, this sum is equal to the sum of the
algebraic lengths. But again using the fact that free modules are stable under
base change, we find that this sum is the local rank provided by
Grothendieck’s theorem. This concludes the proof. ∎

###### Remark 4.6

The tropical intersection number in Theorem 4.5 is an algebraic intersection
number in a suitable toric variety. Indeed, let $X(\Delta)$ be a toric variety
such that $\Delta$ is a compatible compactifying fan for the
$\mathrm{trop}(X_{i,P})$ as in [OR13, Section 3]. Then by [OR13, Proposition
3.12], we find that the closures of the $X_{i,P}$ in $X(\Delta)$ only
intersect in the dense torus. In particular, the tropical intersection number
in Theorem 4.5 is equal to the algebraic intersection number
$\prod_{i=1}^{k}\overline{X}_{i,P}$.

The following examples shows the necessity of tropical flatness in Theorem
4.5:

###### Example 4.7

Let $K=\mathbb{C}\\{\\!\\{t\\}\\!\\}$ be the field of Puiseux series over
$\mathbb{C}$ and let $Y=\mathrm{Spec}(A)$ for $A=K[a]$. Consider the two
subschemes $X_{1}$ and $X_{2}$ of the three-dimensional torus
$T_{Y}=\mathrm{Spec}(A[x^{\pm},y^{\pm},z^{\pm}])$ over $Y$ given by the ideals

$\displaystyle
I_{1}=(ax^{2}+y^{2}+x+2y+3+az,x^{2}+y^{2}+3x-y+5)\quad\text{and}\quad
I_{2}=(z-at).$

We write $P$ for the point in $Y^{\mathrm{an}}$ corresponding to the ring
homomorphism $A\to{K}$ sending $a$ to $1$. The fiberwise tropicalization of
$X_{1}=V(I_{1})$ over $P$ is a tropical quartic curve in $\mathbb{R}^{3}$ and
$\mathrm{trop}(X_{2,P})$ is an affine hyperplane. These intersect
transversally in one point of multiplicity two. The generic root count is
however $4$, so $X_{1}$ cannot be tropically flat at $P$. To illustrate this
change in root counts, consider the points $P_{\lambda}$ given by sending
$a\mapsto{1+t^{\lambda}}$ for $\lambda\in\mathbb{\mathbb{Q}}$ large. The
corresponding fiberwise tropicalizations can be found in Figure 1. We have
that $\mathrm{trop}(X_{1,P})$ and $\mathrm{trop}(X_{2,P})$ intersect in two
double points, one of which diverges as $\lambda\rightarrow\infty$.

$-e_{1}-e_{2}-e_{3}$$e_{1}$$e_{2}$$-e_{1}-e_{2}-e_{3}$$e_{2}$$e_{1}+2e_{3}$$(1-\lambda,1-\lambda,1)$$(0,0,0)$$(0,0,\lambda)$$\mathrm{trop}(V(I_{2})_{P})$$\mathrm{trop}(V(I_{1})_{P})$
Figure 1.  The intersection of two families of tropical varieties with one
family that is not tropically flat. The edges of $\mathrm{trop}(V(I_{1})_{P})$
have multiplicity $2$, as do the two intersection points colored blue.

By combining Theorems 3.22 and 4.5, we obtain the following.

###### Corollary 4.8

Suppose that $Y$, $X$ and the $X_{i}$ are as in Theorem 4.5. Suppose that
$\bigcap_{i=1}^{k}\mathrm{trop}(X_{i,P})$ is bounded for $P$ in a non-empty
open subset of $Y^{\mathrm{an}}$. Then there is a non-empty open subset $U$ of
$Y^{\mathrm{an}}$ such that the generic root count $\ell_{X/Y}$ is
$\prod_{i=1}^{k}\mathrm{trop}(X_{i,Q})$ for $Q\in{U}$.

###### Corollary 4.9

Suppose that $\ell_{X/Y}\neq\prod_{i=1}^{k}\mathrm{trop}(X_{i,Q})$ for $Q$ in
an open dense subset of $Y^{\mathrm{an}}$. Then there is a dense subset $W$ of
$Y^{\mathrm{an}}$ such that the tropical prevariety
$\bigcap_{i=1}^{k}\mathrm{trop}(X_{i,Q})$ is unbounded for $Q\in W$.

We now explore the ramifications of Theorem 4.5 for square decompositions.

###### Corollary 4.10

Suppose that the $X_{i}$ in Theorem 4.5 are hypersurfaces $V(f_{i})$ given by
polynomials $f_{i}=\sum_{\alpha}c_{i,\alpha}x^{\alpha}$ and that the tropical
prevariety $\bigcap_{i=1}^{n}\mathrm{trop}(V(f_{i})_{P})$ is bounded for some
$P\in{Y^{\mathrm{an}}\backslash\bigcup_{\alpha,i}{V(c_{i,\alpha})}}$. Then the
generic root count of $p\colon X\to{Y}$ is the normalized mixed volume
$\mathrm{MV}(f_{1},...,f_{n})$.

###### Proof.

The $V(f_{i})$ are tropically flat, generically Cohen-Macaulay and pure of
relative dimension $n-1$ over the given locus. We can thus use Theorem 4.5 to
conclude that the generic root count is the tropical intersection number. By
[MS15, Theorem 4.6.8], this is the mixed volume, so we obtain the statement of
the corollary. ∎

######  Corollary 4.11 (Bernstein-Kushnirenko)

Let $\phi\colon X\to{T_{Y}}$ be the closed embedding associated to the
universal family of square polynomial systems with fixed monomial supports
$M_{i}$. Then the generic root count $\ell_{X/Y}$ is
$\mathrm{MV}(f_{1},...,f_{n})$.

###### Proof.

One easily finds a point $P\in{Y^{\mathrm{an}}}$ for which the
tropicalizations intersect in finitely many points (using for instance Lemma
4.16). The statement then follows from Corollary 4.10. ∎

For square systems, Corollary 4.9 can be simplified to give the following.

###### Corollary 4.12

Suppose that a square system over $Y$ given by polynomials
$f_{i}=\sum_{\alpha}c_{i,\alpha}x^{\alpha}$ has nonzero generic root count
less than the mixed volume. Then the fiberwise tropical prevariety over every
$P\in{Y^{\mathrm{an}}\backslash\bigcup_{i,\alpha}{V(c_{i,\alpha})}}$ is
unbounded.

###  4.2. Torus-equivariant systems and generic root counts

In this section, we introduce the notion of torus-equivariance, which gives a
natural condition under which Theorem 4.5 holds. It will be used in Sections 5
and 6 to obtain generic root counts for certain classes of systems.

The relative torus is a group scheme with group law

$T_{Y}\times{T_{Y}}\to{T_{Y}}.$

Let $L$ be a field over $K$ and let $\lambda\in{T_{Y}(L)}$ be an $L$-valued
point. We denote the image of $\lambda$ in $Y(L)$ by $P$. The fiber of $T_{Y}$
over $P$ gives a torus $T_{P}$ defined over $L$ and by multiplying by
$\lambda$, we obtain an isomorphism $\lambda\colon T_{P}\to{T_{P}}$. Let
$\phi\colon X\to{T_{Y}}$ be a closed immersion. The fiber of $\phi\colon
X\to{T_{Y}}$ over $P$ gives a closed immersion $\phi_{P}\colon
X_{P}\to{T_{P}}$. We denote the image of $X_{P}$ under $\lambda\circ\phi_{P}$
by $\lambda\cdot{X_{P}}$. We use the same notation for points
$\lambda\in{T^{\mathrm{an}}_{Y}}$. If we have an action
$T_{Y}\times_{Y}Y\to{Y}$, then we denote $L$-valued points of
$T_{Y}\times_{Y}Y$ by $(\lambda,P)$. The image of such a point under the group
action is denoted by $\lambda\cdot P$.

###### Definition 4.13

We say that $\phi\colon X\to{T_{Y}}$ is _torus-equivariant_ if there exists an
action $T_{Y}\times_{Y}{Y}\to{Y}$ such that for every field $L$ and
$(\lambda,P)\in{(T_{Y}\times_{Y}Y)}(L)$, we have
$X_{\lambda\cdot{}P}=\lambda\cdot{X_{P}}$. Here the tori over two $L$-valued
points of $Y$ are canonically identified.

###### Example 4.14

Consider the parameter ring $A=K[a_{0},a_{1},a_{2}]$ with polynomial system
defined by $f=a_{0}x+a_{1}xy+a_{2}y$. We define an action of
$T_{Y}=\mathrm{Spec}(A[x^{\pm},y^{\pm}])$ on $Y=\mathrm{Spec}(A)$ by the Hopf-
algebra maps

$\displaystyle a_{0}\mapsto{x^{-1}a_{0}},\qquad
a_{1}\mapsto{x^{-1}y^{-1}a_{1}},\qquad a_{2}\mapsto{y^{-1}a_{2}}.$

Here we used the identification
$A[x^{\pm},y^{\pm}]\otimes_{A}A=A[x^{\pm},y^{\pm}]$. It is easy to see that
$A\to{A[x^{\pm},y^{\pm}]/(f)}$ is a torus-equivariant system with respect to
this action. In a sense, if the monomials in a polynomial contain free
variables, then these can be used to define a torus-equivariant system.

###### Remark 4.15

We will need the torus-equivariance to eliminate the non-Cohen-Macaulay loci
in Proposition 4.19. For systems that are automatically Cohen-Macaulay
everywhere (e.g., square systems), we can make do with the weaker assumption
of the tropicalizations being torus-equivariant, i.e., if for every
$P\in{Y^{\mathrm{an}}}$ and every $\lambda\in{T}^{\mathrm{an}}_{Y}$ mapping to
$P$, there is a $Q\in{Y^{\mathrm{an}}}$ such that
$\mathrm{trop}(X_{Q})=\mathrm{trop}(\lambda\cdot{X_{P}})=\mathrm{trop}(\lambda)+\mathrm{trop}(X_{P})$.

To prove Proposition 4.19, we will need the following lemma from tropical
geometry, which shows that there are sufficiently many translations that give
the stable intersection of two balanced polyhedral complexes in
$\mathbb{R}^{n}$. A similar result also holds for $k$ polyhedral complexes.

###### Lemma 4.16

Let $\Sigma_{1}$ and $\Sigma_{2}$ be two balanced polyhedral complexes in
$\mathbb{R}^{n}$. There is a dense open subset
$U\subset{\mathbb{R}^{n}\times\mathbb{R}^{n}}$ such that for all
$(\lambda_{1},\lambda_{2})\in{U}$, we have that $\lambda_{1}+\Sigma_{1}$ and
$\lambda_{2}+\Sigma_{2}$ intersect transversally. This set moreover contains
$\\{0\\}\times{U_{1}}$ and $U_{2}\times{\\{0\\}}$ for suitable dense open
subsets $U_{i}$ of $\mathbb{R}^{n}$.

###### Proof.

This follows from the proof of [MS15, Proposition 3.6.12]. ∎

###### Lemma 4.17

Let $X_{1}$,…,$X_{k}$ be closed pure subschemes of an $n$-dimensional torus
$T_{n}$ over a non-archimedean field $K$ with

$\sum_{i=1}^{k}\mathrm{codim}(X_{i})=\mathrm{dim}(T_{n})=n.$

There is a non-empty open subset $U\subset{T^{\mathrm{an}}_{n(k-1)}}$ such
that for $t=(t_{2},...,t_{k})\in{U}$, we have that
$X_{1}$,$t_{2}X_{2}$,…,$t_{k}X_{k}$ meet properly in finitely many points.
Furthermore, for $t_{1}\coloneqq 1$ and $t=(t_{2},...,t_{k})\in{U}$ the
intersection $\bigcap_{i=1}^{k}t_{i}X_{i}$ lies in the Cohen-Macaulay locus of
each $t_{i}X_{i}$ and the tropicalizations of the $t_{i}X_{i}$ meet
transversally and properly.

###### Proof.

Let $Z_{i}$ be the non-Cohen-Macaulay locus of $X_{i}$, which are proper
closed subsets of the $X_{i}$ by [Stacks22, Lemma 00RG]. Consider the stable
intersection of the tropicalizations of $X_{1}$,$X_{2}$,…,$X_{k-1}$ and
$Z_{k}$. By [MS15, Theorem 3.6.10] and the assumption on the codimensions,
this is empty. The other $k-1$ combinations also give empty stable
intersections. For each of these, we obtain a dense open subset in
$\mathbb{R}^{n(k-1)}$ for which the tropicalizations do not intersect by Lemma
4.16. We intersect these dense open subsets to obtain a dense open subset $V$.
We can moreover shrink $V$ so that the $\mathrm{trop}(t_{i}X_{i})$ intersect
properly and transversally everywhere. We then immediately find that
$U=\mathrm{trop}^{-1}(V)$ has the desired properties. ∎

###### Definition 4.18

Let $X_{i}$ be a decomposition of a closed subscheme $X$ of $T_{Y}$. We say
this decomposition is _algebraically independent_ if there exist parameter
rings $A_{i}$ over $K$ with spectra $Y_{i}$ and closed subschemes
$X^{\prime}_{i}\to{T_{Y_{i}}}$ such that

  1. (1)

$\prod_{K}Y_{i}=Y$ and

  2. (2)

$X^{\prime}_{i}\times_{Y_{i}}Y=X_{i}$.

We will identify $X^{\prime}_{i}$ with $X_{i}$.

###### Proposition 4.19

Suppose the $k$ subschemes $X_{i}\to{Y}$ in Theorem 4.5 are algebraically
independent, generically pure of complementary dimension, and that at least
$k-1$ families are torus-equivariant. Then the conclusion of Theorem 4.5 holds
for $P$ in an open dense subset of ${Y^{\mathrm{an}}}$.

###### Proof.

For every $X_{i}$, we take a set of generators $f_{i,j}$ of the corresponding
ideal. We write $U_{0}$ for the open subset of $Y$ over which the coefficients
of the monomials of the $f_{i,j}$ are non-zero. Let $U_{1}\subset{Y}$ be the
open subset provided by Grothendieck’s generic freeness theorem and let
$U_{2}$ be the open dense subset over which the $W_{i}$ are pure. Let
$U=U_{0}\cap{}U_{1}\cap{U_{2}}$ and $Q\in{U^{\mathrm{an}}}$. We can assume
that $Q$ is rational. Since the family is algebraically independent, we can
find points $Q_{i}\in{Y^{\mathrm{an}}_{i}}$ that give rise to $Q$. We write
$X_{i,Q}$ for the fibers of the families and $Z_{i,Q}$ for their non-CM-loci.
By Lemma 4.17, there is an open subset $V$ of
$(t_{1},...,t_{k})\in{T_{K,nk}^{\mathrm{an}}}$ such that the $t_{i}X_{i,Q}$
meet in the CM-locus of each. Moreover, their tropicalizations meet
transversally in finitely many points. Consider the torus-action

$T_{Y_{i},n}\times_{Y_{i}}{Y_{i}}\to{Y_{i}}$

for each $i$. Applying the projection maps $Y\to{Y_{i}}$ to the open dense
subset $U$, we obtain open dense subsets $U_{i}\subset{Y_{i}}$. Consider the
map $T_{K,n}\to{T_{Y_{i},n}\times_{Y_{i}}{Y_{i}}}\to{Y_{i}}$ induced by the
point $\pi(Q_{i})$ and the group action on $Y_{i}$. The inverse image of
$U_{i}$ under this map is open and non-empty, and thus dense. Its
analytification thus intersects the projection $V_{i}$ of $V$. By doing this
for all $i$, we obtain a new point $Q$ such that the $X_{i,Q}$ meet in the CM-
locus of each. We then conclude using Lemma 4.4. ∎

###### Remark 4.20

To find a generic point as in Theorem 4.19, we can use a point $P$ over which
the $X_{i}\to{Y_{i}}$ are tropically flat. Indeed, the given open neighborhood
$V$ of $P$ will then intersect the open set $U^{\mathrm{an}}$, so we can use
the tropicalizations of the given fibers over $P$.

We can use this to prove an invariance of tropical intersection numbers in
torus-equivariant algebraically independent systems.

###### Corollary 4.21

Consider a decomposition $X_{i}$ of an embedding $X\to{T_{Y}}$ as in
Proposition 4.19. Then for a generic pair of points $P,Q\in{Y^{\mathrm{an}}}$,
we have

$\prod\mathrm{trop}(X_{i,P})=\prod\mathrm{trop}(X_{i,Q}).$

###### Remark 4.22

We note here that the results in Section 4 can be generalized to analytic
families of polynomial equations. For many applications in practice this is
important, since the functions in the parameters naturally contain analytic
functions. For instance, one can consider the system

$\displaystyle f_{1}$
$\displaystyle=\sin(a_{1}+a_{2})x^{2}+\sin(a_{1}a_{2})y^{2}+\cos(a_{1})x+a_{4}y+a_{5},$
$\displaystyle f_{2}$
$\displaystyle=\cos(b_{1})x^{2}+\cos(b_{1}+b_{2})y^{2}+b_{3}x+b_{4}y+b_{5}$

over the ring $A=\mathbb{C}[[a_{i},b_{i}]]$, where $\mathbb{C}$ is trivially
valued. This is a $\mathbb{C}$-affinoid domain, so the material in [Ber93,
Section 2] is again applicable. Note that the coefficients of the monomials in
these types of equations can satisfy algebraic relations that are not always
apparent. The tropical material presented here is however directly applicable,
and we can find a ring homomorphism $A\to\mathbb{C}[[t]]$ with corresponding
point $P$ such that the fiberwise tropical prevariety over $P$ is finite. This
then implies that the generic root count is $4$. More generally, one can
consider polynomial equations defined over affinoid $K$-algebras
$\mathcal{A}$, where $K$ is any (complete) non-archimedean field. The material
in this paper directly extends to this more general scenario.

##  5\. Generic root counts of square systems

In this section, we focus on general square systems. In Section 5.1 we
introduce the degeneracy graph of a square system, which characterizes all its
finite degenerations. In Section 5.2 we study a special class of square
systems for which we can express the generic root count in terms of tropical
intersection numbers. This includes the steady-state equations of chemical
reaction networks and the birational intersection indices $[L_{1},...,L_{n}]$
studied by Kaveh and Khovanskii in [KK10][KK12]. These applications will be
studied in more detail in Section 6.

###  5.1. The degeneracy graph of a square system

Let $p\colon X\to{Y}$ be a generically finite square polynomial system given
by polynomials $f_{1},\dots,f_{n}$ in $A[x_{1},\dots,x_{n}]$. We start with a
basic lemma on the quasi-finite locus $\text{QF}(p)\subset{Y}$ for the
morphism $p\colon X\to{Y}$.

###### Lemma 5.1

$\text{QF}(p)$ is open.

###### Proof.

Since $p:X\to Y$ is square, it is flat at every point $x$ lying over
$y\in{\text{QF}(p)}$ by [Stacks22, Lemma 00ST] and [Stacks22, Lemma 00SW]. The
inverse image of $\text{QF}(p)$ in $X$ is thus in the open flat locus $U$.
Consider the restriction $p_{U}$ of $p$ to $U$, which is a flat morphism. By
[Stacks22, Lemma 02NM], the locus of relative dimension zero of $p_{U}$ is
open in $U$. The image of this locus under $p_{U}$ is exactly $\text{QF}(p)$.
Since flat morphisms are open, we conclude that $\text{QF}(p)$ is open. ∎

For every $y\in{\text{QF}(p)}$ with prime ideal $\mathfrak{p}_{y}$, we have
that the induced system

$\mathrm{Spec}(B/\mathfrak{p}_{y}B)\to\mathrm{Spec}(A/\mathfrak{p}_{y})$

is generically finite. We denote the generic root count of this system by
$\ell_{X/Y,y}$. If $y$ is the generic point $\eta$ of $Y$, this gives the
generic root count of $p$. We now show that root counts decrease under
specialization, so that $\ell_{X/Y,y}\leq\ell_{X/Y}$ for every
$y\in{\text{QF}(p)}$.

###### Lemma 5.2

Let $y\in{\text{QF}(p)}$ with prime ideal $\mathfrak{p}_{y}$ and induced
system
$p_{y}\colon\mathrm{Spec}(B/\mathfrak{p}_{y}B)\to\mathrm{Spec}(A/\mathfrak{p}_{y})$.
Then $\ell_{X/Y,y}\leq{\ell_{X/Y}}$.

###### Proof.

Let $D(h)$ be an open dense subset of $\mathrm{Spec}(A)$ over which the root
count is $\ell_{X/Y}$. By [Gub13, Proposition 4.14], the reduction map
$\pi\colon D(h)^{\mathrm{an}}\cap{Y^{\mathrm{an},0}}\to{Y}$ is surjective. We
take a point $Q$ of $Y$ over which the root count is $\ell_{X/Y,y}$. By the
above, we can then find a valuation ring $R$ with residue field $k$ and a
point $P\in{Y(R)}$ such that $P_{\eta}$ lies in $D(h)(K)$ and $P_{s}=Q$. Here
$P_{\eta}$ is the composition of $P\colon\mathrm{Spec}(R)\to{Y}$ and
$\mathrm{Spec}(K)\to\mathrm{Spec}(R)$, and $P_{s}$ is the composition of $P$
and $\mathrm{Spec}(k)\to\mathrm{Spec}(R)$. We can assume that $R$ is
Henselian.

We consider the base change $R\to{B\otimes_{A}R=:M}$. This is quasi-finite by
construction. It is moreover flat by the considerations in Lemma 5.1. Since
$R$ is Henselian, we can write

$M=M_{\mathrm{fin}}\times{M_{\mathrm{nfin}}},$

where $M_{\mathrm{fin}}$ is finite and $M_{\mathrm{nfin}}\otimes_{R}{k}=(0)$.
Moreover, $M_{\mathrm{fin}}$ is flat over $R$, so it is free. We have
$\ell_{X/Y}=\mathrm{rank}(M_{\eta})=\mathrm{rank}(M_{\mathrm{fin},\eta})+\mathrm{rank}(M_{\mathrm{nfin},\eta})$
and $\mathrm{rank}(M_{s})=\mathrm{rank}(M_{\mathrm{fin},s})$. But
$\ell_{X/Y,y}=\mathrm{rank}(M_{\mathrm{fin},s})=\mathrm{rank}(M_{\mathrm{fin},\eta})$,
so we obtain the statement of the lemma. ∎

###### Remark 5.3

In general, if a system does not admit a square decomposition, then root
counts need not decrease under specialization. We give two examples of this
phenomenon. Consider the scheme
$Y=\mathrm{Spec}(\mathbb{C}[x,y]/(y^{2}-x^{3}-x^{2}))$ with normalization
$X\to{Y}$. Its generic root count is one, however its root count over the
maximal ideal $(x,y)$ is two. This can be easily seen using the presentation
$X=\mathrm{Spec}(\mathbb{C}[x,y,z]/(y^{2}-x^{3}-x^{2},xz-y,z^{2}-x-1))$.

For our second example, consider the scheme $X_{t}=V(f_{1},\dots,f_{6})$ over
$Y=\mathrm{Spec}(\mathbb{C}[t])$ given by

$\displaystyle f_{1}$ $\displaystyle=(x-t)(z-t),$ $\displaystyle f_{3}$
$\displaystyle=yz,$ $\displaystyle f_{5}$ $\displaystyle=x-z,$ $\displaystyle
f_{2}$ $\displaystyle=(x-t)w,$ $\displaystyle f_{4}$ $\displaystyle=yw,$
$\displaystyle f_{6}$ $\displaystyle=y-w.$

The generic root count of $X_{t}\to{Y}$ is $2$, but the root count of the
fiber $X_{0}$ over $t=0$ is $3$. We thus see that generic root counts can
increase under specialization. Note that in both cases, we necessarily have
that the families are not flat, since otherwise the proof in Lemma 5.2 would
have gone through verbatim.

By Lemma 5.2 and [Stacks22, Lemma 07RY], there are reduced closed subschemes

$\emptyset=Z_{-1}\subset{}Z_{0}\subset{Z_{1}}\subset{...}\subset{Z_{\ell_{X/Y}}}=\text{QF}(p)$

such that the locus of rank $i$ is $Z_{i}\backslash{Z_{i-1}}$. Every $Z_{i}$
has finitely many generic points which we denote by $y_{i,j}$. These in turn
correspond to minimal prime ideals $\mathfrak{p}_{i,j}$. By abuse of notation,
we write $y\subset{z}$ for points $y,z$ in $Y$ if
$\mathfrak{p}_{y}\subset{\mathfrak{p}_{z}}$, so that $z$ is a specialization
of $y$.

###### Definition 5.4

The _degeneracy graph_ of $p\colon\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ is the
graph with vertices $y_{i,j}$ and edges $(y_{i,j},y_{i-1,k})$ for
$y_{i,j}\subset{y_{i-1,k}}$. For every $y\in{\text{QF}(p)}$, there exists a
$y_{i,j}$ such that $\ell_{X/Y,y}=\ell_{X/Y,y_{i,j}}$ and $y_{i,j}\subset{y}$.
We will call such a point $y$ a _witness_ of $y_{i,j}$.

###### Remark 5.5

Consider the universal family $p\colon X\to{Y}$ of square polynomial systems
with fixed monomial supports $M_{i}$. By Corollary 4.11, the generic root
count is the corresponding mixed volume. Suppose that we take a subset
$M^{\prime}_{i}\subset{M_{i}}$ of every monomial support, giving a new
universal family $X^{\prime}\to{Y^{\prime}}$. The degeneracy graph of
$X^{\prime}\to{Y^{\prime}}$ is then naturally a minor of the degeneracy graph
of $X\to{Y}$. More generally, the degeneracy graph associated to any
irreducible closed subscheme $Y^{\prime}\subset{Y}$ with generic point in the
quasi-finite locus will give a minor of the degeneracy graph of $p\colon X\to
Y$.

###  5.2. Modifications of a square system

In this section, we discuss a general strategy for expressing the generic root
count of a square system $X\to Y$ as a tropical intersection number. This is
done using appropriate re-embeddings, also referred to as _tropical
modifications_ [Kal15]. Similar to the works of Kaveh and Khovanskii [KK10,
KK12], we may need to restrict to an open dense subset $U$ of the relative
torus $T_{Y}$ and consider the generic root count of $X_{U}\coloneqq X\cap
U\to Y$ instead of $X\to Y$.

###### Definition 5.6

Let $f_{1},\dots,f_{n}\in A[x_{1}^{\pm},\dots,x_{n}^{\pm}]$. Choose
$p_{i,j}\in{A}$ and $q_{j}\in K[x_{1}^{\pm},...,x_{n}^{\pm}]$ such that

$f_{i}=\sum_{j=1}^{m}p_{i,j}\cdot q_{j}.$ (2)

Note that some $p_{i,j}$ may be zero and the $q_{j}$ are not necessarily
pairwise distinct.

We write $T_{Y}=\mathrm{Spec}(A[x_{1}^{\pm},...,x_{n}^{\pm}])$ for the torus
and $X=V(f_{1},...,f_{n})$. For any open subset $U$ of $T_{Y}$, we write
$X_{U}=X\cap{U}$, which we again consider as a $Y$-scheme. Let

$\displaystyle\widehat{C}$ $\displaystyle\coloneqq
A[x_{i}^{\pm},w_{j}^{\pm}\mid 1\leq i\leq n,1\leq j\leq m],$ and
$\displaystyle\hat{f}_{i}$
$\displaystyle\coloneqq\sum_{j=1}^{m}p_{i,j}w_{j}\qquad\text{for }1\leq i\leq
n,$ $\displaystyle\hat{h}_{j}$ $\displaystyle\coloneqq w_{j}-q_{j}\qquad\hskip
11.38109pt\text{for }1\leq j\leq m.$

Here the latter two are polynomials in $\widehat{C}$. Set
$\widehat{B}\coloneqq\widehat{C}/(\hat{f}_{i},\hat{h}_{j}\mid 1\leq i\leq
n,1\leq j\leq m)$ and $\widehat{X}\coloneqq\operatorname{Spec}(\widehat{B})$.
The scheme $\widehat{X}$ admits a natural embedding into the relative torus
$\widehat{T}_{Y}\coloneqq\operatorname{Spec}(\widehat{C})$ over
$Y=\mathrm{Spec}(A)$, and the two ideals
$\widehat{I}_{{\mathrm{lin}}}=(\hat{f}_{i}\mid 1\leq i\leq n)$ and
$\widehat{I}_{{\mathrm{nlin}}}=(\hat{h}_{j}\mid 1\leq j\leq m)$ give a
decomposition of $\widehat{X}\to\widehat{T}_{Y}$. We write
$\widehat{X}_{{\mathrm{lin}}}=V(\widehat{I}_{{\mathrm{lin}}})$ and
$\widehat{X}_{{\mathrm{nlin}}}=V(\widehat{I}_{{\mathrm{nlin}}})$. Note that
$\widehat{X}_{{\mathrm{lin}}}$ is linear, and that
$\widehat{X}_{{\mathrm{nlin}}}$ is a constant scheme over $Y$, which makes
$\widehat{X}_{{\mathrm{lin}}}$ and $\widehat{X}_{{\mathrm{nlin}}}$
algebraically independent. We refer to $\widehat{X}\to{Y}$ as the
_modification_ of $X\to Y$ derived from the representation in Equation 2.

Note that the modification of $X\to Y$ depends on the choice of $p_{i,j}$ and
$q_{j}$. If the modification satisfies the conditions of Proposition 4.19,
then we immediately obtain a formula for the root count:

###### Corollary 5.7

Suppose that $\widehat{X}_{\mathrm{lin}}$ is torus-equivariant and of generic
codimension $n$. Then $\widehat{X}\to{Y}$ is generically finite and
$\ell_{\widehat{X}/Y}=\mathrm{trop}(\widehat{X}_{\mathrm{lin},P})\cdot\mathrm{trop}(\widehat{X}_{\mathrm{nlin},P})$
for $P$ in an open dense subset of $Y^{\mathrm{an}}$.

###### Proof.

We only have to verify that $\widehat{X}_{\mathrm{nlin}}$ is generically pure
and of the right codimension. This follows from the fact that
$\widehat{X}_{\mathrm{nlin}}$ is isomorphic to an open subset of the
$n$-dimensional torus over $Y$. ∎

We now focus on three aspects of Corollary 5.7 which will serve as guides for
the remainder of this section:

  1. (a)

The assumption that $\widehat{X}_{\mathrm{lin}}$ is of generic codimension
$n$.

  2. (b)

The assumption that $\widehat{X}_{\mathrm{lin}}$ is torus-equivariant.

  3. (c)

The fact that Corollary 5.7 gives a formula for $\ell_{\widehat{X}/Y}$ and not
$\ell_{X/Y}$.

In general, Assumptions LABEL:enumitem:genericCodimension and
LABEL:enumitem:torusEquivariance need not be satisfied for a representation of
the $f_{i}$ as in Equation 2. However, Lemma 5.8 shows that
LABEL:enumitem:genericCodimension is satisfied in all cases of interest and
Lemma 5.9 shows that LABEL:enumitem:torusEquivariance is guaranteed by a set
of natural algebraic conditions on the $p_{i,j}$. For
LABEL:enumitem:compatibleSubset, we will see that
$\ell_{X_{U}/Y}=\ell_{\widehat{X}/Y}$ for an open dense subset
$U\subset{T_{Y}}$. This open subset also plays an important role in [KK10,
KK12].

###### Lemma 5.8

Suppose that $X_{U}\to{Y}$ is generically finite for some open dense $U$. Then
$\widehat{X}_{\mathrm{lin}}\to{Y}$ is generically a complete intersection of
generic codimension $n$.

###### Proof.

Suppose that $\widehat{X}_{\mathrm{lin}}\to{Y}$ is not generically a complete
intersection of generic codimension $n$. Then there is a linear relation over
the function field of $Y$ among the $\hat{f}_{i}$. But this implies that there
is a linear relation among the $f_{i}$, contradicting the fact that
$X_{U}\to{Y}$ is generically finite. ∎

###### Lemma 5.9

Let $Y=\mathrm{Spec}(A)$, where $A=K[a_{1},...,a_{m}]$. Suppose that the
$p_{i,j}$ from Definition 5.6 are linear and homogeneous. Suppose that there
exist subrings $A_{j}\subset{A}$ generated by linear polynomials in $A$ with
$\bigotimes_{K}A_{j}=A$ and $p_{i,j}\in{A_{j}}$. Then
$\widehat{X}_{\mathrm{lin}}$ is torus-equivariant.

###### Proof.

To define the action, we consider a set of independent linear generators
$z_{j,k}$ of $A_{j}$. The torus action sends a vector $(w_{i})$ to
$(\lambda_{i}w_{i})$. We then send the $z_{j,k}$ to $z_{j,k}/\lambda_{k}$. One
easily checks that $\widehat{X}_{\mathrm{lin}}$ is torus-equivariant with
respect to this action. ∎

###### Assumption 5.10

If $\widehat{X}_{\mathrm{lin}}$ is torus-equivariant and of generic
codimension $n$, then we say that $X\to{Y}$ is _tropically modifiable_. We
will assume for the remainder of this section that $X\to{Y}$ is tropically
modifiable.

We now address point LABEL:enumitem:compatibleSubset. As with the previous
two, there is no guarantee that $\ell_{\widehat{X}/Y}=\ell_{X/Y}$. Instead, we
will see that generic root count of the modification $\widehat{X}$ is the
generic root count of $X_{U}$ for an open dense $U$ in $T_{Y}$. In the
following the we will show what $U$ is, why $U$ is necessary, and when
$X_{U}=X$.

###### Theorem 5.11

Suppose that $X\to{Y}$ is tropically modifiable as in 5.10 and let
$U=\bigcap_{i=1}^{m}D(q_{i})$. Then $X_{U}\to{Y}$ is generically finite and
$\ell_{X_{U}/{Y}}=\ell_{\widehat{X}/{Y}}=\mathrm{trop}(\widehat{X}_{\mathrm{lin},P})\cdot\mathrm{trop}(\widehat{X}_{\mathrm{nlin},P})$
for $P$ in an open dense subset of $Y^{\mathrm{an}}$.

###### Proof.

Note that the solutions of $\widehat{X}\to{Y}$ correspond exactly to solutions
of $X_{U}\to{Y}$ since the $q_{j}$ are nonzero. Moreover, the multiplicities
are preserved by the linear nature of the modification, hence the generic root
counts coincide. ∎

###### Example 5.12

Note that without passing to $X_{U}$, Theorem 5.11 is not true in general. For
instance, consider the system

$\displaystyle f_{1}=a_{1}(x-1)+a_{2}(y-1)\text{ and
}f_{2}=a_{3}(x-1)+a_{4}(y-1)$

over $Y=\mathrm{Spec}(A)$ with $A=K[a_{1},a_{2},a_{3},a_{4}]$. The unique
solution for every non-zero choice of parameters of this system is $(1,1)$.
The tropical intersection number of the modification is however $0$, which
gives the number of solutions with $x\neq{1}$ or $y\neq{1}$.

We would now like to extend the open subset from Theorem 5.11 so that the
tropical intersection number derived from $\widehat{X}_{\mathrm{lin}}$ and
$\widehat{X}_{\mathrm{nlin}}$ still gives a valid generic root count. We start
with a basic tool in extending this root count.

###### Lemma 5.13

Suppose that $U_{i}\supset{U_{0}}={\bigcap_{j=1}^{m}D(q_{j})}$ are open
subsets of $T_{Y}$ such that the $X_{U_{i}}\to{Y}$ are generically finite with
$\ell_{X_{U_{i}}/Y}=\ell_{X_{U_{0}}/Y}$. Let $U=\bigcup U_{i}$. Then
$X_{U}\to{Y}$ is generically finite with $\ell_{X_{U}/Y}=\ell_{X_{U_{0}}/Y}$.

###### Proof.

Any point in the generic fiber of $X_{U}\to{Y}$ lies in the generic fiber of
$X_{U_{i}}\to Y$ for some $i$. But these all lie in the generic fiber of
$X_{U_{0}}\to{Y}$, which quickly gives the desired equality. ∎

It now follows that there is a largest open subset
$U\supset{\bigcap_{j=1}^{m}D(q_{j})}$ such that $X_{U}$ has generic root count
$\ell_{\widehat{X}/{Y}}$.

###### Definition 5.14

Let $U$ be the union of all open subsets
$U_{i}\supseteq\bigcap_{j=1}^{m}D(q_{j})$ such that
$\ell_{X_{U_{i}}/Y}=\ell_{\widehat{X}/Y}$. We call this the maximal compatible
open subset for the modification.

Example 5.12 shows that this $U$ can be a strict subset of $T_{Y}$. In
Proposition 5.16, we will give a criterion that allows us to extend
${\bigcap_{j=1}^{m}D(q_{j})}$ to a larger open set $U$ with
$\ell_{X_{U}/Y}=\ell_{\widehat{X}/Y}$. We first give some preliminary
definitions.

###### Definition 5.15

Write $Z=\mathrm{Spec}(A[x_{i}^{\pm},w_{j}\mid
1\leq{i}\leq{n},1\leq{j}\leq{m}])$. Any subset $J\subseteq[m]$ gives rise to a
toric stratum $H_{J}=\bigcap_{j\notin J}D(w_{j})\cap\bigcap_{j\in J}V(w_{j})$
of $Z$. Note that every $H_{J}$ is isomorphic to a standard torus over $Y$.
Let $\widehat{I}_{\mathrm{lin},a}$ and $\widehat{I}_{\mathrm{nlin},a}$ be the
ideals generated by the $\widehat{f}_{i}$ and $\widehat{h}_{j}$ in
$A[x_{i}^{\pm},w_{j}\mid 1\leq{i}\leq{n},1\leq{j}\leq{m}]$. We write
$\widehat{X}_{\mathrm{lin},a}=V(\widehat{I}_{\mathrm{lin},a})$ and
$\widehat{X}_{\mathrm{nlin},a}=V(\widehat{I}_{\mathrm{nlin},a})$ for the
corresponding subschemes in $Z$.

###### Proposition 5.16

Let $J\subseteq[m]$ be a subset and suppose that

$\mathrm{codim}(\widehat{X}_{\mathrm{lin},a}\cap
H_{J^{\prime}},H_{J^{\prime}})+\mathrm{codim}(\widehat{X}_{\mathrm{nlin},a}\cap{H},H_{J^{\prime}})>\mathrm{dim}(H_{J^{\prime}})$
(3)

for every $J^{\prime}$ with $J^{\prime}\cap J=\emptyset$. Here the
codimensions are taken over the generic point $\eta$ of $Y$. Then
$\ell_{\widehat{X}/Y}=\ell_{X_{U_{J}}/Y}$, where $U_{J}=\bigcap_{j\in
J}D(q_{j})$.

###### Proof.

We first note that every $\widehat{X}_{\mathrm{lin},a}\cap{H}$ is torus-
equivariant. For $H$ as in the statement of the proposition, we can use [MS15,
Theorem 3.6.10] and Proposition 4.2 to conclude that
$(\widehat{X}_{\mathrm{lin},a}\cap{H})\cap(\widehat{X}_{\mathrm{nlin},a}\cap{H})$
is generically empty. Suppose that there is a point $P$ in $X_{U,\eta}$ with
$q_{j}(P)=0$ for some $q_{j}$. Then $q_{j}\notin{T_{1}}$, so we can find a
stratum $H$ such that $P$ gives a point of
$(\widehat{X}_{\mathrm{lin},a}\cap{H})\cap(\widehat{X}_{\mathrm{nlin},a}\cap{H})$.
But the fiber of this scheme over $\eta$ is empty, a contradiction. ∎

###### Remark 5.17

Note that $\mathrm{codim}(\widehat{X}_{\mathrm{lin},a}\cap{H},H)$ is
relatively easy to compute. Computing
$\mathrm{codim}(\widehat{X}_{\mathrm{nlin},a}\cap{H},H)$ on the other hand
generally involves calculating a Gröbner basis, which can be unfeasible. In
certain key cases we can still easily give a a good lower bound for
$\mathrm{codim}(\widehat{X}_{\mathrm{nlin},a}\cap{H},H)$ however, so that we
can again apply Proposition 5.16.

###### Corollary 5.18

Let $J\subset[m]$ be a subset such that for every $f_{i}$ there exists a $j\in
J$ with $p_{i,j}\neq{0}$. Then $\ell_{\widehat{X}/Y}=\ell_{X_{U_{J}}/Y}$,
where $U_{J}=\bigcap_{j\in J}D(q_{j})$.

###### Proof.

Indeed, the condition implies that the codimension of the linear space does
not change. Using Krull’s Hauptidealsatz, one then easily finds that the
inequality from Proposition 5.16 holds for every $H$ with
$S^{\prime}\cap{S_{1}}=\emptyset$. ∎

###### Corollary 5.19

Suppose that for every $f_{i}$, there exists a $j$ with $p_{i,j}\neq{0}$ and
$q_{j}$ monomial. Then $\ell_{X/{Y}}=\ell_{\widehat{X}/{Y}}$.

###### Example 5.20

Consider the system $X=V(f_{1},f_{2})$ over
$Y=\mathrm{Spec}(\mathbb{C}[a_{1},a_{2},a_{3},a_{4}])$ given by the
polynomials

$\displaystyle f_{1}=a_{1}(x-1)+a_{2}(y-2),\qquad\text{and}\qquad
f_{2}=a_{3}(x-1)+a_{4}(y-4).$

The maximal open subsets we obtain from Corollary 5.18 are $D(x-1)$ and
$D(y-2)\cap{D(y-4)}$. Moreover, their union $U$ is not the entire torus. Using
Proposition 5.16 with $J=\emptyset$, we however easily see that the root count
is also valid over $T_{Y}$. We thus see that the maximal compatible subset can
be strictly larger than the ones obtained from Corollary 5.18.

We conclude this section with a few comments on the linear scheme
$\widehat{X}_{\mathrm{lin}}$. In order to make
$\mathrm{trop}(\widehat{X}_{\mathrm{lin},P})$ generic enough, we can use the
following lemma.

###### Lemma 5.21

Consider the non-zero Plücker vectors $z_{i,j}$ of
$\widehat{X}_{\mathrm{lin}}$ as elements of the parameter ring $A$. For any
$P\in{Y^{\mathrm{an}}}$ such that $z_{i,j}(P)\neq{0}$ for all $z_{i,j}$, we
have that

$\ell_{\widehat{X}/Y}=\mathrm{trop}(\widehat{X}_{\mathrm{lin},P})\cdot\mathrm{trop}(\widehat{X}_{\mathrm{nlin},P}).$

###### Proof.

By Lemma 3.7, there is an open neighborhood $V$ of $P$ such that
$|z_{i,j}(Q)|=|z_{i,j}(P)|\neq{0}$ for all $Q\in{V}$. This intersects the open
dense subset from Proposition 4.19, which gives the statement of the lemma. ∎

Let $\Sigma$ be a pure balanced polyhedral complex and let
$\Sigma_{\mathrm{lin}}$ be a tropical linear space of complementary dimension.
We can degenerate $\Sigma_{\mathrm{lin}}$ to its Bergman fan without changing
the intersection number $\Sigma\cdot\Sigma_{\mathrm{lin}}$. This number
therefore only depends on the matroid of the tropical linear space. We can
thus define the following.

###### Definition 5.22

Let $W=V(h_{1},....,h_{m})$ be a closed subscheme determined by linear forms
$h_{i}\in A[x_{1}^{\pm},\dots,x_{n}^{\pm}]$. The _generic matroid_ of $W$ is
the matroid of $W_{\eta}$, where $\eta$ is the generic point of $Y$. Let $M$
be a matroid on $[n]$ and let $\Sigma_{\mathrm{lin}}$ be a tropical linear
space of dimension $\rho(M)$ with matroid $M$. Let $X\subseteq T_{K,n}$ be a
subvariety such that $\mathrm{dim}(X)+\rho(M)=n$. The _matroidal degree_ of
$X$ is

$\deg_{M}X\coloneqq\Sigma_{\mathrm{lin}}\cdot\operatorname{trop}(X).$

###### Corollary 5.23

Let $U\supset\bigcap_{i=1}^{m}{D(q_{i})}$ be the maximal compatible subset.
Then $\ell_{X_{U}/Y}=\deg_{M}(\widehat{X}_{\mathrm{nlin},P})$, where $M$ is
the generic matroid of $\widehat{X}_{\mathrm{lin}}$.

To compute the matroidal degree, it may be advantageous to replace
$\operatorname{trop}(\widehat{X}_{\mathrm{lin}})$ with a different tropical
linear space. For example, it might be beneficial to consider a translate of
its Bergman fan, as the intersection will generally involve fewer maximal
cells.

##  6\. Applications to systems with linear dependencies

In this section, we explore square systems with linear dependencies between
the coefficients of their polynomials. We will focus on two special types of
dependencies which we call vertical and horizontal dependencies. The first is
inspired by the steady-state equations of chemical reaction networks [Dic16],
and the second is inspired by equations with fixed polynomial supports [KK12].
For each of these, either $\widehat{X}_{\text{lin}}$ or
$\widehat{X}_{\text{nlin}}$ is a tropical complete intersection, so that the
tropical intersection product from Section 5.2 simplifies.

###### Assumption 6.1

Throughout this section, the parameter space $Y$ will be the $m$-dimensional
affine space $\mathbb{A}^{m}=\operatorname{Spec}(K[a_{1},\dots,a_{m}])$ and
the coefficients of the polynomials $f_{1},\dots,f_{n}\in
K[a_{1},\dots,a_{m}][x_{1}^{\pm},\dots,x_{n}^{\pm}]$ will be linear and
homogeneous in $a_{1},\dots,a_{m}$.

###  6.1. Square systems with vertical parameter dependencies

In this section, we consider a class of parametrized polynomial systems
inspired by the steady-state equations of chemical reaction networks [Dic16].

###### Definition 6.2

Let $x^{\alpha_{1}},\dots,x^{\alpha_{m}}$ be the monomials of
$f_{1},\dots,f_{n}$. We say $f_{1},\dots,f_{n}$ have _vertical parameter
dependencies_ , if there is a decomposition $A=\bigotimes_{j=1}^{m}A_{j}$ of
the parameter ring $A$, such that each $f_{i}$ is of the form

$f_{i}=\sum_{j=1}^{m}p_{i,j}\cdot x^{\alpha_{j}}\qquad\text{with }p_{i,j}\in
A_{j}.$

In other words, the Macaulay matrix $(p_{i,j})\in A^{n\times m}$ of
$f_{1},\dots,f_{n}$ has algebraic dependencies along its columns, but not
rows.

###### Definition 6.3

Let $f_{1},\dots,f_{n}$ have vertical parameter dependencies. Choosing
$q_{j}=x^{\alpha_{j}}$ in Definition 5.6 we obtain:

$\displaystyle\widehat{C}$ $\displaystyle\coloneqq
A[x_{i}^{\pm},w_{j}^{\pm}\mid 1\leq i\leq n,1\leq j\leq m],$
$\displaystyle\hat{f}_{i}$
$\displaystyle\coloneqq\sum_{j=1}^{m}p_{i,j}w_{j}\qquad\text{for }1\leq i\leq
n,$ $\displaystyle\hat{h}_{j}$ $\displaystyle\coloneqq
w_{j}-x^{\alpha_{j}}\qquad\text{for }1\leq j\leq m.$

By Lemma 5.9, the resulting
$\widehat{X}_{\mathrm{lin}}=V(\langle\hat{f}_{1},\dots,\hat{f}_{n}\rangle)$ is
torus-equivariant. Note moreover that
$\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$ is a fixed linear
subspace independent of $P$. We will refer to the resulting
$\widehat{X}\rightarrow Y$ as the _modification_ for vertical dependencies.

For vertical dependencies, one exceptionally nice property of
$\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$ is that it is a
tropical complete intersection, i.e., it arises as a stable intersection

$\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})=\operatorname{trop}(V(\hat{h}_{1})_{P})\cap_{{\mathrm{st}}}\dots\cap_{{\mathrm{st}}}\operatorname{trop}(V(\hat{h}_{m})_{P}).$

This means that the intersection product with
$\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$ equals the intersection
product with all the $\operatorname{trop}(V(\hat{h}_{j})_{P})$:

###### Proposition 6.4

Let $f_{1},\dots,f_{n}$ have vertical parameter dependencies. Then, for $P$ as
in Lemma 5.21, the generic root count of $X\rightarrow Y$ is

$\mathrm{trop}(\widehat{X}_{{\mathrm{lin}},P})\cdot\prod_{j=1}^{m}\mathrm{trop}(V(\hat{h}_{j})_{P}).$

###### Proof.

Indeed, Corollary 5.19 shows that the maximal compatible subset is $T_{Y}$, so
that $\ell_{X/Y}=\ell_{\widehat{X}/Y}$. ∎

Proposition 6.4 allows us to compute the generic root count via mixed volumes,
eliminating the need to compute any set-theoretic intersections of tropical
hypersurfaces:

###### Remark 6.5

Let $M$ denote the matroid of $\mathrm{trop}(\widehat{X}_{{\mathrm{nlin}},P})$
in Proposition 6.4. If $M$ is _cotransversal_ , then its Bergman fan is a
tropical complete intersection, so that
$B(M)=\operatorname{trop}(V(\hat{\ell}_{1}))\cap_{\mathrm{st}}\dots\cap_{\mathrm{st}}\operatorname{trop}(V(\hat{\ell}_{n}))$
for linear polynomials $\hat{\ell}_{i}$. Consequently, the generic root count
coincides with the mixed volume
$\operatorname{MV}(\hat{\ell}_{1},\dots,\hat{\ell}_{n},\hat{h}_{1},\dots,\hat{h}_{m})$.
If $M$ is not cotransversal, then it can be expressed as a signed linear
combination of cotransversal matroids, see [Ham17]. Consequently, the generic
root count coincides with a signed linear combination of mixed volumes.

We conclude this section with an example that showcases how our techniques can
be applied to steady-state equations of reaction networks.

###### Example 6.6

Consider the following reaction network:

$\small X_{i-1}+E\mathrel{\mathop{\mathrel{\mathop{\xrightleftharpoons{\hskip
19.91692pt}}\limits^{\vbox to0.0pt{\kern-2.0pt\hbox{$\scriptstyle
a_{1,i}$}\vss}}}}\limits_{\vbox to0.0pt{\kern-7.0pt\hbox{$\scriptstyle
d_{1,i}$}\vss}}}Y_{1,i}\mathrel{\mathop{\xrightarrow{\hskip
19.91692pt}}\limits^{\vbox to0.0pt{\kern-2.0pt\hbox{$\scriptstyle
k_{1,i}$}\vss}}}X_{i}+E\hskip
14.22636ptX_{i}+F\mathrel{\mathop{\mathrel{\mathop{\xrightleftharpoons{\hskip
19.91692pt}}\limits^{\vbox to0.0pt{\kern-2.0pt\hbox{$\scriptstyle
a_{2,i}$}\vss}}}}\limits_{\vbox to0.0pt{\kern-7.0pt\hbox{$\scriptstyle
d_{2,i}$}\vss}}}Y_{2,i}\mathrel{\mathop{\xrightarrow{\hskip
19.91692pt}}\limits^{\vbox to0.0pt{\kern-2.0pt\hbox{$\scriptstyle
k_{2,i}$}\vss}}}X_{i-1}+F\hskip 14.22636pti=1,...,n.$

It describes the standard model of distributive sequential phosphorylation on
$n$ sites, see for example [FRW20, Section 2]. Under the assumption of mass-
action kinematics, the corresponding evolution equations are given by

$\displaystyle\dot{x}_{i}$ $\displaystyle=f_{i}\coloneqq-
a_{1,i+1}x_{i}x_{E}-a_{2,i}x_{i}x_{F}+d_{1,i+1}y_{1,i+1}$ $\displaystyle\hskip
71.13188pt+d_{2,i}y_{2,i}+k_{1,i}y_{1,i}+k_{2,i+1}y_{2,i+1},$ $\displaystyle
i$ $\displaystyle=0,\dots,n,$ $\displaystyle\dot{y}_{1,i}$
$\displaystyle=f_{1,i}\coloneqq a_{1,i}x_{i-1}x_{E}-(d_{1,i}+k_{1,i})y_{1,i},$
$\displaystyle i$ $\displaystyle=1,\dots,n,$ $\displaystyle\dot{y}_{2,i}$
$\displaystyle=f_{2,i}\coloneqq a_{2,i}x_{i}x_{F}-(d_{2,i}+k_{2,i})y_{2,i},$
$\displaystyle i$ $\displaystyle=1,\dots,n,$ $\displaystyle\dot{x}_{E}$
$\displaystyle=f_{E}\coloneqq\sum_{i=1}^{n}-a_{1,i}x_{i-1}x_{E}+(d_{1,i}+k_{1,i})y_{1,i},$
$\displaystyle\dot{x}_{F}$
$\displaystyle=f_{F}\coloneqq\sum_{i=1}^{n}-a_{2,i}x_{i}x_{F}+(d_{2,i}+k_{2,i})y_{2,i},$

where $a_{2,0}=d_{2,0}=k_{1,0}=0$ and $a_{1,n+1}=d_{1,n+1}=k_{2,n+1}=0$. The
solutions of the evolution equations describe a three-dimensional set of
steady states. In order to obtain finitely many solutions, observe that the
following quantities are conserved:

$\displaystyle E_{tot}$ $\displaystyle=g_{E}\coloneqq
x_{E}+\sum_{i=1}^{n}y_{1,i}-c_{E},$ $\displaystyle F_{tot}$
$\displaystyle=g_{F}\coloneqq x_{F}+\sum_{i=1}^{n}y_{2,i}-c_{F},$
$\displaystyle X_{tot}$
$\displaystyle=g_{X}\coloneqq\sum_{i=0}^{n}x_{i}+\sum_{i=1}^{n}(y_{1,i}+y_{2,i})-c_{F}$

To make the system square, we can omit a suitable subset of the evolution
equations depending on the conservation equations. In the system above, we may
omit $f_{0},f_{E},f_{F}$ and consider the system consisting of
$f_{i},f_{1,i},f_{2,i}$ for $i=1,\dots,n$ and $g_{E},g_{F},g_{X}$.

If the conservation equations $g_{E}$, $g_{F}$ and $g_{X}$ were generic, in
the sense that each of its monomials come with a unique parameter as in the
evolution equations, then the modifications in Definition 6.3 would be
immediately applicable, so that the generic root count is given by the
tropical intersection number in Proposition 6.4.

To address non-generic conservation equations, a slight adaptation of the
modification and decomposition in Definition 6.3 suffices: Let
$\hat{f}_{i},\hat{f}_{1,i},\hat{f}_{2,i}$ and $\hat{h}_{j}$ be defined as in
Definition 6.3. This means that the $\hat{f}_{i},\hat{f}_{1,i},\hat{f}_{2,i}$
are linear and that there is an $\hat{h}_{j}$ per vertex in the reaction
network. Set

$\displaystyle\widehat{X}_{\mathrm{lin}}$ $\displaystyle\coloneqq
V(\hat{f}_{i},\hat{f}_{1,i},\hat{f}_{2,i}\mid i=1,\dots,n),$
$\displaystyle\widehat{X}_{\text{con}}$ $\displaystyle\coloneqq
V(g_{E},g_{F},g_{X}),$ $\displaystyle\widehat{X}_{\mathrm{nlin}}$
$\displaystyle\coloneqq V(\hat{h}_{j}\mid j=1,\dots,m).$

One can show that $\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$ and
$\operatorname{trop}(\widehat{X}_{\text{con},P})$ intersect transversally, and
that $\widehat{X}_{\mathrm{lin}}$ is equivariant and of complementary
dimension. Hence the conditions in Proposition 4.19 are satisfied, so that

$\ell_{X/Y}=\operatorname{trop}(\widehat{X}_{\mathrm{lin},P})\cdot\operatorname{trop}(\widehat{X}_{\mathrm{nlin},P})\cdot\operatorname{trop}(\widehat{X}_{\mathrm{con},P})$

for $P$ in an open dense subset of $Y^{\mathrm{an}}$. We thus see that the
generic root count remains expressible as a tropical intersection product
between a tropical linear space
$\operatorname{trop}(\widehat{X}_{\mathrm{lin},P})\cap\operatorname{trop}(\widehat{X}_{\mathrm{con},P})$
and a classical linear space
$\operatorname{trop}(\widehat{X}_{\mathrm{nlin},P})$. The above technique
works for arbitrary chemical reaction networks, provided that the generic
codimension of $\widehat{X}_{\mathrm{lin}}$ is as expected as in Lemma 5.8.

###  6.2. Square systems with horizontal parameter dependencies

In this section, we consider a class of parametrized square systems inspired
by the work of Kaveh and Khovanskii [KK12].

###### Definition 6.7

Let $x^{\alpha_{1}},\dots,x^{\alpha_{s}}$ be the monomials of
$f_{1},\dots,f_{n}$. We say $f_{1},\dots,f_{n}$ have _horizontal parameter
dependencies_ , if there is a decomposition of the parameter ring
$A=\bigotimes_{K}A_{i}$, such that each $f_{i}$ is of the form

$f_{i}=\sum_{j=1}^{s}p_{i,j}\cdot x^{\alpha_{j}}\qquad\text{with }p_{i,j}\in
A_{i}.$

In other words, the Macaulay matrix $(p_{i,j})\in A^{n\times s}$ of
$f_{1},\dots,f_{n}$ has algebraic dependencies along its rows, but not
columns. Note that we can also write $f_{i}$ as

$f_{i}=\sum_{j=1}^{m}a_{i,j}\cdot q_{j}\qquad\text{with }a_{i,j}\in A_{i}.$

where the non-zero $a_{i,j}$ are algebraically independent.

###### Definition 6.8

Let $f_{1},\dots,f_{n}$ have horizontal parameter dependencies. Choosing the
$q_{j}$ in Definition 5.6 to be the $q_{i,j}$ of Definition 6.7, we obtain:

$\displaystyle\widehat{C}$ $\displaystyle\coloneqq
A[x_{i}^{\pm},w_{j}^{\pm}\mid 1\leq i\leq n,1\leq j\leq m],$
$\displaystyle\hat{f}_{i}$
$\displaystyle\coloneqq\sum_{j=1}^{m}a_{i,j}w_{j}\qquad\text{for }1\leq i\leq
n,$ $\displaystyle\hat{h}_{j}$ $\displaystyle\coloneqq
w_{j}-q_{j}\qquad\text{for }1\leq i\leq n,1\leq j\leq m.$

As before by Lemma 5.9, the resulting
$\widehat{X}_{\mathrm{lin}}=V(\langle\hat{f}_{1},\dots,\hat{f}_{n}\rangle)$ is
torus-equivariant and $\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})$
is independent of $P$. We will refer to the resulting $\widehat{X}\rightarrow
Y$ as the _modification_ for horizontal dependencies.

Recall that, for vertical dependencies,
$\operatorname{trop}(\widehat{X}_{\mathrm{lin}})$ can be a general tropical
linear space while $\operatorname{trop}(\widehat{X}_{\mathrm{nlin}})$ is a
tropical complete intersection. In a reversal of roles, for horizontal
dependencies, $\operatorname{trop}(\widehat{X}_{\mathrm{lin}})$ is a tropical
complete intersection while $\operatorname{trop}(\widehat{X}_{\mathrm{nlin}})$
can be a general tropical variety:

###### Proposition 6.9

Let $f_{1},\dots,f_{n}$ have horizontal parameter dependencies and let $U$ be
the maximal compatible subset. Then, for $P$ as in Lemma 5.21, the generic
root count of $X_{U}\rightarrow Y$ is

$\mathrm{trop}(\widehat{X}_{{\mathrm{nlin}},P})\cdot\prod\mathrm{trop}(V(\widehat{f}_{i})_{P}).$

######  Remark 6.10 (Comparison to the works of Kaveh and Khovanskii)

Let $\mathcal{X}$ be an $n$-dimensional irreducible variety over the complex
numbers, and let $L_{1},\dots,L_{n}\subset\mathbb{C}(\mathcal{X})$ be linear
subspaces of the function field. In [KK10], Kaveh and Khovanskii define the
_birational intersection index_ $[L_{1},...,L_{n}]$, which records the generic
number of solutions to the system given by $h_{1},...,h_{n}$, where $h_{i}$
are generic elements of $L_{i}$, outside a suitable support of the $L_{i}$. In
[KK12], a suitable higher-rank valuation
$\mathbb{C}(X)\backslash\\{0\\}\to\mathbb{Z}^{n}$ is used to attach a convex
body $\Delta_{L}$, the _Newton-Okounkov body_ , to any subspace $L$. In [KK12,
Theorem 4.9] it is then proved that

$[L,...,L]=\dfrac{n!\cdot\mathrm{deg}(\Phi_{L})}{\mathrm{ind}(A_{L})}\mathrm{vol}(\Delta_{L}),$

where $\Phi_{L}$ is the Kodaira map and $\mathrm{ind}(A_{L})$ is the index of
a certain subgroup, see [KK10, KK12] for more details. If the Kodaira map is a
birational isomorphism, then both $\mathrm{deg}(\Phi_{L})$ and
$\mathrm{ind}(A_{L})$ are equal to $1$, so that the formula reduces to
$[L,...,L]=n!\mathrm{vol}(\Delta_{L})$.

In comparison, our paper always assumes that the ambient variety $\mathcal{X}$
is a torus, though it may be over any field $K$. Let
$q_{i,1},\dots,q_{i,k_{i}}$ be a basis of $L_{i}$ and consider

$f_{i}=\sum_{j=1}^{k_{i}}a_{i,j}q_{i,j}\qquad\text{for }i=1,\dots,n.$

These give a square system $X\to{Y}$ with horizontal parameter dependencies.
The open subset used in [KK12, Definition 4.5 (1)] is a subset of the maximal
compatible subset. Namely, if we write
$Z_{i}=\bigcap_{j:p_{i,j}\neq{0}}V(q_{j})$, then $Z_{\mathbf{L}}$ from [KK12,
Section 4.2] is $\bigcup_{i=1}^{n}Z_{i}$. Its complement is then easily seen
to be a union of open subsets $U$ as in Corollary 5.18.

In [KK12, Definition 4.5 (2)], it is furthermore required that the solutions
are non-degenerate in the sense that the generic fiber of the morphism
$X\to{Y}$ is étale. This will generally not be the case if the characteristic
of the base field $K$ is positive. For example, the parametrized polynomial
$f=a_{1}+a_{2}x^{p}$ over the ring $A=\mathbb{F}_{p}[a_{1},a_{2}]$ has generic
root count $p$, but the number of generic solutions where the morphism is
étale is zero. If the characteristic of the base field $K$ is zero, then the
proof of [KK10, Proposition 5.7] implies that $X_{U}\to{Y}$ is generically
étale for the maximal compatible subset $U$, from which we obtain

$[L_{1},\dots,L_{n}]=\deg_{M}(\widehat{X}_{{\mathrm{nlin}},P})=\ell_{X_{U}/Y}$

using Corollary 5.23 and our earlier observation on the open subset used in
the definition of $[L_{1},...,L_{n}]$.

We end this section with two examples from literature, in which we highlight
two different ideas to simplify the tropical intersection product in
Proposition 6.9:

  1. (1)

In Example 6.11, $\widehat{X}_{\mathrm{nlin}}$ is quasi-linear in the sense
that it is the preimage of a linear space under a finite toric morphism.

  2. (2)

In Example 6.12, $\widehat{X}_{\mathrm{nlin}}$ is a tropical complete
intersection and we showcase how one can simplify the resulting mixed volume.

######  Example 6.11 (Kuramoto model)

Consider the following polynomials whose zero sets give the stationary
equations of the Kuramoto model [CMMN19, Equation F3]:

$f_{i}\coloneqq\sum_{j=1}^{N}a_{i,j}(x_{i}x_{j}^{-1}-x_{j}x_{i}^{-1})-b_{i}\quad\text{for
}i=1,\dots,N-1\text{ and }x_{N}\coloneqq 1.$ (4)

Note that the parameters were renamed to $a_{i,j}$ and $b_{i}$, and some
constants which do not change the generic root count were omitted. Moreover,
[CMMN19] also studies specializations of (4) where some $a_{i,j}$ are set to
$0$. In our set-up, this degenerates the matroid corresponding to
$\widehat{X}_{\mathrm{lin}}$ and leaves $\widehat{X}_{\mathrm{nlin}}$
unchanged.

The modification in Definition 6.8 yields

$\displaystyle\widehat{C}$ $\displaystyle\coloneqq
A[x_{i}^{\pm},w_{i,j}^{\pm}\mid 1\leq i,j<N,i\neq j],$
$\displaystyle\hat{f}_{i}$ $\displaystyle\coloneqq\sum_{j\neq
i}a_{i,j}w_{i,j}-b_{i}\qquad\text{for }1\leq i<N,$
$\displaystyle\hat{h}_{i,j}$ $\displaystyle\coloneqq
w_{i,j}-(x_{i}x_{j}^{-1}-x_{j}x_{i}^{-1})\qquad\text{for }1\leq i,j<N,i\neq
j.$

Note that the maximal compatible subset here is $T_{Y}$. By Proposition 6.9,
the generic root count equals
$\operatorname{trop}(\widehat{X}_{{\mathrm{nlin}},P})\cdot\prod_{i=1}^{N-1}\operatorname{trop}(V(\widehat{f}_{i})_{P})$.
We determine the polyhedral complex
$\mathrm{trop}(\widehat{X}_{\mathrm{nlin},P})$ by pulling back the
tropicalization of a suitable linear space under a toric map.

We first define an automorphism $\kappa_{1}$ of the torus by sending $x_{i}$
to $x_{i}$, and $w_{i,j}$ to $w^{\prime}_{i,j}\coloneqq x_{i}x_{j}w_{i,j}$. We
then define a Kummer map $\kappa_{2}$ by sending $x_{i}$ to $z_{i}\coloneqq
x_{i}^{2}$ and $w^{\prime}_{i,j}$ to $w^{\prime}_{i,j}$. The composition
$\kappa=\kappa_{2}\circ\kappa_{1}$ defines a finite map of tori of degree
$2^{N-1}$ and the space $X_{\mathrm{nlin}}$ is the inverse image of the linear
space $\widehat{X}_{\mathrm{lin}^{\prime}}=\bigcap
V(w^{\prime}_{i,j}-(z_{i}-z_{j}))$ under $\kappa$. One checks that the
circuits of this linear space are given by

$\displaystyle w^{\prime}_{i,j}+w^{\prime}_{j,i},$ $\displaystyle
w^{\prime}_{i,j}-w^{\prime}_{i,k}-w^{\prime}_{k,j},$ $\displaystyle
w^{\prime}_{i,j}+w^{\prime}_{j,k}-w^{\prime}_{m,k}-w^{\prime}_{i,m},$
$\displaystyle w^{\prime}_{i,j}-z_{i}+z_{j},$ $\displaystyle
w^{\prime}_{i,j}-w^{\prime}_{i,k}-z_{k}+z_{j}.$

These form a tropical basis by [MS15, Proposition 4.1.6]. One then obtains a
tropical basis for $\widehat{X}_{\mathrm{nlin}}$ by pulling these forms back
under $\kappa$. This yields a formula for the generic root count in terms of
two explicit polyhedral complexes.

###### Example 6.12

Fix $N>0$, and consider the following polynomials for $i=1,\dots,N$:

$\displaystyle f_{i}$
$\displaystyle=a_{1,i}u_{i}(u_{i}^{2}+v_{i}^{2})+a_{2,i}u_{i}+a_{3,i}v_{i}+a_{4,i}+\sum_{j\neq{i}}c_{j,i}v_{j},$
$\displaystyle g_{i}$
$\displaystyle=b_{1,i}v_{i}(u_{i}^{2}+v_{i}^{2})+b_{2,i}u_{i}+b_{3,i}v_{i}+b_{4,i}+\sum_{j\neq{i}}d_{j,i}u_{j}.$

Here, the $a_{j,i},b_{j,i},c_{j,i},d_{j,i}$ are the parameters and the
$u_{i},v_{i}$ are the variables. This polynomial system describes the steady
states of coupled Duffing oscillators. In [BMMT22], Breiding, Michałek, Monid
and Telen use Newton-Okounkov bodies and Khovanskii bases to show that the
generic root count of this system is $5^{N}$. We will show here how the same
root count can be obtained using our results.

Applying our modifications in Definition 6.8 yields:

$\displaystyle\hat{f}_{i}=$ $\displaystyle
a_{1,i}w_{i,1}+a_{2,i}u_{i}+a_{3,i}v_{i}+a_{4,i}+\sum_{j\neq{i}}c_{j,i}v_{j},$
$\displaystyle\hat{h}_{i,1}=$ $\displaystyle
w_{i,1}-u_{i}(u_{i}^{2}+v_{i}^{2}),$ $\displaystyle\hat{g}_{i}=$
$\displaystyle
b_{1,i}w_{i,2}+b_{2,i}u_{i}+b_{3,i}v_{i}+b_{4,i}+\sum_{j\neq{i}}d_{j,i}u_{j},$
$\displaystyle\hat{h}_{i,2}=$ $\displaystyle
w_{i,2}-v_{i}(u_{i}^{2}+v_{i}^{2}),$

which we can reformulate to the following generating set

$\displaystyle\hat{f}_{i}=$ $\displaystyle
a_{1,i}w_{i,1}+a_{2,i}u_{i}+a_{3,i}v_{i}+a_{4,i}+\sum_{j\neq{i}}c_{j,i}v_{j},$
$\displaystyle\hat{h}_{i}=$ $\displaystyle
w_{i,1}-u_{i}(u_{i}^{2}+v_{i}^{2}),$ (5) $\displaystyle\hat{g}_{i}=$
$\displaystyle
b_{1,i}w_{i,2}+b_{2,i}u_{i}+b_{3,i}v_{i}+b_{4,i}+\sum_{j\neq{i}}d_{j,i}u_{j},$
$\displaystyle\hat{\rho}_{i}=$ $\displaystyle v_{i}w_{i,1}-u_{i}w_{i,2}.$

As before, we have that the maximal compatible subset is $T_{Y}$, so that
$\ell_{X/Y}=\ell_{\widehat{X}/Y}$. We will first show that the
$\operatorname{trop}(V(\hat{h}_{i})_{P})$ and
$\operatorname{trop}(V(\hat{\rho}_{i})_{P})$ intersect transversally, which
combined with Proposition 6.9 implies that the generic root count is the mixed
volume of the polynomials in System (5). We then use previous work by Bihan
and Soprunov [BS19] to reduce this mixed volume to $5^{N}$.

To see that the aforementioned tropical hypersurfaces intersect transversally,
observe that $\operatorname{trop}(V(\hat{h}_{i})_{P})$ consists of three
maximal cells, while $\operatorname{trop}(V(\hat{\rho}_{i})_{P})$ consists of
only one. Letting $\sigma_{i}$ denote a maximal cell of
$\operatorname{trop}(V(\hat{h}_{i})_{P})$ and $\tau_{i}$ the maximal cells of
$\operatorname{trop}(V(\hat{\rho}_{i})_{P})$, we have

$\sigma_{i}\subseteq\begin{cases}(e_{w_{i,1}}-3e_{u_{i}})^{\perp}&\text{or}\\\
(e_{w_{i,1}}-e_{u_{i}}-2e_{v_{i}})^{\perp}&\text{or}\\\
(2e_{u_{i}}-2e_{v_{i}})^{\perp},\end{cases}$

and

$\tau_{i}=(e_{v_{i}}+e_{w_{i,1}}-e_{u_{i}}+e_{w_{i,2}})^{\perp}.$

It is straightforward to check that, regardless of the choice of $\sigma_{i}$,
the normal vectors of $\sigma_{1},\tau_{1},\dots,\sigma_{N},\tau_{N}$
specified above will always be linearly independent, which in turn implies
that the cells intersect transverally. This can for example be done by
constructing a matrix of normal vectors, where the rows are indexed by the
maximal cells and the columns are indexed by the unit vectors in the following
ordering:

${e_{w_{1,2}}}$${\cdots}$${e_{w_{N,2}}}$${e_{w_{1,1}}\;e_{u_{1}}\;e_{v_{1}}}$${\cdots}$${e_{w_{N,1}}\;e_{u_{N}}\;e_{v_{N}}}$${\tau_{1}}$${1}$
${\vdots}$${\tau_{N}}$${1}$ ${\sigma_{1}}$${1}$
${\vdots}$${\sigma_{N}}$${1}$$\ast$$\ast$$\ast$$\ast$$\ast$$\ast$$\ast$$\left(\vbox{\hrule
height=52.1612pt,depth=52.1612pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=52.1612pt,depth=52.1612pt,width=0.0pt}\right)$

Regardless of the choice of $\sigma_{i}$, the matrix of normal vectors will
always be in row-echelon form, and hence of full rank.

To show that the mixed volume of the Newton polytopes of the polynomials in
Equation 5 is $5^{N}$, recall [BS19, Proposition 3.2] which states that for
polytopes $Q_{1},\dots,Q_{n}$ and $P_{1}\subseteq Q_{1}$ in $\mathbb{R}^{n}$:

$\displaystyle\operatorname{MV}(P_{1},Q_{2},\dots,Q_{n})=\operatorname{MV}(Q_{1},Q_{2},\dots,Q_{n})$
$\displaystyle\quad\Longleftrightarrow\quad\forall u\in\mathbb{R}^{n}\text{
with }\operatorname{MV}(Q_{2}^{u},\dots,Q_{n}^{u})>0:\;P_{1}\cap
Q_{1}^{u}\neq\emptyset.$ (6)

Here $P^{u}$ denotes the face of $P$ minimizing scalar product by $u$ and the
polytopes in $\operatorname{MV}(Q_{2}^{u},\dots,Q_{n}^{u})$ are considered to
be polytopes of $u^{\perp}\cong\mathbb{R}^{n-1}$. We will use this result to
show that the Newton polytopes of $\hat{f}_{i}$ and $\hat{g}_{i}$ can be
replaced by the Newton polytopes of

$a_{1,i}w_{i,1}+a_{2,i}u_{i}+a_{3,i}v_{i}+a_{4,i}\qquad\text{and}\qquad
b_{1,i}w_{i,2}+b_{2,i}u_{i}+b_{3,i}v_{i}+b_{4,i}$

without changing the mixed volume. A quick computation then reveals the mixed
volume to be $5^{N}$.

Let $(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ be a vector whose minimum on
the Newton polytope of $\hat{f}_{i}$ is uniquely attained at a vertex
corresponding to a monomial in $\sum_{j\neq i}c_{i,j}v_{j}$. Here,
$\dutchcal{u}_{i},\dutchcal{v}_{i},\dutchcal{w}_{i,j}$ represent weights on
the variables $u_{i},v_{i},w_{i,j}$. Without loss of generality, we may assume
that $i=1$ and that the monomial is $v_{2}$, so that the assumption implies:

$\dutchcal{v}_{2}<\dutchcal{w}_{1,1},\quad\dutchcal{v}_{2}<\dutchcal{u}_{1},\quad\dutchcal{v}_{2}<0,\quad\text{and}\quad\dutchcal{v}_{2}<\dutchcal{v}_{j}\text{
for }j\neq 2.$

We will show that the mixed volume in Expression (6) is zero. This is done by
assuming that none of the polytopes in it are vertices and proving that it
either leads to a contradiction or to mixed volume $0$.

In the following, we will use ($p$) as a shorthand for a tropical equation. In
other words, ($p$) means that the Newton polytope of $p$ minimizing
$(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is not a vertex, or equivalently
that the minimum of
$\operatorname{trop}(p)(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is attained
at least twice. We distinguish between three cases:

$\dutchcal{u}_{2}<\dutchcal{v}_{2}$:

From $(\hat{h}_{2})$ and $(\hat{f}_{2})$ we obtain
$\dutchcal{w}_{2,1}=3\dutchcal{u}_{2}$ and
$\dutchcal{w}_{2,1}=\dutchcal{u}_{2}$, respectively. Together, they imply
$\dutchcal{u}_{2}=0$, which contradicts $\dutchcal{u}_{2}<\dutchcal{v}_{2}<0$.

$\dutchcal{u}_{2}>\dutchcal{v}_{2}$:

From $(\hat{\rho}_{2})$ we obtain
$\dutchcal{w}_{2,2}=3\dutchcal{v}_{2}<\dutchcal{v}_{2}$, the last inequality
simply following from $\dutchcal{v}_{2}<0$. By $(\hat{g}_{2})$ we then get
that $\dutchcal{w}_{2,2}=\dutchcal{u}_{j}$ for some $j\neq 2$. We therefore
have $\dutchcal{u}_{j}<\dutchcal{v}_{2}$. Considering $(\hat{f}_{j})$, we thus
obtain $\dutchcal{w}_{j,1}=\dutchcal{u}_{j}<\dutchcal{v}_{j}$. From
$\dutchcal{u}_{j}<\dutchcal{v}_{j}$ and $(\hat{h}_{j})$ we get
$\dutchcal{w}_{j,1}=3\dutchcal{u}_{j}$, which together with the previous
$\dutchcal{w}_{j,1}=\dutchcal{u}_{j}$ implies $\dutchcal{u}_{j}=0$. This
contradicts $\dutchcal{u}_{j}<\dutchcal{v}_{2}<0$.

$\dutchcal{u}_{2}=\dutchcal{v}_{2}$:

From $(\hat{\rho}_{2})$ and $(\hat{h}_{2})$ we obtain
$\dutchcal{w}_{2,1}=\dutchcal{w}_{2,2}$ and $\dutchcal{w}_{2,1}\geq
3\dutchcal{u}_{2}=\dutchcal{u}_{2}+2\dutchcal{v}_{2}$ respectively. We again
distinguish between three cases:

$\dutchcal{u}_{2}=\dutchcal{v}_{2}>\dutchcal{w}_{2,1}=\dutchcal{w}_{2,2}$:

The minimum of
$\operatorname{trop}(f_{2})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is
attained uniquely at $\dutchcal{w}_{2,1}$, hence the mixed volume is $0$.

$\dutchcal{u}_{2}=\dutchcal{v}_{2}<\dutchcal{w}_{2,1}=\dutchcal{w}_{2,2}$:

Suppose that $\dutchcal{v}_{2}<\dutchcal{u}_{i}$ for all $i\neq 2$. Then the
minimum of both tropical polynomials $\operatorname{trop}(f_{2})$ and
$\operatorname{trop}(g_{2})$ evaluated at
$(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is attained at the monomials
$u_{2}$ and $v_{2}$, hence the mixed volume is $0$.

Suppose that $\dutchcal{v}_{2}\geq\dutchcal{u}_{i}$ for some $i\neq 2$. Then
$\dutchcal{u}_{i}\leq\dutchcal{v}_{2}<\dutchcal{v}_{i}$ and from
$(\hat{h}_{i})$ we obtain
$\dutchcal{w}_{i,1}=3\dutchcal{u}_{i}<\dutchcal{u}_{i}$. This implies that the
minimum of $\operatorname{trop}(\hat{f}_{i})$ evaluated at
$(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is uniquely attained at the
monomial $w_{i,1}$, contradicting $(\hat{f}_{i})$.

$\dutchcal{u}_{2}=\dutchcal{v}_{2}=\dutchcal{w}_{2,1}=\dutchcal{w}_{2,2}$:

From $(\hat{h}_{2})$ we obtain $\dutchcal{w}_{2,1}\geq 3\dutchcal{u}_{2}$.
Combined with the assumptions $\dutchcal{u}_{2}=\dutchcal{w}_{2,1}$ and
$\dutchcal{w}_{2,1}=\dutchcal{v}_{2}<0$, this implies
$\dutchcal{w}_{2,1}>3\dutchcal{u}_{2}$. Hence the minimum of
$\operatorname{trop}(\hat{h}_{2})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is
attained uniquely at $u_{2}^{3}$ and $u_{2}v_{2}^{2}$. Our assumptions on
$\hat{f}_{1}$ imply that the minimum in
$\operatorname{trop}(\hat{f}_{2})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is
attained at $w_{2,1}$, $u_{2}$, and $v_{2}$. We again distinguish between
three cases:

If the minimum in
$\operatorname{trop}(\hat{g}_{2})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is
attained at $v_{2}$ and $u_{j}$ for $j\neq{2}$, then from our initial
assumption, we obtain $\dutchcal{u}_{j}<\dutchcal{v}_{j}$ and thus
$\dutchcal{w}_{j,1}=3\dutchcal{u}_{j}<\dutchcal{u}_{j}$. But then the minimum
in $\operatorname{trop}(\hat{f}_{j})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$
is uniquely attained at $w_{j,1}$, contradicting $(\hat{f}_{j})$.

If the minimum in
$\operatorname{trop}(\hat{g}_{2})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is
attained at $u_{j}$ and $u_{k}$ for $j,k\neq{2}$ and $j\neq k$, then the
minimum in
$\operatorname{trop}(\hat{f}_{j})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is
attained at $w_{j,1}$ and $u_{j}$. As before, this implies that
$\dutchcal{w}_{j,1}=3\dutchcal{u}_{j}=0$, which contradicts $(\hat{f}_{j})$.

The remaining case is where the minimum in
$\operatorname{trop}(\hat{g}_{2})(\dutchcal{u},\dutchcal{v},\dutchcal{w})$ is
attained at $w_{2,2}$, $u_{2}$, and $v_{2}$. But then the mixed volume of the
Newton polytopes of $\operatorname{in}(h_{i,1})=u_{2}(u_{2}^{2}+v_{2}^{2})$,
$\operatorname{in}(\rho_{2})$, $\operatorname{in}(f_{2})$ and
$\operatorname{in}(g_{2})$ is zero, since they contain a common lineality
space.

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