# Dimension and degeneracy of solutions of parametric polynomial systems
arising from reaction networks

Elisenda Feliu, Oskar Henriksson, and Beatriz Pascual-Escudero

###### Abstract.

We study the generic dimension of the solution set over $\mathbb{C}^{*}$,
$\mathbb{R}^{*}$ and $\mathbb{R}_{>0}$ of parametric polynomial systems that
consist of linear combinations of monomials scaled by free parameters. We
establish a relation between this dimension, Zariski denseness of the set of
parameters for which the system has solutions, and the existence of
nondegenerate solutions, which enables fast dimension computations. Systems of
this form are used to describe the steady states of reaction networks modeled
with mass-action kinetics, and as a corollary of our results, we prove that
weakly reversible networks have finitely many steady states for generic
reaction rate constants and total concentrations.

##  1\. Introduction

In this work we study parametric (Laurent) polynomial systems of the form

$g(\alpha,x)={0},\quad{x}\in\mathcal{X}\subseteq(\mathbb{C}^{*})^{n},\quad\alpha\in\mathcal{A}\subseteq\mathbb{C}^{\ell},$
(1.1)

where $g(\alpha,x)\in\mathbb{C}[\alpha,x^{\pm}]^{s}$ is a tuple of $s$
linearly independent polynomials in parameters
$\alpha=(\alpha_{1},\ldots,\alpha_{\ell})$ restricted to some set
$\mathcal{A}\subseteq\mathbb{C}^{\ell}$, and variables
$x=(x_{1},\ldots,x_{n})$ restricted to some set
$\mathcal{X}\subseteq(\mathbb{C}^{*})^{n}$.

Such systems arise naturally in many applications. Important examples include
reaction network theory, where $\mathcal{X}=\mathbb{R}^{n}_{>0}$ and $x$
represents concentrations or abundances at steady states [Dic16, Fei19],
algebraic statistics, where one often takes $\mathcal{X}$ to be the interior
of the probability simplex and $x$ to be parameters of discrete probability
distributions [Sul18], or robotics, where $\mathcal{X}=(\mathbb{R}^{*})^{n}$
and $x$ represents configurations of various mechanisms [CR04].

A fundamental problem is to determine the _generic dimension_ of the algebraic
variety defined by the solutions to the system of interest, and, in
particular, whether it agrees with the lower bound $n-s$ derived from the
number of linearly independent equations.

For sparse polynomial systems with fixed support and freely varying parametric
coefficients, this lower bound is always the generic dimension. As an
illustration of this, consider the simplest example, namely a system of linear
equations with free coefficients:

$\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}=0,\quad\alpha_{4}x_{1}+\alpha_{5}x_{2}+\alpha_{6}=0,\quad{x}\in(\mathbb{C}^{*})^{2},\quad\alpha\in\mathbb{C}^{4}.$
(1.2)

The generic dimension of the solution set is $0$, as there will be a unique
solution in $(\mathbb{C}^{*})^{2}$ whenever
$(\alpha_{1}\alpha_{5}-\alpha_{2}\alpha_{4})(\alpha_{1}\alpha_{6}-\alpha_{3}\alpha_{4})(\alpha_{2}\alpha_{6}-\alpha_{3}\alpha_{5})\neq
0$. On the other hand, when there are algebraic dependencies between the
coefficients, the situation can be more complicated. Consider for example the
system

$\alpha_{1}x_{1}-\alpha_{2}x_{2}=0,\quad\alpha_{1}^{2}x_{1}^{2}-\alpha_{2}^{2}x_{2}^{2}=0,\quad{x}\in(\mathbb{C}^{*})^{2},\quad\alpha\in\mathbb{C}^{2}.$
(1.3)

As the first polynomial divides the second, the generic complex dimension is
$1$, despite the system having two (linearly independent) equations, and hence
a lower dimension bound of $0$.

The motivation of this work comes from the theory of reaction networks, where
the parametric systems of interest are of the form

$f_{\kappa}(x)=N\operatorname{diag}(\kappa){x}^{B},\quad{x}\in\mathbb{R}_{>0}^{n},\quad\kappa\in\mathbb{R}^{r}_{>0}\mbox{,}$
(1.4)

with $N\in\mathbb{Z}^{s\times r}$, $\operatorname{rk}(N)=s$, and
$B\in\mathbb{Z}^{n\times r}$. This system describes the steady states of a
reaction network under the assumption of mass-action kinetics (more details
are given in Section 4). Even though the study of reaction networks in the
current mathematical formalism goes back at least to Feinberg, Horn and
Jackson in the 70’s [Fei72, HJ72], many fundamental questions about the system
(1.4) remain unclear, including the question about the generic dimension of
the solution sets.

Since each parameter $\kappa_{i}$ might appear as a coefficient in several
equations, it is at first glance not obvious whether the generic dimension can
deviate from $n-s$, similarly to (1.3). There are well-known examples of
networks where the solution set always has a higher dimension than $n-s$
whenever it is nonempty (see e.g. [Fei87, Appendix IV]), but for all such
networks, the solution set has the additional pathology of being empty for
almost all parameter values.

Analogous observations have been made for a second parametric system arising
in reaction network theory, which describes the steady states constrained by
the linear first integrals of the underlying system of differential equations:

$N\operatorname{diag}(\kappa){x}^{B}=0,\quad Wx=c,\quad
x\in\mathbb{R}^{n}_{>0},\quad(\kappa,c)\in\mathbb{R}^{r}_{>0}\times\mathbb{R}^{n-s},$
(1.5)

with $W\in\mathbb{Z}^{(n-s)\times n}$ of full rank. Here, the empirical
observation is that for all realistic networks, the solution sets are finite
(i.e., 0-dimensional) for generic parameters $\kappa$ and $c$, and if not,
then the solution sets are generically empty.

In this work, we settle these questions and confirm that the pathologies of
higher dimension and generic emptiness always go hand in hand. If the complex
solutions to (1.4) do not form a variety of dimension $n-s$ for no
$\kappa\in\mathbb{R}^{r}_{>0}$, then at the same time, it holds that the
variety is empty for all $\kappa$ outside a proper Zariski closed subset.
Likewise, if (1.5) has either infinitely many or no solutions for all
$(\kappa,c)$, then it also has no solution for any $(\kappa,c)$ outside of a
proper Zariski closed subset.

Our second main contribution is that deciding whether the set of parameters
for which the system has solutions is Zariski dense reduces to checking
whether the system has a nondegenerate solution (in the sense that the rank of
the Jacobian agrees with the number of equations) for _at least one choice_ of
parameters. This can be easily checked computationally, by inspecting the
generic rank of a family of matrices. In practice, this turns out to be a
significantly cheaper computation than the standard methods for computing the
generic dimension based on Gröbner bases (see e.g. [CLO15, Section 9.3] for a
discussion of such methods).

To obtain these results, we study first general families of parametric systems
of the form (1.1). Background theory establishes that the generic dimension of
the complex variety of solutions to $g(\alpha,x)=0$ for a fixed $\alpha$ can
be determined by employing the Theorem on the Dimension of Fibers, after
checking that the incidence variety of the parametric system is irreducible.
Then, the generic dimension depends on the dimension of the incidence variety
as well as on whether the set of parameters $\mathcal{D}$ for which the
variety is nonempty is Zariski dense in parameter space. This theory applies
to the examples in (1.2) and (1.3) above. The underlying arguments are
reviewed in Section 2, where we also compare the condition of $\mathcal{D}$
being Zariski dense to having nonempty Euclidean interior in $\mathcal{A}$.
The latter can be studied by means of the concept of nondegeneracy.

With these tools in place, we proceed in Section 3 to study the systems of the
form (1.4) and (1.5), where we allow the coefficient matrix to take complex
values. We show that for these systems, the existence or lack of nondegenerate
solutions have very strong consequences in terms of the generic dimension of
the associated complex varieties. Our results are gathered in the main theorem
of this work, Theorem 3.7, which gives several equivalent conditions
guaranteeing that the generic dimension is $n-s$. The theorem is stated for
sets $\mathcal{X}$ and $\mathcal{A}$ satisfying certain mild algebraic and
topological conditions (which are satisfied for the positive orthant, and the
real and complex torus). Furthermore, the equivalent conditions of Theorem 3.7
hold for a pair of subsets $\mathcal{A}$ and $\mathcal{X}$ satisfying the
conditions if and only if they hold for any other such pair, in particular for
complex parameters and solutions in the complex torus.

In Section 4 we connect the results of Section 3 to the theory of reaction
networks. This section is presented with less technicalities, to make it
accessible to the broader reaction network community. Furthermore, we
specialize our conclusions to show that for the well-known and well-studied
class of so-called _weakly reversible_ networks, the generic dimension of the
complex varieties of the two systems (1.4) and (1.5) is _always_ the lower
bound, namely $n-s$ and $0$, respectively. This settles a question posed by
Boros, Craciun and Yu in [BCY20]. We illustrate further the simplicity of our
conditions by exploring the database ODEbase of biologically relevant reaction
networks [LSR22], and are able to verify within minutes that for the vast
majority of them, the systems of interest attain the lower bound on the
generic dimension, whereas for a handful of exceptions, the steady state
variety is generically empty. Given the size of the networks in the database,
these computations are nontrivial with standard Gröbner basis techniques.

### Notation and conventions

We let $\circ\colon\mathbb{C}^{n}\times\mathbb{C}^{n}\to\mathbb{C}^{n}$ denote
the Hadamard product, given by $({x}\circ{y})_{i}=x_{i}y_{i}$. For a field
$\mathbb{F}$, we let $\mathbb{F}^{*}=\mathbb{F}\setminus\\{{0}\\}$ be the
group of units of $\mathbb{F}$. For a matrix $A=(a_{ij})\in\mathbb{Z}^{n\times
m}$ and a vector ${x}\in(\mathbb{F}^{*})^{n}$, we let
${x}^{A}\in\mathbb{F}^{m}$ be defined by $({x}^{A})_{j}=x_{1}^{a_{1j}}\cdots
x_{n}^{a_{nj}}$ for $j\in\\{1,\ldots,m\\}$.

In this work we will consider both the Euclidean and Zariski topologies on
$\mathbb{C}^{n}$, and their restrictions to subsets. By default, we use the
Zariski topology, and for a set $S\subseteq\mathbb{C}^{n}$, we let
$\overline{S}$ denote the Zariski closure of $S$ in $\mathbb{C}^{n}$. When we
say that a set $U\subseteq X$ is open or has nonempty interior _in_ $X$, for
some $X\subseteq\mathbb{C}^{n}$, we mean with respect to the subspace topology
topology on $X$. For example, if we say that $U$ has nonempty Euclidean
interior in $X$, we mean there exists an open ball $B\subseteq\mathbb{C}^{n}$
such that $B\cap X\subseteq U$.

When saying that a property holds _generically_ in a family indexed by some
parameters $\alpha\in\mathcal{A}$, we mean that it holds outside a proper
Zariski closed subset of $\mathcal{A}$.

### Acknowledgements

EF and OH have been funded by the Novo Nordisk project with grant reference
number NNF20OC0065582. BP has been funded by the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie IF grant
agreement No 794627 and the Spanish Ministry of Economy project with reference
number PGC2018-095392-B-I00.

##  2\. The dimension of solution sets of parametric polynomial systems

In this section we study the generic dimension of the solution set of
parametric systems for which the incidence variety is irreducible of known
dimension. This sets the background theory for the application to the systems
of polynomials indicated in the introduction. We start by stating the
framework for solutions in the complex torus, and then proceed to restrict the
set of solutions to subsets such as the real torus or the positive orthant.

We restrict from the beginning to solutions in a subset of the complex torus
$(\mathbb{C}^{*})^{n}$ (and thus also allow Laurent polynomials with negative
exponents). This will be required to apply the results in Section 3. However,
most of this section also extends to $\mathbb{C}^{n}$ if one restricts to
polynomials with nonnegative exponents.

###  2.1. Framework

We consider a parametric family of systems of (Laurent) polynomials of the
form

$g(\alpha,{x})={0},\quad\alpha=(\alpha_{1},\ldots,\alpha_{\ell})\in\mathbb{C}^{\ell},\quad
x=(x_{1},\ldots,x_{n})\in(\mathbb{C}^{*})^{n},$ (2.1)

where $g\in\mathbb{C}[\alpha,{x}^{\pm}]^{s}$ for some $s\in\mathbb{Z}_{>0}$,
and where we view $\alpha=(\alpha_{1},\ldots,\alpha_{\ell})$ as parameters,
and $x=(x_{1},\ldots,x_{n})$ as variables. We consider the _incidence variety_

$\mathcal{E}_{g}:=\\{(\alpha,{x})\in\mathbb{C}^{\ell}\times(\mathbb{C}^{*})^{n}:g(\alpha,{x})={0}\\},$
(2.2)

and the projection map to parameter space

$\displaystyle\pi\colon\mathcal{E}_{g}\rightarrow\mathbb{C}^{\ell},\quad(\alpha,{x})\mapsto\alpha.$
(2.3)

For each choice of parameters $\alpha\in\mathbb{C}^{\ell}$, we get a
polynomial $g_{\alpha}:=g(\alpha,\cdot)\in\mathbb{C}[{x}^{\pm}]^{s}$, which
leads to the specialized system

$g_{\alpha}({x})={0},\quad{x}\in(\mathbb{C}^{*})^{n}.$ (2.4)

We will identify the algebraic variety
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ consisting of solutions to (2.4)
with the fiber $\pi^{-1}(\alpha)$ of the projection map. We will sometimes
restrict the parameter space to a subset
$\mathcal{A}\subseteq\mathbb{C}^{\ell}$. (The main example will be
$\mathcal{A}=\mathbb{R}^{\ell}_{>0}$.) As there is no guarantee that system
(2.4) has solutions, we introduce the following set:

$\mathcal{D}_{g,\mathcal{A}}=\pi\big{(}\mathcal{E}_{g}\cap(\mathcal{A}\times(\mathbb{C}^{*})^{n})\big{)}=\operatorname{im}(\pi)\cap\mathcal{A}=\\{\alpha\in\mathcal{A}:\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\neq\varnothing\\}\mbox{.}$
(2.5)

In what follows we will distinguish between the coefficient matrix $N$ with
entries in $\mathbb{C}$ obtained by viewing $g$ as a polynomial in
$\mathbb{C}[\alpha,x^{\pm}]$, and the coefficient matrix $\Sigma_{\alpha}$
with entries in $\mathbb{C}[\alpha]$, obtained by viewing $g$ as a polynomial
in $\mathbb{C}[\alpha][x^{\pm}]$.

###  2.2. Dimension over $\mathbb{C}^{*}$

The first concept of interest is the dimension of the family of complex
varieties $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ for
$\alpha\in\mathcal{A}$. An immediate first observation is that for each fixed
$\alpha\in\mathcal{A}$, the principal ideal theorem [Eis95, Thm. 10.2] gives
that

$\dim\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\geq n-s,$ (2.6)

since $g_{\alpha}$ is a tuple of $s$ polynomials. Moreover, all irreducible
components of $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ have dimension at
least $n-s$. Hence, if $\dim(\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha}))=n-s$ for
a given $\alpha\in\mathcal{A}$, then all irreducible components have dimension
$n-s$, and we say that $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is
_equidimensional_ of dimension $n-s$. With this in mind, it makes sense to
remove redundancies in the entries of $g(\alpha,x)$ such that the coefficient
matrix $N$ of $g$ as a polynomial in $\mathbb{C}[\alpha,{x}^{\pm}]^{s}$ has
full rank.

For a fixed $\alpha$, the codimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is additionally bounded above by the
rank of the coefficient matrix $\Sigma_{\alpha}$. Therefore, if the generic
rank of $\Sigma_{\alpha}$ is strictly smaller than $s$, the dimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is strictly larger than $n-s$.

When the incidence variety $\mathcal{E}_{g}$ is irreducible, the generic
dimension of $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is well understood. We
will use the following version of the Theorem on the Dimension of Fibers. A
proof of the result can be found in [SR13, Thm. 1.25], [Vak17, Section 11.4].

######  Theorem 2.1 (Dimension of fibers).

Let $\varphi\colon X\to Y$ be a dominant morphism of irreducible varieties.
Then, for any $y\in\varphi(X)$, it holds that

$\dim(\varphi^{-1}(y))\geqslant\dim(X)-\dim(Y).$ (2.7)

Moreover, there exists a nonempty Zariski open subset $U\subseteq Y$ such that
(2.7) holds with equality for all $y\in U\cap\varphi(X)$.

With this in place, we state the first theorem on the generic dimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$.

######  Theorem 2.2 (Generic dimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$).

Let $g\in\mathbb{C}[\alpha,{x}^{\pm}]^{s}$ and consider the specialized
polynomials $g_{\alpha}$ for parameters $\alpha\in\mathcal{A}$, with
$\mathcal{A}\subseteq\mathbb{C}^{\ell}$ being Zariski dense. Assume that the
incidence variety $\mathcal{E}_{g}$ is irreducible, and let
$d=\dim(\mathcal{E}_{g})-\ell$. Then the following holds:

  1. (i)

If $\mathcal{D}_{g,\mathcal{A}}$ is Zariski dense in $\mathbb{C}^{\ell}$, then
there exists a nonempty Zariski open subset $\mathcal{U}$ of
$\mathcal{D}_{g,\mathcal{A}}$ such that for all $\alpha\in\mathcal{U}$, it
holds that

$\dim\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})=d.$

Furthermore, if $d=n-s$, then $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is
equidimensional.

  2. (ii)

If $\mathcal{D}_{g,\mathcal{A}}$ is not Zariski dense in $\mathbb{C}^{\ell}$,
then it holds for all $\alpha\in\mathcal{D}_{g,\mathcal{A}}$ that

$\dim\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})>d.$

###### Proof.

The projection map $\pi$ over parameter space defined in (2.3) is a regular
map of irreducible varieties. Let $\overline{\operatorname{im}(\pi)}$ denote
the Zariski closure of the image of $\pi$. We can now apply the Theorem on the
Dimension of Fibers (Theorem 2.1), to conclude that nonempty fibers
$\pi^{-1}(\alpha)$ have dimension at least

$
e=\dim(\mathcal{E}_{g})-\dim(\overline{\,\operatorname{im}(\pi)}\,)=\ell+d-\dim(\,\overline{\operatorname{im}(\pi)}\,),$
(2.8)

with generic dimension equal to $e$. That is, there exists a nonempty Zariski
open subset $U\subseteq\overline{\operatorname{im}(\pi)}$ such that, for any
$\alpha\in U$, either $\dim(\pi^{-1}(\alpha))=e$ or
$\pi^{-1}(\alpha)=\varnothing$.

Let us consider scenario (i). Note that
$\mathcal{D}_{g,\mathcal{A}}\subseteq\operatorname{im}(\pi)$. Hence, taking
Zariski closures in $\mathbb{C}^{\ell}$ gives
$\mathbb{C}^{\ell}=\overline{\mathcal{D}_{g,\mathcal{A}}}\subseteq\overline{\operatorname{im}(\pi)}$,
and hence $\overline{\operatorname{im}(\pi)}=\mathbb{C}^{\ell}$. Therefore,
the generic dimension of the fibers of $\pi$ is $e=\ell+d-\ell=d$. It follows
that

$\dim(\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha}))=d$

for all $\alpha\in U\cap\mathcal{D}_{g,\mathcal{A}}$, since in this case
$\pi^{-1}(\alpha)=\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\neq\varnothing$.
Noting that $U\cap\mathcal{D}_{g,\mathcal{A}}\neq\varnothing$ as
$\mathcal{D}_{g,\mathcal{A}}\subseteq\mathbb{C}^{\ell}$ is Zariski dense, we
obtain statement (i) by taking $\mathcal{U}=U\cap\mathcal{D}_{g,\mathcal{A}}$.

Consider now scenario (ii). By (2.8), the result follows if
$\overline{\operatorname{im}(\pi)}\subseteq\mathbb{C}^{\ell}$ is proper, and
hence of lower dimension than $\ell$. By Chevalley’s theorem [Sta23, 00F5],
the image of a constructible set under a projection map is constructible.
Hence $\operatorname{im}(\pi)$ is a constructible set, and it can be written
as $\operatorname{im}(\pi)=\bigcup_{i=1}^{m}Z_{i}\cap U_{i}$, for some
$m\in\mathbb{Z}_{>0}$, irreducible Zariski closed subsets
$Z_{i}\subseteq\mathbb{C}^{\ell}$ and nonempty Zariski open subsets
$U_{i}\subseteq\mathbb{C}^{\ell}$. It follows that

$ \overline{\operatorname{im}(\pi)}=\bigcup_{i=1}^{m}Z_{i}.$

Assume for a contradiction that
$\overline{\operatorname{im}(\pi)}=\mathbb{C}^{\ell}$. Then
$Z_{i}=\mathbb{C}^{\ell}$ for some $i\in\\{1,\dots,m\\}$, and we have
$U_{i}\subseteq\operatorname{im}(\pi)$. Hence,

$U_{i}\cap\mathcal{A}\subseteq\operatorname{im}(\pi)\cap\mathcal{A}=\mathcal{D}_{g,\mathcal{A}},$

where $U_{i}\cap\mathcal{A}\neq\varnothing$, since $\mathcal{A}$ is Zariski
dense. Taking Zariski closures, as
$\overline{U_{i}\cap\mathcal{A}}=\mathbb{C}^{\ell}$, we have
$\overline{\mathcal{D}_{g,\mathcal{A}}}=\mathbb{C}^{\ell}$, which is a
contradiction. This shows that $\overline{\operatorname{im}(\pi)}$ is proper
and concludes the proof of (ii). ∎

Combining cases (i) and (ii) of Theorem 2.2, we obtain that the dimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is $d$ for some $\alpha$ if and only
if $\mathcal{D}_{g,\mathcal{A}}$ is Zariski dense in $\mathbb{C}^{\ell}$.
Furthermore, if the dimension is $d$ for one value of $\alpha$, then it is
generically $d$ in $\mathcal{D}_{g,\mathcal{A}}$.

######  Example 2.3.

Consider $g(\alpha,x)$ with $n=\ell=s=2$ giving rise to the system

$\displaystyle\alpha_{1}x_{1}-\alpha_{2}x_{2}=0,\quad\alpha_{1}^{2}x_{1}^{2}-\alpha_{2}^{2}x_{2}^{2}=0,$

which was briefly discussed in the Introduction in (1.3). Let
$\mathcal{A}=\mathbb{C}^{2}$. In this case,
$\mathcal{D}_{g,\mathcal{A}}=\mathbb{C}^{2}$. Furthermore, the incidence
variety is defined by the equation $\alpha_{1}x_{1}-\alpha_{2}x_{2}=0$ as
$\alpha_{1}^{2}x_{1}^{2}-\alpha_{2}^{2}x_{2}^{2}=(\alpha_{1}x_{1}-\alpha_{2}x_{2})(\alpha_{1}x_{1}+\alpha_{2}x_{2})$.
Hence, $\mathcal{E}_{g}$ is irreducible of dimension $3$ and by Theorem
2.2(i), the generic dimension of $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is
$d=3-2=1$.

In the previous example, the generic dimension is not the minimal one from
(2.6), which would be $n-s=0$. However, in the systems of interest in the
coming sections, the goal will be to determine when the generic dimension is
exactly $n-s$, in which case the variety also is equidimensional.

######  Remark 2.4.

Recall that $N$ denotes the coefficient matrix of $g$, and $\Sigma_{\alpha}$
the coefficient matrix of $g_{\alpha}$. Let us assume for simplicity that $N$
has rank $s$. Then the dimension of $\mathcal{E}_{g}$ cannot be smaller than
$n+\ell-s$. Under the setting of Theorem 2.2, if $\mathcal{D}_{g,\mathcal{A}}$
is Zariski dense and additionally $\dim(\mathcal{E}_{g})=n+\ell-s$, then the
generic dimension of $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is exactly
$n-s$. In this case, the generic rank of $\Sigma_{\alpha}$ is necessarily $s$,
as the corank of $\Sigma_{a}$ is a lower bound for the dimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$.

###  2.3. Restricting the ambient space

Let $\mathbb{F}\subseteq\mathbb{C}$ be a subfield, and consider
$\mathcal{A}\subseteq\mathbb{F}^{\ell}$. Then, for $\alpha\in\mathcal{A}$, we
let $\mathbb{V}_{\mathbb{F}^{*}}(g_{\alpha})$ denote the set of solutions to
system (2.4) in $(\mathbb{F}^{*})^{n}$. Additionally, for a set
$\mathcal{X}\subseteq(\mathbb{F}^{*})^{n}$ we let

$ \mathbb{V}_{\mathbb{F}^{*}}^{\mathcal{X}}(g_{\alpha})$

denote the variety defined as the union of the irreducible components of
$\mathbb{V}_{\mathbb{F}^{*}}(g_{\alpha})$ that intersect $\mathcal{X}$. We
will also consider the set of solutions in $\mathcal{X}$, that is,
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\cap\mathcal{X}$. The main example we
have in mind, which will become relevant in the next section, is the case
$\mathbb{F}=\mathbb{R}$ and $\mathcal{X}=\mathbb{R}^{n}_{>0}$, hence of real
varieties. We generalize (2.5), and introduce the set

$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})=\pi\big{(}\mathcal{E}_{g}\cap(\mathcal{A}\times\mathcal{X})\big{)}=\\{\alpha\in\mathcal{A}:\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\cap\mathcal{X}\neq\varnothing\\}\mbox{.}$
(2.9)

With this notation,
$\mathcal{D}_{g,\mathcal{A}}((\mathbb{C}^{*})^{n})=\mathcal{D}_{g,\mathcal{A}}$
and we have the following inclusions:

$
\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\subseteq\mathcal{D}_{g,\mathcal{A}},\quad\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\subseteq\mathcal{D}_{g,\mathbb{C}^{\ell}}(\mathcal{X}).$

######  Remark 2.5.

The fact that $\mathbb{V}^{\mathcal{X}}_{\mathbb{C}^{*}}(g_{\alpha})$ is a
Zariski closed set gives that

$\overline{\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\cap\mathcal{X}}\subseteq\mathbb{V}^{\mathcal{X}}_{\mathbb{C}^{*}}(g_{\alpha})$
(2.10)

but the reverse inclusion is not necessarily true. The inclusion (2.10) holds
with equality if $\mathcal{X}$ is an Euclidean open subset of
$(\mathbb{C}^{*})^{n}$, since, in that case,
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\cap\mathcal{X}$ is, whenever
nonempty, Zariski dense in $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$. When
$\mathcal{X}$ is an Euclidean open subset of $(\mathbb{R}^{*})^{n}$, a
sufficient condition for equality in (2.10) is that each irreducible component
of $\mathbb{V}_{\mathbb{C}^{*}}^{\mathcal{X}}(g_{\alpha})$ has a nonsingular
point (see [BCR98], [PEF22, Thm. 6.5], [HHS21] and [Mar08, Thm. 12.6.1]). On
the same line of thought, the lower bound for the complex dimension in (2.6)
does not hold over $\mathbb{R}$, as the next example illustrates.

######  Example 2.6.

A simple example that illustrates the previous remark is the polynomial

$g(\alpha,x)=\alpha_{1}x_{1}^{2}-\alpha_{2}x_{1}+\alpha_{3}x_{2}^{2}-\alpha_{4}x_{2}+\alpha_{5}$

with $\mathcal{X}=\mathbb{R}^{2}$ and $\alpha\in\mathbb{R}^{5}$. For
$\alpha=(1,2,1,2,2)$, we get

$\mathbb{V}^{\mathcal{X}}_{\mathbb{C}^{*}}(g_{\alpha})=\mathbb{V}_{\mathbb{C}^{*}}((x_{1}-1)^{2}+(x_{2}-1)^{2}))\supsetneq\\{(1,1)\\}=\overline{\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})\cap\mathcal{X}}.$

Hence the real variety $\mathbb{V}_{\mathbb{R}^{*}}(g_{\alpha})$ is zero-
dimensional, while the lower bound for the dimension of
$\dim(\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha}))$ is $n-s=1$. Note that the
point $(1,1)$ is singular in this case.

From Theorem 2.2 we obtain the following result on the generic dimension of
$\mathbb{V}_{\mathbb{C}^{*}}^{\mathcal{X}}(g_{\alpha})$, which is in general
bounded above by the generic dimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$.

######  Corollary 2.7 (Generic dimension of
$\mathbb{V}_{\mathbb{C}^{*}}^{\mathcal{X}}(g_{\alpha})$).

Let $g\in\mathbb{C}[\alpha,{x}^{\pm}]^{s}$ and $g_{\alpha}$ as in (2.4),
${x}\in(\mathbb{C}^{*})^{n}$, $\alpha\in\mathcal{A}\subseteq\mathbb{C}^{\ell}$
with $\mathcal{A}$ being Zariski dense. Assume that the incidence variety
$\mathcal{E}_{g}$ is irreducible and has dimension $\ell+d$ for some $d\geq
0$.

For $\mathcal{X}\subseteq(\mathbb{C}^{*})^{n}$, if
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\subseteq\mathcal{A}$ is Zariski
dense in $\mathbb{C}^{\ell}$, then there exists a nonempty Zariski open set
$\mathcal{U}\subseteq\mathbb{C}^{\ell}$ such that
$\mathcal{U}\cap\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\neq\varnothing$ and

$\dim\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})=\dim\mathbb{V}_{\mathbb{C}^{*}}^{\mathcal{X}}(g_{\alpha})=d,\quad\text{for
all }\quad\alpha\in\mathcal{U}\cap\mathcal{D}_{g,\mathcal{A}}(\mathcal{X}).$

If $d=n-s$, then $\mathbb{V}_{\mathbb{C}^{*}}^{\mathcal{X}}$ is
equidimensional.

###### Proof.

This is an immediate consequence of Theorem 2.2(i) as
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\subseteq\mathcal{D}_{g,\mathcal{A}}$,
with the set $\mathcal{U}$ in the statement of the theorem, after noting that
$\mathcal{U}\cap\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\neq\varnothing$ as
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ is Zariski dense. ∎

Observe that Corollary 2.7 extends only Theorem 2.2(i) to $\mathcal{X}$. As
discussed in Remark 2.5, some special care needs to be taken for certain
instances of $\mathcal{X}$.

###  2.4. Zariski denseness and nonempty Euclidean interior

One of the hypotheses of Theorem 2.2 and Corollary 2.7 relies on checking that
$\mathcal{D}_{g,\mathcal{A}}$ or $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$
are Zariski dense. In this subsection, we relate this hypothesis to the
concept of nondegeneracy, under the assumption that $\mathcal{A}$ has the
following property.

######  Definition 2.8.

A subset $X\subseteq\mathbb{C}^{n}$ is said to be locally Zariski dense (in
$\mathbb{C}^{n}$) if for any Euclidean open subset $U\subseteq\mathbb{C}^{n}$
with $U\cap X\neq\varnothing$, it holds that $\overline{U\cap
X}=\mathbb{C}^{n}$ with respect to the Zariski topology on $\mathbb{C}^{n}$.

Simple examples of locally Zariski dense sets in $\mathbb{C}^{n}$ include
$\mathbb{R}^{n}_{>0}$ and $\mathbb{R}^{n}$. Note that any locally Zariski
dense set is in particular Zariski dense, but the converse is not true: for
instance, $\mathbb{Z}^{n}$ is Zariski dense in $\mathbb{C}^{n}$, but not
locally Zariski dense.

######  Lemma 2.9.

Let $\mathcal{X}\subseteq(\mathbb{C}^{*})^{n}$ and suppose that
$\mathcal{A}\subseteq\mathbb{C}^{\ell}$ is locally Zariski dense. If
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ has nonempty Euclidean interior in
$\mathcal{A}$, then $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ is Zariski
dense in $\mathbb{C}^{\ell}$.

###### Proof.

By hypothesis, there exists an open (Euclidean) ball
$B\subseteq\mathbb{C}^{\ell}$ such that $\varnothing\neq
B\cap\mathcal{A}\subseteq D_{g,\mathcal{A}}(\mathcal{X})$. In particular, by
local Zariski denseness of $\mathcal{A}$, the Zariski closures satisfy
$\mathbb{C}^{\ell}=\overline{B\cap\mathcal{A}}\subseteq\overline{\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})}$.
Hence $\mathbb{C}^{\ell}=\overline{\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})}$.
∎

######  Example 2.10.

The set $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ might be nonempty but not
(Euclidean) dense in $\mathcal{A}$. Consider the single parametric polynomial

$g(\alpha,{x})=\alpha_{1}x-\alpha_{2}x+\alpha_{3}$

with $n=1$, $\ell=3$, $\mathcal{X}=\mathbb{R}_{>0}$ and
$\mathcal{A}=\mathbb{R}^{3}_{>0}$. Then

$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})=\\{\alpha\in\mathbb{R}^{3}_{>0}:\alpha_{1}<\alpha_{2}\\}.$

We have that $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ is full dimensional in
$\mathbb{R}^{3}_{>0}$ and so is its complement. As $\mathbb{R}^{3}_{>0}$ is
locally Zariski dense in $\mathbb{C}^{3}$ and
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ has nonempty Euclidean interior in
$\mathbb{R}^{3}_{>0}$, by Lemma 2.9 we obtain that
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ is Zariski dense and hence we are
in the scenario (i) of Corollary 2.7.

The next results show that for most interesting sets, the converse of Lemma
2.9 also holds, that is, being Zariski dense or having nonempty Euclidean
interior are equivalent.

######  Lemma 2.11.

  1. (i)

A constructible set $S\subseteq\mathbb{C}^{n}$ is Zariski dense in
$\mathbb{C}^{n}$ if and only if it has nonempty Euclidean interior in
$\mathbb{C}^{n}$.

  2. (ii)

A semialgebraic set $S\subseteq\mathbb{R}^{n}$ is Zariski dense in
$\mathbb{C}^{n}$ if and only if it has nonempty Euclidean interior in
$\mathbb{R}^{n}$.

###### Proof.

For part (i), note that $S$ being constructible means that it can be written
in the form $S=\bigcup_{i=1}^{m}Z_{i}\cap U_{i}$, for irreducible Zariski
closed sets $Z_{i}\subseteq\mathbb{C}^{n}$ and Zariski open subsets
$U_{i}\subseteq\mathbb{C}^{n}$, with $Z_{i}\cap U_{i}\neq\varnothing$ for each
$i=1,\ldots,m$. The statement now follows from the fact that the Zariski
closure satisfies $\overline{S}=\bigcup_{i=1}^{n}Z_{i}$. In case (ii), note
that [BCR98, Prop. 2.8.2] gives that $S$ is Zariski dense in $\mathbb{C}^{n}$
if and only if its semialgebraic dimension is $n$, which in turn is equivalent
to $S$ having nonempty Euclidean interior in $\mathbb{R}^{n}$. ∎

We now apply Lemma 2.11 to $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$.

######  Proposition 2.12.

Assume that $\mathcal{A}$ and $\mathcal{X}$ satisfy any of the following
cases:

  1. (i)

$\mathcal{A}\subseteq\mathbb{R}^{\ell}$ and
$\mathcal{X}\subseteq(\mathbb{R}^{*})^{n}$ are both semialgebraic.

  2. (ii)

$\mathcal{A}\subseteq\mathbb{C}^{\ell}$ and
$\mathcal{X}\subseteq(\mathbb{C}^{*})^{n}$ are both constructible.

  3. (iii)

$\mathcal{A}\subseteq\mathbb{R}^{\ell}$ is semialgebraic and
$\mathcal{X}\subseteq(\mathbb{C}^{*})^{n}$ is constructible.

Assume additionally that $\mathcal{A}$ is locally Zariski dense in
$\mathbb{C}^{\ell}$. Then $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ is
Zariski dense in $\mathbb{C}^{\ell}$ if and only if it has nonempty Euclidean
interior in $\mathcal{A}$.

###### Proof.

The reverse implication is Lemma 2.9. The forward implication is a consequence
of Lemma 2.11. Note that if $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ has
nonempty Euclidean interior in $\mathbb{R}^{\ell}$ in cases (i) and (iii) and
in $\mathbb{C}^{\ell}$ in case (ii), then it has nonempty Euclidean interior
in $\mathcal{A}$. In case (i), the Tarski–Seidenberg theorem [BCR98, Thm.
2.2.1] gives that
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\subseteq\mathbb{R}^{\ell}$ is
semialgebraic. In case (ii) the set
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})\subseteq\mathbb{C}^{\ell}$ is
constructible by Chevalley’s theorem. Finally, in case (iii), Chevalley’s
theorem gives that
$\mathcal{D}_{g,\mathbb{C}^{\ell}}(\mathcal{X})\subseteq\mathbb{C}^{\ell}$ is
constructible. This in turn implies that

$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})=\mathcal{D}_{g,\mathbb{C}^{\ell}}(\mathcal{X})\cap\mathcal{A}=(\mathcal{D}_{g,\mathbb{C}^{\ell}}(\mathcal{X})\cap\mathbb{R}^{\ell})\cap\mathcal{A}$

is an intersection of semialgebraic sets and therefore semialgebraic (note
that the real points of a constructible set in $\mathbb{C}^{\ell}$ form a
semialgebraic set in $\mathbb{R}^{\ell}$). ∎

###  2.5. Nondegeneracy and real dimension.

An approach to decide whether $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ has
nonempty Euclidean interior is to use the concept of _nondegeneracy_. In turn,
nondegenerate solutions will allow us to compare the real and complex
dimension of the varieties in an easier way.

######  Definition 2.13.

Given a polynomial system $g({x})={0}$ with
$g=(g_{1},\ldots,g_{s})\in\mathbb{C}[{x^{\pm}}]^{s}$, and
${x}=(x_{1},\dots,x_{n})$, a solution ${x}^{*}$ is called nondegenerate if
$\operatorname{rk}(J_{g}({x}^{*}))=s$. Otherwise, it is called degenerate.

Note that the concept of nondegeneracy refers to the system and not to the
variety defined by the system. Note also that if the coefficient matrix of $g$
does not have full rank $s$, then all solutions to the system will be
degenerate. Hence, a natural preprocessing step of any system $g(x)=0$ is to
remove redundant equations by keeping a maximal set of linearly independent
equations. In particular, we could assume that the rank of the coefficient
matrix of $g$ is $s$.

The following proposition gathers well-known results about nondegenerate
solutions, which will become relevant later on.

######  Proposition 2.14.

Let $g=(g_{1},\ldots,g_{s})\in\mathbb{C}[x_{1}^{\pm},\dots,x_{n}^{\pm}]^{s}$.

  1. (i)

If $x^{*}\in(\mathbb{C}^{*})^{n}$ is a nondegenerate solution to $g({x})=0$,
then $x^{*}$ is a nonsingular point of the variety
$\mathbb{V}_{\mathbb{C}^{*}}(g)$, and belongs to a unique irreducible
component of dimension $n-s$.

  2. (ii)

The set of nondegenerate solutions of $g(x)=0$ is either empty or a Zariski
open subset of $\mathbb{V}_{\mathbb{C}^{*}}(g)$.

  3. (iii)

If $g\in\mathbb{R}[x_{1}^{\pm},\dots,x_{n}^{\pm}]^{s}$ and $x^{*}$ is a
nondegenerate real solution to $g({x})=0$, then the irreducible component of
$\mathbb{V}_{\mathbb{R}^{*}}(g)$ containing ${x}^{*}$ has dimension $n-s$.

###### Proof.

Statement (i) is Theorem 9 in [CLO15, Section 9.6]. For statement (ii), note
that the degenerate locus is given by the vanishing of all $s\times s$ minors
of $J_{g}(x)$ for $x\in\mathbb{V}_{\mathbb{C}^{*}}(g)$, and therefore is
Zariski closed. For (iii), consider (i) and Remark 2.5. ∎

We note that nonsingular points of a variety are not necessarily nondegenerate
solutions of a defining polynomial set.

######  Lemma 2.15.

Let $g\in\mathbb{C}[\alpha,{x}^{\pm}]^{s}$, and consider the specialized
polynomials $g_{\alpha}$ for $\alpha\in\mathcal{A}$ for some subset
$\mathcal{A}\subseteq\mathbb{C}^{\ell}$. Suppose that there is some parameter
value $\alpha^{*}\in\mathcal{A}$ such that $g_{\alpha^{*}}(x)=0$ has a
nondegenerate solution in $(\mathbb{C}^{*})^{n}$. Then the following holds:

  1. (i)

$\mathcal{D}_{g,\mathcal{A}}$ has nonempty Euclidean interior in
$\mathcal{A}$.

  2. (ii)

If in addition $\mathcal{A}$ is locally Zariski dense in $\mathbb{C}^{\ell}$,
then $\mathcal{D}_{g,\mathcal{A}}$ is Zariski dense in $\mathbb{C}^{\ell}$.

###### Proof.

For part (i), suppose that
$(\alpha^{*},x^{*})\in\mathcal{A}\times(\mathbb{C}^{*})^{n}$ is such that
$\operatorname{rk}(J_{g_{\alpha^{*}}}(x^{*}))=s$. Let
$A\in\mathbb{C}^{(n-s)\times n}$ be a matrix whose rows extend the rows of
$J_{g_{\alpha^{*}}}(x^{*})$ to a basis for $\mathbb{C}^{n}$. Then $x^{*}$ is a
nondegenerate solution of the square system $\tilde{g}_{\alpha^{*}}(x)=0$,
where

$\tilde{g}(\alpha^{*},x)=\begin{bmatrix}g(\alpha^{*},x)\\\ Ax-
Ax^{*}\end{bmatrix}\in\mathbb{C}[\alpha,x^{\pm}]^{n}.$

The complex implicit function theorem [Huy05, Prop. 1.1.11] now gives that
there exists an open Euclidean neighborhood of $\alpha^{*}$ contained in
$\mathcal{D}_{g,\mathbb{C}^{\ell}}$. Intersecting this open set with
$\mathcal{A}$, we obtain the statement. Part (ii) is immediate from (i) and
Lemma 2.9. ∎

######  Corollary 2.16.

Let $g\in\mathbb{C}[\alpha,{x}^{\pm}]^{s}$ and $g_{\alpha}$ be as in (2.4).
Let $\mathcal{A}\subseteq\mathbb{C}^{\ell}$ be locally Zariski dense. Assume
that the incidence variety $\mathcal{E}_{g}$ is irreducible of dimension
$n+\ell-s$, and that $g_{\alpha^{*}}(x)=0$ has a nondegenerate solution in
$(\mathbb{C}^{*})^{n}$ for some $\alpha^{*}\in\mathcal{A}$.

Then $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is generically equidimensional
of dimension $n-s$, that is, this holds for all $\alpha$ in a nonempty Zariski
open subset of $\mathcal{D}_{g,\mathcal{A}}$.

If in addition $\mathcal{A}\subseteq\mathbb{R}^{\ell}$, then
$\mathbb{V}_{\mathbb{R}^{*}}(g_{\alpha})$ is generically of dimension $n-s$.

###### Proof.

Lemma 2.15 gives that $\mathcal{D}_{g,\mathcal{A}}$ has nonempty Euclidean
interior in $\mathcal{A}$ and by Lemma 2.9, it is Zariski dense in
$\mathcal{C}^{\ell}$. As $\mathcal{E}_{g}$ is irreducible, Theorem 2.2 gives
that $\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ has dimension
$\dim(\mathcal{E}_{g})-\ell=n-s$ for generic
$\alpha\in\mathcal{D}_{g,\mathcal{A}}$. For the last statement, we use
Proposition 2.14(iii). ∎

Note that in the setting of Corollary 2.16, the real dimension being $n-s$
does not directly imply equidimensionality.

Corollary 2.16 gives a condition for the generic dimension of
$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ to be the smallest possible, namely
$n-s$. This condition is not satisfied in Example 2.3, where there are no
nondegenerate solutions. Consistently, we saw that the generic dimension was
strictly higher than $n-s$ in that case.

In the next section we introduce three families of polynomial systems where
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ has nonempty Euclidean interior if
and only if there exists a nondegenarate solution, and if and only if the
generic dimension is the minimal, namely $n-s$. This phenomenon does not need
to happen for general families, as Example 2.3 shows.

We conclude this subsection by noting that in certain scenarios within the
setting of Theorem 2.2, the system $g_{\alpha}(x)=0$ will only have
nondegenerate solutions for all $\alpha$ in a nonempty Zariski open subset of
$\mathcal{A}$.

######  Theorem 2.17.

Let $g\in\mathbb{C}[\alpha,{x}^{\pm}]^{s}$ and $g_{\alpha}$ as in (2.4), for
parameters $\alpha\in\mathcal{A}$, with
$\mathcal{A}\subseteq\mathbb{C}^{\ell}$ being Zariski dense. Assume that the
incidence variety $\mathcal{E}_{g}$ is irreducible of dimension $\ell$ and
that there exists a nondegenerate solution to $g_{\alpha}(x)=0$ for some
$\alpha\in\mathcal{A}$. Then there exists a nonempty Zariski open subset
$\mathcal{U}\subseteq\mathcal{A}$ such that for all $\alpha\in\mathcal{U}$,
any solution to $g(x)=0$ is nondegenerate.

###### Proof.

The points $(\alpha,x)\in\mathcal{E}_{g}$ for which $x$ is a degenerate
solution of $g_{\alpha}$ form a proper Zariski closed subset $Z$ of
$\mathcal{E}_{g}$, as they are the zeroes of a collection of polynomial
equations and there exists a nondegenerate solution. As
$\dim(\mathcal{E}_{g})=\ell$, $\dim(Z)<\ell$, and hence
$\overline{\pi(Z)}\subseteq\mathbb{C}^{\ell}$ has dimension strictly smaller
than $\ell$. For any $\alpha$ in the nonempty Zariski open set
$U:=\mathbb{C}^{\ell}\setminus\overline{\pi(Z)}$, all solutions to
$g_{\alpha}(x)=0$ are nondegenerate. As $\mathcal{A}$ is Zariski dense, the
Zariski open set $\mathcal{U}$ in the statement is
$U\cap\mathcal{A}\neq\varnothing$. ∎

##  3\. Generic dimension for the three parametric systems

We consider now three polynomial systems of a specific form, motivated by the
application to the study of steady states of reaction networks. These systems
are built from parametric systems that are linear in the parameters, and, in
addition, each parameter accompanies always the same monomial in $x$.

###  3.1. The parametric systems.

The first parametric system we consider takes the form

$f(\kappa,{x})=N(\kappa\circ{x}^{B}),\quad
x\in\mathcal{X}\subseteq(\mathbb{C}^{*})^{n},\quad\kappa\in\mathcal{K}\subseteq\mathbb{C}^{r}\mbox{,}$
(3.1)

with $N\in\mathbb{C}^{s\times r}$ of full rank $\operatorname{rk}(N)=s$, and
$B\in\mathbb{Z}^{n\times r}$. We let $f_{\kappa}$ denote the specialization of
$f$ to a chosen value of $\kappa$. In the notation of the previous section,
$\mathcal{A}=\mathcal{K}$ and $\ell=r$.

For a vector subspace $S\subseteq\mathbb{C}^{n}$ of dimension $s$, we are also
interested in the intersection of
$\mathbb{V}_{\mathbb{C}^{*}}(f_{\kappa})\cap\mathcal{X}$ with parallel
translates of $S$. To this end, consider any (full rank) matrix
$W\in\mathbb{C}^{d\times n}$ such that $S=\ker(W)$ and $d=n-s$. Then we
consider the family of linear varieties parametrized by
${c}=(c_{1},\ldots,c_{d})\in\mathcal{C}\subseteq\mathbb{C}^{d}$ as the
solutions to the polynomial family

$W{x}-{c}={0}\mbox{.}$

The intersections of the parallel translates of $S$ with
$\mathbb{V}_{\mathbb{C}^{*}}(f_{\kappa})$ are the solutions to the extended
polynomial system $F_{\kappa,{c}}({x})={0}$ with

$F(\kappa,c,x)=\begin{bmatrix}f(\kappa,{x})\\\
W{x}-{c}\end{bmatrix},\quad{x}\in\mathcal{X}\subseteq(\mathbb{C}^{*})^{n},\quad\kappa\in\mathcal{K}\subseteq\mathbb{C}^{r},\quad{c}\in\mathcal{C}\subseteq\mathbb{C}^{d}.$
(3.2)

For this second parametrized system, the parameter vector is
$\alpha=(\kappa,{c})$ and the parameter space becomes
$\mathcal{A}=\mathcal{K}\times\mathcal{C}\subseteq\mathbb{C}^{r}\times\mathbb{C}^{d}$
and $\ell=r+d$.

Note that if $\mathcal{C}\cap W(\mathcal{X})=\varnothing$, then the system
$F_{\kappa,{c}}({x})=0$ does not have a solution in
$\mathcal{X}\subseteq\mathbb{C}^{n}$. Since $W$ has full rank $d$, it defines
a continuous and surjective map. If $\mathcal{X}\subseteq\mathbb{C}^{n}$ is
locally Zariski dense, then so is $W(\mathcal{X})$, and in particular,

$\overline{W(\mathcal{X})}=\mathbb{C}^{d}.$

Finally, the third parametric system we consider is given by the restriction
of $F$ to a value of $c$:

$ F_{\cdot,{c}}(\kappa,{x})=F(\kappa,{c},{x}).$ (3.3)

###  3.2. The incidence varieties

We now consider the incidence varieties (2.2) for the polynomial functions
$f$, $F$ and $F_{\cdot,c}$, and derive some basic facts about their geometry.

######  Proposition 3.1.

Let $N\in\mathbb{C}^{s\times r}$ of rank $s$, $W\in\mathbb{C}^{(n-s)\times n}$
of rank $d=n-s$, and $B\in\mathbb{Z}^{n\times r}$. Construct $f$, $F$ and
$F_{\cdot,c}$ as in (3.1), (3.2) and (3.3) for $\kappa\in\mathbb{C}^{r}$ and
$c\in\mathbb{C}^{d}$.

  * (i)

The incidence varieties
$\mathcal{E}_{f}\subseteq\mathbb{C}^{r}\times(\mathbb{C}^{*})^{n}$ and
$\mathcal{E}_{F}\subseteq\mathbb{C}^{r+d}\times(\mathbb{C}^{*})^{n}$ admit
injective rational parametrizations

$\mathbb{C}^{r-s}\times(\mathbb{C}^{*})^{n}\rightarrow\mathcal{E}_{f},\quad\mathbb{C}^{r-s}\times(\mathbb{C}^{*})^{n}\rightarrow\mathcal{E}_{F}.$

Hence, $\mathcal{E}_{f}$ and $\mathcal{E}_{F}$ are irreducible varieties of
dimension $r+d$.

  * (ii)

For any $c\in\mathbb{C}^{d}$, the incidence variety
$\mathcal{E}_{F_{\cdot,{c}}}\subseteq\mathbb{C}^{r}\times(\mathbb{C}^{*})^{n}$
admits an injective rational parametrization

$\mathbb{C}^{r-n}\times(\mathbb{C}^{*})^{n}\rightarrow\mathcal{E}_{F_{\cdot,{c}}}.$

Hence, $\mathcal{E}_{F_{\cdot,{c}}}$ is an irreducible variety of dimension
$r$.

  * (iii)

The incidence varieties $\mathcal{E}_{f}$, $\mathcal{E}_{F}$ and
$\mathcal{E}_{F_{\cdot,{c}}}$ (for any $c\in\mathbb{C}^{d}$) have no singular
points.

###### Proof.

Viewing $f$ as a linear function in $\kappa_{1},\dots,\kappa_{r}$, the
coefficient matrix is $N\operatorname{diag}({x}^{B})$. As
${x}\in(\mathbb{C}^{*})^{n}$, the rank of the coefficient matrix is
$\operatorname{rk}(N)=s$. Hence, the equation $f(\kappa,{x})=0$ can be solved
for $s$ of the $\kappa_{i}$’s, giving a parametrization
$\varphi(\kappa^{\prime},{x})$ of the solution set in terms of the $r-s$
remaining $\kappa_{i}$’s (forming the vector
$\kappa^{\prime}\in\mathbb{C}^{r-s}$) and the entries of ${x}$.

Similarly, as the first $s$ components of $F(\kappa,{c},{x})$ agree with $f$,
we consider the parametrization $\varphi(\kappa^{\prime},{x})$ above. The last
$d$ components are linear in ${x}$, with coefficient matrix $W$ of rank $d$.
Hence, $d$ of the $x_{i}$’s can be solved in terms of the rest of the
$x_{i}$’s and ${c}$, yielding a function $\psi({x}^{\prime},{c})$. The desired
parametrization is $\varphi(\kappa^{\prime},\psi({x}^{\prime},{c}))$, which
has $r-s+n-d+d=r+d$ free parameters.

Finally, the parametrization of $\mathcal{E}_{F_{\cdot,{c}}}$ arises by
specializing $\varphi(\kappa^{\prime},\psi({x}^{\prime},{c}))$ to the chosen
${c}$, and hence has $r+d-d=r$ free parameters.

To prove (iii) it suffices to show that all points in the variety are
nondegenerate solutions of the equations defining the variety as in (2.2). To
this end, we find a minor of each of the Jacobian matrices $J_{f}(\kappa,x)$,
$J_{F}(\kappa,c,x)$, $J_{F_{\cdot,c}}(\kappa,x)$ with maximal rank. For $f$,
the submatrix arising from the partial derivatives of $f$ with respect to
$\kappa$ is $N\operatorname{diag}(x^{B})$, which has maximal rank $s$. For $F$
and $F_{\cdot,c}$, the Jacobian matrix has block form

$\begin{bmatrix}N\operatorname{diag}(x^{B})&*\\\ 0&W\end{bmatrix},$

where $0$ denotes the zero matrix of size $d\times r$. As
$N\operatorname{diag}(x^{B})$ and $W$ have maximal rank, so has the Jacobian
matrix. ∎

Proposition 3.1 tells us that the hypothesis of Theorem 2.2 on the incidence
variety being irreducible holds for the three parametrized systems considered
here. Therefore, for each of the varieties, the minimal dimension is attained
if scenario (i) of Theorem 2.2 holds.

######  Corollary 3.2.

Let $g=F$ as in (3.2), $g=F_{\cdot,c}$ as in (3.3), or $g=f$ as in (3.1) with
additionally $s=n$. If there exists a nondegenerate solution to
$g_{\alpha^{*}}(x)=0$ for some parameter value $\alpha^{*}$, then, for
$\alpha$ in a nonempty Zariski open subset of the respective parameter spaces,
all solutions to $g_{\alpha}(x)=0$ are nondegenerate.

###### Proof.

This is a consequence of Theorem 2.17 and Proposition 3.1. ∎

###  3.3. Nondegenerate solutions and nonempty Euclidean interior.

By Corollary 2.16, if any of the polynomial functions $g$ in play have a
nondegenerate solution, then $\mathcal{D}_{g,\mathcal{A}}$ has nonempty
Euclidean interior and the complex algebraic varieties have the minimal
dimension given by the number of equations. We will see next that the converse
also holds for the three families $g$ under consideration, that is, if
$\mathcal{D}_{g,\mathcal{A}}$ has nonempty Euclidean interior for
$\mathcal{A}$ the largest possible parameter space (in each case), then
necessarily the system has a nondegenerate solution.

We start with a description of the sets
$\mathcal{D}_{f,\mathcal{K}}(\mathcal{X})$,
$\mathcal{D}_{F_{\cdot,c},\mathcal{K}}(\mathcal{X})$ and of
$\mathcal{D}_{F,\mathcal{A}}(\mathcal{X})$ with
$\mathcal{A}\subseteq\mathbb{C}^{r}\times\mathbb{C}^{d}$. We proceed then to
study nondegenerate solutions of the systems given by these families and
conclude with the main theorem relating nondegenerate solutions to the
Euclidean interior of $\mathcal{D}_{g,\mathcal{A}}$ for $\mathcal{A}$ begin
the maximal complex subspace of parameters under consideration.

######  Proposition 3.3.

With $f,F,F_{\cdot,c}$ as in (3.1), (3.2) and (3.3), let $\mathbb{M}$ be a
multiplicative subgroup of $\mathbb{C}^{*}$,
$\mathcal{K}\in\\{\mathbb{M}^{r},(\mathbb{M}\cup\\{0\\})^{r}\\}$, and
$\mathcal{X}\subseteq\mathbb{M}^{n}$ a multiplicative subgroup. Let
$\mathcal{A}\subseteq\mathcal{K}\times\mathbb{C}^{d}$. Then it holds that

$\displaystyle\mathcal{D}_{f,\mathcal{K}}(\mathcal{X})$
$\displaystyle=\big{\\{}{w}\circ{h}^{B}:{w}\in\ker(N)\cap\mathcal{K},\
{h}\in\mathcal{X}\big{\\}},$
$\displaystyle\mathcal{D}_{F_{\cdot,c},\mathcal{K}}(\mathcal{X})$
$\displaystyle=\big{\\{}{w}\circ
h^{B}:{w}\in\ker(N)\cap\mathcal{K},h\in\mathcal{X},\ c=Wh^{-1}\big{\\}},$
$\displaystyle\mathcal{D}_{F,\mathcal{A}}(\mathcal{X})$
$\displaystyle=\big{\\{}({w}\circ
h^{B},Wh^{-1}):{w}\in\ker(N)\cap\mathcal{K},h\in\mathcal{X}\big{\\}}.$

###### Proof.

We note that $\kappa\in\mathcal{D}_{f,\mathcal{K}}(\mathcal{X})$ if and only
if $\kappa\circ{x}^{B}\in\ker(N)\cap\mathcal{K}$ for some ${x}\in\mathcal{X}$.
This holds if and only if $\kappa={w}\circ(x^{-1})^{B}$ for some
${w}\in\ker(N)\cap\mathcal{K}$ and ${x}\in\mathcal{X}$. By letting
${h}=x^{-1}$ we obtain the desired equality. Similarly, for $F$ and
$F_{\cdot,c}$, all we need is to include the extra equation $c=Wx$, which
translates into $c=Wh^{-1}$ as $h=x^{-1}$. ∎

######  Proposition 3.4.

With $f,F,F_{\cdot,c}$ as in (3.1), (3.2) and (3.3), let $\mathbb{M}$ be a
multiplicative subgroup of $\mathbb{C}^{*}$,
$\mathcal{K}\in\\{\mathbb{M}^{r},(\mathbb{M}\cup\\{0\\})^{r}\\}$, and
$\mathcal{X}\subseteq\mathbb{M}^{n}$ a multiplicative subgroup. Let
$\mathcal{A}\subseteq\mathcal{K}\times\mathbb{C}^{r}$. Then the following
holds:

  * (i)

There exists a nondegenerate solution of $f_{\kappa}(x)=0$ in $\mathcal{X}$
for some $\kappa\in\mathcal{K}$ if and only if
$N\operatorname{diag}(w)B^{\top}\in\mathbb{C}^{s\times n}$ has rank $s$ for
some $w\in\ker(N)\cap\mathcal{K}$.

  * (ii)

There exists a nondegenerate solution of $F_{\kappa,c}(x)=0$ in $\mathcal{X}$
for some $(\kappa,c)\in\mathcal{A}$ (resp. for some $\kappa\in\mathcal{K}$,
$c$ fixed) if and only if the matrix

$\left[\begin{array}[]{c}N\operatorname{diag}(w)B^{\top}\operatorname{diag}(h)\\\
W\end{array}\right]\in\mathbb{C}^{n\times n}$

has rank $n$ for some $w\in\ker(N)\cap\mathcal{K}$ and some $h\in\mathcal{X}$
(resp. some $h\in\mathcal{X}$ such that $c=Wh^{-1}$).

###### Proof.

An easy computation shows that

$J_{f_{\kappa}}({x})=N\operatorname{diag}({\kappa}\circ{x}^{B})B^{\top}\operatorname{diag}(x^{-1}).$

Statement (i) follows directly from this, by noting that the set of vectors
${\kappa}\circ{x}^{B}$ for which $f(\kappa,{x})=0$, $\kappa\in\mathcal{K}$ and
${x}\in\mathcal{X}$ is exactly $\ker(N)\cap\mathcal{K}$ as
$(1,\dots,1)\in\mathcal{X}$. Statement (ii) follows similarly by noting that

$J_{F_{\kappa,c}}({x})=\begin{bmatrix}N\operatorname{diag}({\kappa}\circ{x}^{B})B^{\top}\operatorname{diag}(x^{-1})\\\
W\end{bmatrix}$

and letting $h=x^{-1}$. ∎

######  Remark 3.5.

Suppose $\mathbb{M}\subseteq\mathbb{C}^{*}$ is a multiplicative subgroup that
is Zariski dense in $\mathbb{C}$ (for instance $\mathbb{R}_{>0}$ or
$\mathbb{R}^{*}$). If in addition $\ker(N)\cap\mathbb{M}^{r}\neq\varnothing$,
then $\ker(N)\cap\mathbb{M}^{r}$ is Zariski dense in $\ker(N)$. As a
consequence, (i) in Proposition 3.4 is equivalent to the existence of some
$w\in\ker(N)$ such that
$\operatorname{rk}(N\operatorname{diag}(w)B^{\top})=s$. Similarly, (ii) is
equivalent to the existence of some $w\in\ker(N)$ and $h\in\mathcal{X}$ such
that

$\operatorname{rk}\left[\begin{array}[]{c}N\operatorname{diag}(w)B^{\top}\operatorname{diag}(h)\\\
W\end{array}\right]=s.$

This makes it significantly easier to verify the conditions in (i) and (ii)
computationally in concrete examples, as we will see in Subsection 3.5.

We now use these propositions to show the main result of this subsection. The
statement is given in the full setting where
$\mathcal{X}=(\mathbb{C}^{*})^{n}$ and the parameter space is the maximal
complex space considered in each situation. We will see later on that the
statement also holds for common sets $\mathcal{A},\mathcal{X}$ (Theorem 3.7).

######  Theorem 3.6.

Consider $f$ as in (3.1) and $F$ as in (3.2). Let $(g,\mathcal{A})$ be either

  1. (i)

$(f,\mathbb{C}^{r})$,

  2. (ii)

$(F,\mathbb{C}^{r}\times\mathbb{C}^{d})$, or

  3. (iii)

$(F_{\cdot,c},\mathbb{C}^{r})$ for some fixed $c$.

For each of these cases, it holds that, under the assumption that
$\mathcal{D}_{g,\mathcal{A}}\neq\varnothing$, all solutions
${x}^{*}\in(\mathbb{C}^{*})^{n}$ of $g_{\alpha}(x)=0$ for all
$\alpha\in\mathcal{D}_{g,\mathcal{A}}$ are degenerate if and only if
$\mathcal{D}_{g,\mathcal{A}}$ has empty Euclidean interior in $\mathcal{A}$.

###### Proof.

For the three polynomial systems, the reverse implication follows from Lemma
2.15.

For the forward implication, we will use Proposition 3.3 and 3.4 with
$\mathbb{M}=\mathbb{C}^{*}$ and $\mathcal{X}=(\mathbb{C}^{*})^{n}$. Let us
start with (i). We assume all solutions ${x}^{*}\in(\mathbb{C}^{*})^{n}$ of
$f_{\kappa}(x)=0$ degenerate. Proposition 3.3 gives that
$\mathcal{D}_{f,\mathcal{K}}(\mathcal{X})$ is contained in the Zariski closure
in $\mathbb{C}^{r}$ of the image of the map

$\varphi\colon\mathbb{C}^{r-s}\times(\mathbb{C}^{*})^{n}\to\mathbb{C}^{r},\quad({u},{h})\mapsto
G{u}\circ{h}^{B},$ (3.4)

where $G\in\mathbb{C}^{r\times(r-s)}$ is any matrix whose columns form a basis
for $\ker(N)$ (recall we are assuming $N$ has full rank). We show now that
$\overline{\operatorname{im}(\varphi)}$ has dimension strictly less than $r$,
by proving that the rank of the Jacobian of $\varphi$,

$J_{\varphi}({u},{h})=\big{[}\,\operatorname{diag}({h}^{B})G\mid\operatorname{diag}(G{u}\circ{h}^{B})B^{\top}\operatorname{diag}(h^{-1})\,\big{]}\in\mathbb{C}^{r\times(r-s+n)},$
(3.5)

is strictly smaller than $r$ for all $({u},{h})$ Specifically, we will do this
by exhibiting more than $n-s$ linearly independent vectors in
$\ker(J_{\varphi}({u},{h}))$. The rank of $J_{\varphi}({u},{h})$ agrees with
the rank of

$C(u,h)=\Big{[}\,\operatorname{diag}({h}^{B})G\mid\operatorname{diag}(G{u}\circ{h}^{B})B^{\top}\,\Big{]}\in\mathbb{C}^{r\times(r+n-s)}$

so we focus on this matrix instead. The first block of $C(u,h)$ has $r-s$
columns and the second blok has $n$ columns.

We can immediately find $n-\operatorname{rk}(B)$ linearly independent vectors
${\delta}_{j}\in\mathbb{C}^{n}$ for $j=1,\dots,n-\operatorname{rk}(B)$ from
$\ker(B^{\top})$. Then for each $j$, $(0,\delta_{j})\in\mathbb{C}^{r-s+n}$
belongs to $\ker(C(u,h))$. If $\operatorname{rk}(B)<s$, we are done. If not,
then our assumption that all solutions are degenerate gives us that
$\operatorname{rk}(N\operatorname{diag}({w})B^{\top})<s$ for all
${w}\in\ker(N)\cap(\mathbb{C}^{*})^{r}$ by Proposition 3.4(i). This in turn
implies that $\operatorname{rk}(N\operatorname{diag}({w})B^{\top})<s$ for all
${w}\in\ker(N)$ as $(\mathbb{C}^{*})^{r}$ is Zariski dense in
$\mathbb{C}^{r}$. Hence,

$\displaystyle n-s$
$\displaystyle<\dim(\ker(N\operatorname{diag}(G{u})B^{\top}))$
$\displaystyle=\dim(\ker(B^{\top}))+\dim(\ker(N)\cap\operatorname{im}(\operatorname{diag}(G{u})B^{\top}))$
$\displaystyle=n-\operatorname{rk}(B)+\dim(\operatorname{im}(G)\cap\operatorname{im}(\operatorname{diag}(G{u})B^{\top})),$

for any ${u}\in\mathbb{C}^{r-s}$. From this we conclude that there are
$p>\operatorname{rk}(B)-s\geq 0$ linearly independent vectors

$
{\gamma}_{i}\in\operatorname{im}(G)\cap\operatorname{im}(\operatorname{diag}(G{u})B^{\top}),\quad\text{for
}i=1,\dots,p.$

For each $i$,
${\gamma}_{i}=G{\alpha}_{i}=\operatorname{diag}(G{u})B^{\top}{\beta}_{i}$ for
some $\alpha_{i}\in\mathbb{C}^{r-s}$, $\beta_{i}\in\mathbb{C}^{n}$. Now, for
$i=1,\dots,p$, $(-{\alpha}_{i},{\beta}_{i})\in\ker(C({u},{h}))$ by
construction.

All that is left is to see that the collection of vectors $(0,{\delta}_{j})$
and $(-{\alpha}_{i},{\beta}_{i})$ is linearly independent. Assume that we have
a linear combination

$\sum_{i=1}^{p}a_{i}(-{\alpha}_{i},{\beta}_{i})+\sum_{j=1}^{n-\operatorname{rk}(B)}b_{j}(0,{\delta}_{j})=0.$

Then $\sum a_{i}{\alpha}_{i}=0$, which after multiplication by $G$ gives $\sum
a_{i}{\gamma}_{i}=0$, and we conclude that $a_{i}=0$ for all $i$. Then from
$0=\sum b_{j}{\delta}_{j}$ we conclude that $b_{j}=0$ for all $j$. All in all,
we have now found
$p+n-\operatorname{rk}(B)>\operatorname{rk}(B)-s+n-\operatorname{rk}(B)=n-s$
linearly independent vectors in the kernel of $C({u},{h})$. This concludes the
proof for case (i).

To show the forward implication in (ii), Proposition 3.3 gives that
$\mathcal{D}_{F,\mathcal{A}}(\mathcal{X})$ is contained in the Zariski closure
in $\mathbb{C}^{r}\times\mathbb{C}^{d}$ of the image of the map

$\psi\colon\mathbb{C}^{r-s}\times(\mathbb{C}^{*})^{n}\to\mathbb{C}^{r}\times\mathbb{C}^{d},\quad({u},h)\mapsto(G{u}\circ
h^{B},Wh^{-1}),$ (3.6)

where $G\in\mathbb{C}^{r\times(r-s)}$ is any matrix whose columns form a basis
for $\ker(N)$. We proceed as in case (i). The Jacobian matrix of $\psi$ is the
square matrix

$J_{\psi}(u,h)=\left[\begin{array}[]{cc}\operatorname{diag}(h^{B})G&\operatorname{diag}(G{u}\circ
h^{B})B^{\top}\operatorname{diag}(h^{-1})\\\
0&-W\operatorname{diag}(h^{-2})\end{array}\right]\in\mathbb{C}^{(r+d)\times(r-s+n)}.$

By multiplying this matrix from the right by
$\operatorname{diag}((1,\dots,1),h^{-2})\in\mathbb{C}^{r-s}\times\mathbb{C}^{n}$,
the rank of this matrix agrees with the rank of

$C(u,h):=\left[\begin{array}[]{cc}\operatorname{diag}(h^{B})G&\operatorname{diag}(G{u}\circ
h^{B})B^{\top}\operatorname{diag}(h)\\\
0&W\end{array}\right]\in\mathbb{C}^{(r+d)\times(r+d)}.$

We now show that $\overline{\operatorname{im}(\psi)}$ has dimension strictly
less than $r+d$, by finding a nonzero vector in the kernel of $C(u,h)$ for all
$(u,h)\in\mathbb{C}^{r-s}\times(\mathbb{C}^{*})^{n}$. Proceeding as above, by
Proposition 3.4(ii), as all solutions of $F_{\kappa,c}(x)=0$ are degenerate,
the kernel of the matrix

$\left[\begin{array}[]{c}N\operatorname{diag}(Gu)B^{\top}\operatorname{diag}(h)\\\
W\end{array}\right]$

is nonzero for all $(u,h)\in\mathbb{C}^{r-s}\times(\mathbb{C}^{*})^{n}$. Let
$\beta\in\mathbb{C}^{n}$ be a nonzero vector in the kernel of this matrix.
Hence $W\beta=0$ and
$N\operatorname{diag}(Gu)B^{\top}\operatorname{diag}(h)(\beta)=0$ and it
follows that $\operatorname{diag}(Gu)B^{\top}(h\circ\beta)=G\alpha$ for some
$\alpha\in\mathbb{C}^{r-s}$. Then, the vector $(-\alpha,\beta)$, which is
nonzero, belongs to $\ker(C(u,h))$. This shows case (ii).

To show the forward implication for (iii), we note that the set
$\\{h\in(\mathbb{C}^{*})^{n}:Wh^{-1}=c\\}$ admits an injective rational
parametrization

$ \Phi\colon U\rightarrow(\mathbb{C}^{*})^{n}$

where $U$ is a nonempty Zariski open subset of $\mathbb{C}^{s}$. This
parametrization is obtained as the composition of a parametrization of the
linear variety $\\{x\in(\mathbb{C}^{*})^{n}:Wx=c\\}$ with the inverse map from
$(\mathbb{C}^{*})^{n}$ to $(\mathbb{C}^{*})^{n}$ sending $x$ to $x^{-1}$. It
follows that $J_{\Phi}(z)$ has full rank $s$ for all $z\in U$.

Using this and Proposition 3.3, we obtain that
$\mathcal{D}_{F_{\cdot,c},\mathcal{A}}(\mathcal{X})$ is contained in the
Zariski closure in $\mathbb{C}^{r}$ of the image of the composition

$\widetilde{\varphi}:=\varphi\circ({\rm
Id}_{r-s}\times\Phi)\colon\>\mathbb{C}^{r-s}\times
U\rightarrow\mathbb{C}^{r},$

with $\varphi$ as given in (3.4) and ${\rm Id}_{r-s}$ the identify map on
$\mathbb{C}^{r-s}$, corresponding to the $u$ variables. We show that the
Jacobian of $\widetilde{\varphi}$ has rank strictly smaller than $r$, and from
this the statement follows as in the previous cases.

By the multivariate chain rule, we have that

$\displaystyle
J_{\widetilde{\varphi}}(u,z)=J_{\varphi}(u,\Phi(z))\left[\begin{array}[]{cc}{\rm
Id}_{(r-s)\times(r-s)}&0_{(r-s)\times s}\\\
0_{n\times(r-s)}&J_{\Phi}(z)\end{array}\right].$ (3.9)

As $W\Phi(z)=0$ identically, implicit differentiation gives $WJ_{\Phi}(z)=0.$
Furthermore, as by construction $J_{\Phi}(z)\in\mathbb{C}^{n\times s}$ has
full rank $s$, its columns form a basis of $\ker(W)$.

We proceed as in case (ii) and use that the kernel of the matrix
$\left[\begin{array}[]{c}N\operatorname{diag}(Gu)B^{\top}\operatorname{diag}(\Phi(z))\\\
W\end{array}\right]$ is nonzero for all $u$ and all $\Phi(z)$ by assumption,
to construct a nonzero vector $(-\alpha,\beta)$ in the kernel of
$J_{\varphi}(u,\Phi(z))$, see (3.5), which additionally satisfies $W\beta=0$.
This last condition implies that $\beta\in\operatorname{im}J_{\Phi}(z)$.
Hence,

$(-\alpha,\beta)^{\top}=\left[\begin{array}[]{cc}{\rm
Id}_{(r-s)\times(r-s)}&0_{(r-s)\times s}\\\
0_{n\times(r-s)}&J_{\Phi}(z)\end{array}\right]\gamma$

for some $\gamma\in\mathbb{C}^{r}$. By (3.9), we found a vector $\gamma$ in
the kernel of $J_{\widetilde{\varphi}}(u,z)$. As this holds for all
$(u,z)\in\mathbb{C}^{r-s}\times U$, the closure of the image of
$\widetilde{\varphi}$ is a proper Zariski closed set of $\mathbb{C}^{r}$. ∎

###  3.4. The main theorem on generic dimension.

Putting all this together, we have obtained the following theorem.

######  Theorem 3.7.

Consider $f$ and $\mathcal{K}$ as in (3.1) and $F$ and $\mathcal{C}$ as in
(3.2). Let the objects $g,\mathcal{A}\subseteq\mathbb{C}^{\ell},\ell,\delta$
be either

  1. (a)

$g=f$, $\mathcal{A}=\mathcal{K}$, $\ell=r$, $\delta=d$,

  2. (b)

$g=F$, $\mathcal{A}=\mathcal{K}\times\mathcal{C}$, $\ell=r+d$, $\delta=0$, or

  3. (c)

$g=F_{\cdot,c}$, $\mathcal{A}=\mathcal{K}$, $\ell=r$, $\delta=0$ for some
fixed $c\in\mathcal{C}$.

We consider any such pair $(g,\mathcal{A})$, and
$\mathcal{X}\in(\mathbb{C}^{*})^{n}$ satisfying:

  * •

$\mathcal{A}$ is locally Zariski dense in $\mathbb{C}^{\ell}$,

  * •

$(\mathcal{A}\times\mathcal{X})\cap\mathcal{E}_{g}$ is Zariski dense in
$\mathcal{E}_{g}$,

  * •

$\mathcal{A},\mathcal{X}$ are such that Proposition 2.12 applies.

Then the following are equivalent:

  1. (i)

There is a nondegenerate solution of $g_{\alpha}({x})=0$ in
$(\mathbb{C}^{*})^{n}$ for some $\alpha\in\mathbb{C}^{\ell}$.

  2. (ii)

There is a nondegenerate solution of $g_{\alpha}({x})=0$ in $\mathcal{X}$ for
some $\alpha\in\mathcal{A}$.

  3. (iii)

There exists a nonempty Zariski open subset
$\mathcal{U}\subseteq\mathcal{E}_{g}$ such that for any
$(\alpha^{*},x^{*})\in\mathcal{U}\cap(\mathcal{A}\times\mathcal{X})$, $x^{*}$
is a nondegenerate solution of $g_{\alpha^{*}}(x^{*})=0$.

  4. (iv)

$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ is Zariski dense in
$\mathbb{C}^{\ell}$.

  5. (v)

$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ has nonempty Euclidean interior in
$\mathcal{A}$.

  6. (vi)

$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is equidimensional of dimension
$\delta$ for generic $\alpha\in\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$, that
is, for all $\alpha$ in a nonempty Zariski open subset
$U\cap\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ of
$\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$.

  7. (vii)

$\mathbb{V}_{\mathbb{C}^{*}}(g_{\alpha})$ is equidimensional of dimension
$\delta$ for at least one $\alpha\in\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$.

Furthermore, any of these imply:

  1. (1)

$\mathbb{V}_{\mathbb{C}^{*}}^{\mathcal{X}}(g_{\alpha})$ is equidimensional of
dimension $\delta$ for generic
$\alpha\in\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$.

  2. (2)

If $\mathcal{A}\subseteq\mathbb{R}^{\ell}$ and
$\mathcal{X}\subseteq(\mathbb{R}^{*})^{n}$, then
$\mathbb{V}_{\mathbb{R}^{*}}^{\mathcal{X}}(g_{\alpha})$ and
$\mathbb{V}_{\mathbb{R}^{*}}(g_{\alpha})$ have dimension $\delta$ for generic
$\alpha\in\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$.

###### Proof.

First, we show that (i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iii). We have
that (iii)$\Rightarrow$(ii), as $\mathcal{U}$ is a nonempty Zariski open set
and $(\mathcal{A}\times\mathcal{X})\cap\mathcal{E}_{g}$ is Zariski dense in
$\mathcal{E}_{g}$, guaranteeing that the intersection
$\mathcal{U}\cap(\mathcal{A}\times\mathcal{X})$ is nonempty.
(ii)$\Rightarrow$(i) holds trivially. To prove (i)$\Rightarrow$(iii), note
that the subset of points $(\alpha,x)$ of $\mathcal{E}_{g}$ for which $x$ is a
nondegenerate solution of $g_{\alpha}(x)=0$ is Zariski open if nonempty.

The equivalence (iv)$\Leftrightarrow$(v) is Proposition 2.12, which applies by
the assumptions on $\mathcal{A}$ and $\mathcal{X}$.

We now show (iv)$\Rightarrow$(vi)$\Rightarrow$(vii)$\Rightarrow$(i). The
implication (iv)$\Rightarrow$(vi) is Corollary 2.7. The implication
(vi)$\Rightarrow$(vii) is clear. For the implication (vii)$\Rightarrow$(i), we
note that (vii) implies that $\mathcal{D}_{g,\mathcal{A}}$ is Zariski dense in
$\mathbb{C}^{\ell}$ by Theorem 2.2 but, as
$\mathcal{D}_{g,\mathcal{A}}\subseteq\mathcal{D}_{g,\mathbb{C}^{\ell}}$, so is
$\mathcal{D}_{g,\mathbb{C}^{\ell}}$. Hence $\mathcal{D}_{g,\mathbb{C}^{\ell}}$
has nonempty Euclidean interior in $\mathbb{C}^{\ell}$ by Proposition 2.12,
which leads to (i) by Theorem 3.6.

All that is left is to show that (ii)$\Rightarrow$(iv). By Proposition 2.12
and Theorem 3.6, (ii) gives that $\mathcal{D}_{g,\mathbb{C}^{\ell}}$ is
Zariski dense in $\mathbb{C}^{\ell}$. The assumption that
$(\mathcal{A}\times\mathcal{X})\cap\mathcal{E}_{g}$ is Zariski dense in
$\mathcal{E}_{g}$, gives that

$
\overline{\mathcal{D}_{g,\mathcal{A}}(X)}=\overline{\pi((\mathcal{A}\times\mathcal{X})\cap\mathcal{E}_{g})}=
\overline{\pi(\overline{(\mathcal{A}\times\mathcal{X})\cap\mathcal{E}_{g}})}=\overline{\pi(\mathcal{E}_{g})}=\overline{\mathcal{D}_{g,\mathbb{C}^{\ell}}}=\mathbb{C}^{\ell}.$

Hence, $\mathcal{D}_{g,\mathcal{A}}(\mathcal{X})$ is Zariski dense in
$\mathbb{C}^{\ell}$, which gives (iv). This concludes the first part of the
proof.

Finally, (iv)$\Rightarrow$(1) follows by Corollary 2.7, and
(ii)$\Rightarrow$(2) follows by Corollary 2.16. ∎

######  Remark 3.8.

For any $(g,\mathcal{A})$ and $\mathcal{X}$ under the assumptions of Theorem
3.7, any statement (ii)-(vii) in Theorem 3.7 is equivalent to (i), and hence
to any of the statements for $\mathcal{A}=\mathbb{C}^{\ell}$ and
$\mathcal{X}=(\mathbb{C}^{*})^{n}$ instead.

######  Remark 3.9.

We note that the equivalences (i)$\Leftrightarrow$(ii)$\Leftrightarrow$(iii)
in Theorem 3.7 work in the more general setting of Proposition 3.3 and 3.4,
but for the equivalences (iv)$\Leftrightarrow$(v) and (vii)$\Rightarrow$(v),
Proposition 2.12 is required, which relies on Lemma 2.11.

######  Remark 3.10.

The following scenarios that are common in applications satisfy the hypotheses
in Theorem 3.7:

  1. (i)

$\mathcal{K}\in\\{\mathbb{R}^{r}_{>0},(\mathbb{R}^{*})^{r},\mathbb{R}^{r}\\}$,
$\mathcal{C}\in\\{\mathbb{R}^{d}_{>0},(\mathbb{R}^{*})^{d},\mathbb{R}^{d}\\}$
and
$\mathcal{X}\in\\{\mathbb{R}^{n}_{>0},(\mathbb{R}^{*})^{n},(\mathbb{C}^{*})^{n}\\}$.

  2. (ii)

$\mathcal{K}\in\\{(\mathbb{C}^{*})^{r},\mathbb{C}^{r}\\}$,
$\mathcal{C}\in\\{(\mathbb{C}^{*})^{d},\mathbb{C}^{d}\\}$ and
$\mathcal{X}=(\mathbb{C}^{*})^{n}$.

In general, we note that if $\mathcal{A}$ and $\mathcal{X}$ are Euclidean open
subsets of $\mathbb{R}^{\ell}$ and $\mathbb{R}^{n}$ respectively, or of
$\mathbb{C}^{\ell}$ and $\mathbb{C}^{n}$ respectively, then
$(\mathcal{A}\times\mathcal{X})\cap\mathcal{E}_{g}$ is Zariski dense in
$\mathcal{E}_{g}$ if this intersection is nonempty. In the real case, this
follows from the fact that $\mathcal{E}_{g}$ is nonsingular by Proposition
3.1(iii), together with the fact that for a complex irreducible variety
defined by polynomials with real coefficients that has at least one
nonsingular point, the real points form a Zariski dense subset [Sot19, Thm.
5.1].

###  3.5. Computational considerations

In the case when the matrix $N$ has rational entries, the criterion (i) in
Theorem 3.7 can be checked computationally through Proposition 3.4.

The idea is the following: Let $G\in\mathbb{Q}^{r\times(n-s)}$ be a matrix
whose columns form a basis for $\ker(N)$. For $f$, we want to check whether
there exists some $u\in\mathbb{Q}^{n-s}$ such that

$\operatorname{rk}(N\operatorname{diag}(Gu)B^{\top})=s.$

Note that the set of $u$ for which the rank is $s$ forms a Zariski open subset
of $\mathbb{Q}^{n-s}$, so if it is nonempty, a randomly chosen
$u\in\mathbb{Q}^{n-s}$ will give
$\operatorname{rk}(N\operatorname{diag}(Gu)B^{\top})=s$ with probability 1.
Hence, we pick a random $u\in\mathbb{Q}^{n-s}$, and compute
$N\operatorname{diag}(Gu)B^{\top}$ with exact arithmetic. If the rank is $s$,
we conclude that criterion (i) in Theorem 3.7 holds. If not, we can suspect
that it does not hold, and to conclusively prove this, we view
$N\operatorname{diag}(Gu)B^{\top}$ as a symbolic matrix with indeterminate $u$
and computationally verify that all $s\times s$ minors are the zero
polynomial. For $F$, we instead want to determine whether there is some $u$
such that

$\operatorname{rk}\begin{bmatrix}N\operatorname{diag}(Gu)B^{\top}\\\
W\end{bmatrix}=n,$

which is done analogously.

In the setting of Proposition 3.4 for $f$, where
$\mathcal{K}\in\\{\mathbb{M}^{r},(\mathbb{M}\cup\\{0\\})^{r}\\}$ and
$\mathbb{M}$ is a multiplicative subgroup of $\mathbb{C}^{*}$, we know by
Remark 3.10 that to check denseness of
$(\mathcal{K}\times\mathcal{X})\cap\mathcal{E}_{g}$ in $\mathcal{E}_{g}$ it is
enough to check that
$(\mathcal{K}\times\mathcal{X})\cap\mathcal{E}_{g}\neq\varnothing$. This, in
turn, is equivalent to $\ker(N)\cap\mathcal{K}\neq\varnothing$. Indeed, if
$\ker(N)\cap\mathcal{K}=\varnothing$, then
$\mathbb{V}_{\mathbb{C}^{*}}(f)=\varnothing$ for all $\kappa\in\mathcal{K}$.
On the other hand, if $\ker(N)\cap\mathcal{K}\neq\varnothing$, then
$(1,\dots,1)\in\mathbb{V}_{\mathbb{C}^{*}}(f)$ for any
$\kappa\in\ker(N)\cap\mathcal{K}$. In the special case
$\mathbb{M}=\mathbb{R}_{>0}$, checking
$\ker(N)\cap\mathbb{R}_{>0}^{r}\neq\varnothing$ corresponds to showing the
existence of an interior point of the polyhedral cone
$\ker(N)\cap\mathbb{R}_{\geq 0}^{r}$, which is a straightforward computation
using either linear programming, or satisfiability modulo theories solvers
[BFC17].

##  4\. Application to Reaction Networks

As explained in the introduction, the main motivation behind the development
of the results in Section 3 comes from the study of reaction networks. This
connection will be explained in this section. To make this section accessible
to readers familiar with reaction networks but not necessarily acquainted with
the background and language of algebraic geometry, we will use a less
technical approach. In particular, when saying that a property holds for
almost all parameters in a set $\mathcal{A}\subseteq\mathbb{C}^{\ell}$, we
mean that the property holds generically in the set, that is, in a nonempty
Zariski open subset of $\mathcal{A}$. In the reaction network scenario where
$\mathcal{A}$ is $\mathbb{R}^{\ell}$ or $\mathbb{R}^{\ell}_{>0}$, this implies
that the property holds outside a subset of $\mathcal{A}$ of Lebesgue measure
zero (so for almost all parameters, measure theoretically speaking, it holds
too).

###  4.1. Reaction networks

In what follows we present some generalities, necessary to establish the
connection of our results with some questions arising in this field, but we
refer to [Fei19] for further details.

A reaction network is normally pictured by means of a graph, whose vertices
correspond to linear combinations of the species involved (complexes),
connected by a directed edge whenever an interaction is assumed to lead from
any of these complexes (source) to another one (product). A simple reaction
network modeling the enzymatic transfer of calcium ions is

$\displaystyle 0$ $\displaystyle\ce{<=>[\kappa_{1}][\kappa_{2}]}X_{1}$ (4.1)
$\displaystyle X_{1}+X_{2}$ $\displaystyle\ce{->[\kappa_{3}]}2X_{1}$
$\displaystyle X_{1}+X_{3}$
$\displaystyle\ce{<=>[\kappa_{4}][\kappa_{5}]}X_{4}\ce{->[\kappa_{6}]}X_{2}+X_{3}.$

Here, $X_{1}$ stands for cytosolic calcium, $X_{2}$ for calcium in the
endoplasmic reticulum, and $X_{3}$ is an enzyme catalyzing the transfer via
the formation of an intermediate protein complex $X_{4}$ [GES05]. The labels
of the reactions are positive real numbers called reaction rate constants.
Given any reaction network with species $X_{1},\dots,X_{n}$ and reactions

$
b_{1i}X_{1}+\dots+b_{ni}X_{n}\ce{->[\kappa_{i}]}a_{1i}X_{1}+\dots+a_{ni}X_{n},\quad
i=1,\dots,r,$

under the assumption of mass-action kinetics, the concentration
$x=(x_{1},\dots,x_{n})$ of the species in time is modeled by means of a system
of ordinary differential equations (ODEs) of the form

$ \frac{dx}{dt}=\Gamma(\kappa\circ x^{B}),\quad x\in\mathbb{R}^{n}_{\geq 0},$
(4.2)

for reaction rate constants
$\kappa=(\kappa_{1},\dots,\kappa_{r})\in\mathbb{R}^{r}_{>0}$, stoichiometric
matrix $\Gamma\in\mathbb{Z}^{n\times r}_{\geq 0}$ with entries
$(a_{ji}-b_{ji})$, and reactant matrix $B\in\mathbb{Z}^{n\times r}$ with
entries $b_{ji}$. The trajectories of the ODE system are confined into
parallel translates of the image of $\Gamma$, which can be described by linear
equations $Wx-c=0$ with $c$ depending on the initial condition and
$W\in\mathbb{R}^{d\times n}$ with $d=n-\operatorname{rk}(\Gamma)$. The
intersection of each such linear subspace with $\mathbb{R}^{n}_{\geq 0}$ is
called a stoichiometric compatibility class. We note that for common networks
in biochemistry, the rank of $\Gamma$ is not maximal.

The reader might already notice the similarities with the parametric systems
(3.1) and (3.2). These arise naturally in this context as follows:

  * •

The positive steady state variety $V_{\kappa}$ for a given
$\kappa\in\mathbb{R}^{r}_{>0}$ is the set of steady states of (4.2) in
$\mathbb{R}^{n}_{>0}$. By letting $N\in\mathbb{Z}^{s\times n}$ be a matrix of
full rank $s=\operatorname{rk}(\Gamma)$ and
$\operatorname{im}(N^{\top})=\operatorname{im}(\Gamma^{\top})$, the positive
steady state variety is the solution set to

$N(\kappa\circ x^{B})=0,\quad x\in\mathbb{R}^{n}_{>0}.$ (4.3)

This corresponds to the parametric system $f(\kappa,x)$ from (3.1) with
$\mathcal{K}=\mathbb{R}^{r}_{>0}$ and $\mathcal{X}=\mathbb{R}^{n}_{>0}$. The
complex (resp. real) steady state variety is the set of solutions to (4.3) in
$(\mathbb{C}^{*})^{n}$ (resp. $(\mathbb{R}^{*})^{n}$).

  * •

The set of positive steady states within stoichiometric compatibility classes
$P_{\kappa,c}$ for a given $\kappa\in\mathbb{R}^{r}_{>0}$ and
$c\in\mathbb{R}^{d}$ is the set of steady states of (4.2) in
$\mathbb{R}^{n}_{>0}$ that additionally belong to the stoichiometric
compatibility class defined by $c$, that is, solutions to

$N(\kappa\circ x^{B})=0,\quad Wx-c=0.$ (4.4)

This corresponds to the parametric system $F(\kappa,c,x)$ from (3.2) with
$\mathcal{K}=\mathbb{R}^{r}_{>0}$ and $\mathcal{C}=\mathbb{R}^{d}$ with
$d=n-s$. By $P_{\kappa,c}^{\mathbb{C}}$ we denote the set of solutions to
(4.3) in $(\mathbb{C}^{*})^{n}$.

Since the reaction rate constants are normally unknown, studying the dynamics
of a reaction network often implies understanding the behavior of the system
for all parameters, maybe generically, or being able to describe the parameter
regions leading to different behaviors.

In the reaction network literature, a steady state $x^{*}$ is said to be
degenerate if it is degenerate as a solution to the system
$F_{\kappa,Wx^{*}}(x)=0$ (equivalently, if
$\ker(J_{f_{\kappa}}(x^{*}))\cap\operatorname{im}(\Gamma)\neq\\{0\\}$.)

The existence of nondegenerate steady states is assumed to be common, although
not guaranteed for all networks as we will see below. The existence of
nondegenerate steady states is often a requirement to “lift” steady states
from one network to another with arguments relying on the implicit function
theorem or homotopy continuation [JS13, FW13, CF06, BP18]. The relation
between nondegenerate steady states and nonsingularity pointed out in
Proposition 2.14 allows also to employ machinery from algebraic geometry to
the study of steady states [PEF22].

###  4.2. Theorems on dimension and finiteness of steady states

In the language of reaction network theory, Theorem 3.7 gives rise to the
following results.

######  Theorem 4.1 (Expected dimension of the complex steady state variety).

Consider a reaction network with $n$ species, $r$ reactions, and
stoichiometric matrix $\Gamma$. Consider the polynomial function $f(\kappa,x)$
defining $V_{\kappa}$ as in (4.3) with $B$ the reactant matrix, and
$N\in\mathbb{Z}^{s\times n}$ of full rank $s=\operatorname{rk}(\Gamma)$ and
such that $\operatorname{im}(N^{\top})=\operatorname{im}(\Gamma^{\top})$. The
following are equivalent:

  1. (i)

The system $f_{\kappa}(x)=0$ has a nondegenerate solution in
$\mathbb{R}^{n}_{>0}$ for some $\kappa\in\mathbb{R}^{r}_{>0}$.

  2. (ii)

The set of parameters $\kappa$ for which $V_{\kappa}\neq\varnothing$ has
nonempty Euclidean interior in $\mathbb{R}^{r}$.

  3. (iii)

The dimension of the complex steady state variety is $n-s$ for at least one
$\kappa\in\mathbb{R}^{r}_{>0}$ for which $V_{\kappa}\neq\varnothing$.

  4. (iv)

The matrix $N\operatorname{diag}(w)B^{\top}$ has rank $s$ for some
$w\in\ker(N)\in\mathbb{R}^{r}_{>0}$.

If any of these hold, then the dimension of the complex and real steady state
varieties is $n-s$ for almost all $\kappa$ for which
$V_{\kappa}\neq\varnothing$.

If none of these holds, then the complex steady state variety is either empty
or has dimension strictly larger than $n-s$.

###### Proof.

The statement follows from Proposition 3.4(i) and from Theorem 3.7 case (a)
together with Remark 3.10 as $\mathcal{K}=\mathbb{R}^{r}_{>0}$ and
$\mathcal{X}=\mathbb{R}^{n}_{>0}$. ∎

######  Corollary 4.2.

A reaction network such that for all $\kappa\in\mathbb{R}^{r}_{>0}$ the
positive steady state variety is either empty or has dimension higher than
$n-s$, has no positive steady states for almost all
$\kappa\in\mathbb{R}^{r}_{>0}$.

######  Example 4.3.

For the calcium network in (4.1), the stoichiometric matrix and the reactant
matrix are given by

$\Gamma=\left[\begin{array}[]{rrrrrr}1&-1&1&-1&1&0\\\ 0&0&-1&0&0&1\\\
0&0&0&-1&1&1\\\ 0&0&0&1&-1&-1\end{array}\right]\quad\text{and}\quad
B=\begin{bmatrix}0&1&1&1&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\
0&0&0&0&1&1\end{bmatrix}.$

Here, $n=4$, $r=6$, $s=3$, and we can choose

$W=\begin{bmatrix}0&0&1&1\end{bmatrix}.$

We note that $w=(1,1,1,2,1,1)\in\ker(\Gamma)\cap\mathbb{R}^{r}_{>0}$, and that

$\operatorname{rk}\begin{bmatrix}\Gamma\operatorname{diag}(w)B^{\top}\\\
W\end{bmatrix}=4.$

Hence Theorem 4.1(iv) holds and from this we conclude that there exist
positive steady states for $\kappa$ in a set with nonempty Euclidean interior,
and that the complex and real steady state varieties are 1-dimensional for
almost all such reaction rate constants.

######  Example 4.4.

Consider the network

${{\ce{Y}}}$${{\ce{X+Y+Z}}}$${{\ce{X}}}$${{\ce{Y+Z}}}$${{\ce{Z}}}$${\varnothing\hbox
to0.0pt{.\hss}}$$\scriptstyle{\kappa_{1}}$$\scriptstyle{\kappa_{2}}$$\scriptstyle{\kappa_{3}}$$\scriptstyle{\kappa_{4}}$

We have

$f_{\kappa}(x)=\begin{bmatrix}-\kappa_{1}x-\kappa_{2}x+\kappa_{3}yz\\\
\kappa_{1}x-\kappa_{4}yz\\\ \kappa_{2}x-\kappa_{4}yz\end{bmatrix},$

and $V_{\kappa}\neq\varnothing$ precisely when $\kappa_{1}=\kappa_{2}$ and
$\kappa_{3}=2\kappa_{4}$. So Theorem 4.1(ii) does not hold and hence, the
complex steady state variety has dimension strictly larger than $n-s=1$ in
these cases. In fact, for these particular parameter values, the complex
steady state variety is defined by the equation $\kappa_{1}x-\kappa_{4}yz=0$,
and hence has complex dimension $2$.

######  Theorem 4.5 (Finiteness of the number of steady states in
stoichiometric compatibility classes).

Consider a reaction network with $n$ species, $r$ reactions, and
stoichiometric matrix $\Gamma$. Consider the polynomial function
$F(\kappa,c,x)$ defining $P_{\kappa,c}$ as in (4.4) with $B$ the reactant
matrix, $N\in\mathbb{Z}^{s\times n}$ of full rank
$s=\operatorname{rk}(\Gamma)$ such that
$\operatorname{im}(N^{\top})=\operatorname{im}(\Gamma^{\top})$, and $W$ a full
matrix whose rows form a basis of $\ker(\Gamma^{\top})$.

The following are equivalent:

  1. (i)

The network has a nondegenerate steady state in $\mathbb{R}^{n}_{>0}$ for some
$\kappa\in\mathbb{R}^{r}_{>0}$.

  2. (ii)

The set of parameters $(\kappa,c)$ for which $P_{\kappa,c}\neq\varnothing$,
has nonempty Euclidean interior in $\mathbb{R}^{r+d}$.

  3. (iii)

$P_{\kappa,c}^{\mathbb{C}}\neq\varnothing$ is finite for at least one
$\kappa\in\mathbb{R}^{r}_{>0}$ and $c\in\mathbb{R}^{d}$ for which
$P_{\kappa,c}\neq\varnothing$.

  4. (iv)

The matrix

$
\left[\begin{array}[]{c}N\operatorname{diag}(w)B^{\top}\operatorname{diag}(h)\\\
W\end{array}\right]$

has full rank $n$ for some $w\in\ker(N)\in\mathbb{R}^{r}_{>0}$ and
$h\in\mathbb{R}_{>0}^{n}$.

If any of these hold, then $P_{\kappa,c}$ is finite for almost all
$(\kappa,c)\in\mathbb{R}^{r}_{>0}\times\mathbb{R}^{d}$. Additionally, for
almost all values of $(\kappa,c)$, the elements of $P_{\kappa,c}$ are
nondegenerate steady states.

###### Proof.

This follows from Proposition 3.4(ii) and from Theorem 3.7 case (b) together
with Remark 3.10 as $\mathcal{K}=\mathbb{R}^{r}_{>0}$,
$\mathcal{C}=\mathbb{R}^{d}$ and $\mathcal{X}=\mathbb{R}^{n}_{>0}$. The last
part follows from Corollary 3.2. ∎

We note that condition (iv) in Theorem 4.1 and 4.5 holds if
$\ker(N)\cap\mathbb{R}^{r}_{>0}\neq\varnothing$ and the respective conditions
hold over $\mathbb{C}^{*}$ (see Subsection 3.5). We also note that Theorem
4.5(i) implies Theorem 4.1(i), and we obtain the following corollary.

######  Corollary 4.6.

If $P_{\kappa,c}^{\mathbb{C}}\neq\varnothing$ is finite for at least one
$\kappa\in\mathbb{R}^{r}_{>0}$ and $c\in\mathbb{R}^{d}$ for which
$P_{\kappa,c}\neq\varnothing$, then the complex (and the real) steady state
variety has dimension $n-s$ for almost all $\kappa$ for which it is not empty,
and moreover it is nonempty for a set of parameters with nonempty Euclidean
interior.

The converse is not necessarily true, as the following example shows.

######  Example 4.7.

Consider the simple reaction network

$X_{1}+X_{2}\ce{->[\kappa_{1}]}X_{1}\quad X_{2}\ce{->[\kappa_{2}]}2X_{2}.$

The first row of the stoichiometric matrix $\Gamma$ is zero, and hence the
steady states are described by the single parametric polynomial

$f_{\kappa}(x_{1},x_{2})=-\kappa_{1}x_{1}x_{2}+\kappa_{2}=x_{2}(-\kappa_{1}x_{1}+\kappa_{2}).$

We obtain
$V_{\kappa}=\\{(x_{1},x_{2})\in\mathbb{R}^{2}_{>0}:x_{1}=\tfrac{\kappa_{2}}{\kappa_{1}}\\}$,
which is nonempty for all $\kappa$.

On the other hand, the stoichiometric compatibility classes are defined by the
equation $x_{1}=c$ for $c\in\mathbb{R}$. The set $P_{\kappa,c}$ is described
by $x_{1}=\tfrac{\kappa_{2}}{\kappa_{1}}=c$, and hence is empty unless
$\tfrac{\kappa_{2}}{\kappa_{1}}=c$. This shows that Theorem 4.1(i) does not
imply Theorem 4.5(i).

In the next corollaries, we rewrite some of the implications of Theorem 4.5 to
emphasize the consequences that our results have on the existence and number
of positive (nondegenerate) steady states.

######  Corollary 4.8.

If a reaction network has infinitely many steady states within stoichiometric
compatibility classes for all $(\kappa,c)$ for which
$P_{\kappa,c}\neq\varnothing$, then $P_{\kappa,c}=\varnothing$ for almost all
$(\kappa,c)\in\mathbb{R}^{r}_{>0}\times\mathbb{R}^{d}$.

######  Corollary 4.9.

A reaction network that has a nondegenerate steady state for some
$\kappa\in\mathbb{R}^{r}_{>0}$ cannot have infinitely many steady states in
all (not even in most) stoichiometric compatibility classes.

We note that even when the conditions of Theorem 4.5 hold, there might be
values of $\kappa$ for which all steady states are degenerate. Moreover, all
steady states can be degenerate even if the dimension of the steady state
variety has dimension $n-s$. This is illustrated in the following example.

######  Example 4.10.

Consider the reaction network with $s=1$

$3X_{1}+X_{2}\ce{->[\kappa_{1}]}4X_{1}\quad
2X_{1}+X_{2}\ce{->[\kappa_{2}]}3X_{2}\quad
X_{1}+X_{2}\ce{->[\kappa_{3}]}2X_{1},$

giving rise to the polynomial function
$f(\kappa,x)=\kappa_{1}x_{1}^{2}-2\kappa_{2}x_{1}+\kappa_{3}$, after suitably
choosing $N$. For any choice of $\kappa$ such that
$\kappa_{2}^{2}=\kappa_{1}\kappa_{3}$, all positive steady states are
degenerate. For any other $\kappa$, the steady states are all nondegenerate
and therefore, the conditions in Theorem 4.1 and 4.5 apply: for almost all
$\kappa$, the dimension of the complex steady state variety is $n-s=1$ when
nonempty, and for almost all $(\kappa,c)$, there is a finite number of
positive steady states in the corresponding stoichiometric compatibility
class.

######  Example 4.11 (Real networks and steady states).

As an illustration of the applicability of our methods to realistic networks,
we applied the computational steps described in Subection 3.5 to the networks
from the database ODEbase [LSR22], under the assumption of mass-action
kinetics. Out of a total of 610 networks appearing in the database, we found
that precisely 368 have positive steady states. Out of these, 6 networks only
have steady states that are degenerate solutions to $f_{\kappa}(x)=0$, whereas
362 networks have a nondegenerate steady state, and therefore satisfy the
equivalent properties in Theorems 4.1 and 4.5.

The largest network in the database which admits a positive nondegenerate
steady state is BIOMD0000000014, which has 86 species. The computation
verifying this took <2 seconds in Maple***All computations were run in Maple
2023.0 on a 2.70 GHz Intel Core i5 with 8GB RAM.. For comparison, an attempt
of computing the dimension for a fixed random choice of the rate constants via
PolynomialIdeals[HilbertDimension] in Maple failed due to memory problems.

###  4.3. Weakly reversible networks and other particularities

Weakly reversible networks are those for which all connected components of the
underlying digraph are strongly connected. These networks are known to admit
positive steady states for all choices of $\kappa$ [Bor19] and are conjectured
to display some strong dynamical behavior such as persistence [Fei87] or
bounded trajectories [And11].

In [BCY20], the authors considered the following reaction network for which
$n=s=2$

${{\ce{2Y}}}$${{\ce{X+3Y}}}$${{\ce{2X+Y}}}$${{\ce{Y}}}$${{\ce{X+Y}}}$${{\ce{X+2Y}}}$${{\ce{2X+2Y}}}$${{\ce{2X}}}$${{\ce{3X}\hbox
to0.0pt{,\hss}}}$$\scriptstyle{\kappa_{1}}$$\scriptstyle{\kappa_{2}}$$\scriptstyle{\kappa_{4}}$$\scriptstyle{\kappa_{3}}$$\scriptstyle{\kappa_{7}}$$\scriptstyle{\kappa_{8}}$$\scriptstyle{\kappa_{6}}$$\scriptstyle{\kappa_{5}}$$\scriptstyle{\kappa_{10}}$$\scriptstyle{\kappa_{9}}$$\scriptstyle{\kappa_{12}}$$\scriptstyle{\kappa_{11}}$

and fine tuned the reaction rate constants in such a way that the two
equations defining the steady states had a common factor and the other factors
did not admit positive solutions. Hence the complex steady state variety has
dimension 1, while $d=n-s=0$. So the dimension is higher than expected.
Specifically, all parameters were set to one except for
$\kappa_{3}=\kappa_{6}=\kappa_{12}=4$. With this trick, they illustrated that
even for weakly reaction networks, the dimension of the steady state variety
could be larger than expected for some choice of parameter values. Motivated
by this, the authors posed the following question [[BCY20, Section 5]]:

> Is it possible for a weakly reversible network to have infinitely many
> positive steady states [within a stoichiometric class] for each choice of
> reaction rate constants in a [Euclidean] open set of the parameter space
> $\mathbb{R}^{r}_{>0}$?

Theorem 4.5 gives an answer in the negative to this question: at least for
almost all stoichiometric classes, this cannot happen if the network is weakly
reversible. We note that weakly reversible networks admit so called complex
balanced steady states for some choice of reaction rate constants ([Hor72]),
and these are always nondegenerate (see [Fei19, Section 15.2.2]). It follows
that weakly reversible networks admit nondegenerate steady states, and hence
both Theorem 4.1 and Theorem 4.5, as well as Corollary 4.9 apply. This leads
to the following result.

######  Corollary 4.12.

For a weakly reversible network, and with notation as in Theorem 4.5, it holds
that $P_{\kappa,c}$ is finite for almost all
$(\kappa,c)\in\mathbb{R}^{r}_{>0}\times\mathbb{R}^{d}$

Other classes of networks that are known to have nondegenerate steady states
are those that satisfy the criteria in the Deficiency One Theorem of Feinberg
[Fei19, Section 17.1], and injective networks [MFR+15, Thm. 1.4].

To conclude this subsection, we make a connection to some classical objects in
the study of reaction networks: the kinetic and the stoichiometric subspaces.

The stoichiometric subspace $S$ is simply $\operatorname{im}(\Gamma)$, while
the kinetic subspace $S_{\kappa}$ for a given $\kappa$ is given by
$\operatorname{im}(\Sigma_{\kappa})$, with $\Sigma_{\kappa}$ the coefficient
matrix of $f_{\kappa}$ from (4.3). In [FH77], criteria for when the two
subspaces agree are given, but one can construct networks for which they do
not agree for any $\kappa$ (e.g. Example 4.4), or for some values of $\kappa$.
By Remark 2.4, if the network admits a nondegenerate solution to (4.3), then
the generic rank of $\Sigma_{\kappa}$ agrees with $\operatorname{rk}(\Gamma)$
and hence the two subspaces agree generically. It follows that if the two
subspaces do not agree generically, then there is no nondegenerate solution
and consequently, by Corollary 4.2, for almost all values of $\kappa$ there
are no positive steady states.

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Authors’ addresses:

Elisenda Feliu, University of Copenhagen<EMAIL_ADDRESS>
Oskar Henriksson, University of Copenhagen<EMAIL_ADDRESS>
Beatriz Pascual Escudero, Universidad Politécnica de Madrid
<EMAIL_ADDRESS>