Multistationarity in Structured Reaction Networks

Abstract

Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. First, we build on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain “critical function” changes sign. Here, we allow for more general parametrizations, which make it much easier to determine the existence of a sign change. This is particularly simple when the steady-state equations are linearly equivalent to binomials; we give necessary conditions for this to happen, which hold for many networks studied in the literature. We also give a sufficient condition for multistationarity of networks whose steady-state equations can be replaced by equivalent triangular-form equations. Finally, we present methods for finding witnesses to multistationarity, which we show work well for certain structured reaction networks, including those common to biological signaling pathways. Our work relies on results from degree theory, on the existence of explicit rational parametrizations of the steady states, and on the specialization of Gröbner bases.

Appendices

Appendix A: Proof of Theorem 5.3

We now prove Theorem 5.3. We also illustrate the proof in Example A.1. We assume the reader is familiar with the notion of the Laplacian ${\mathcal {L}}(G)$ of a digraph G and its main properties. One important observation is that mass-action kinetics associated with a digraph G with vertices labeled by variables $x_1, \dots , x_s$ equals $\dot{x} ={\mathcal {L}}(G) x$. Another important observation is that when G is strongly connected, the kernel of ${\mathcal {L}}(G)$ has dimension one and there is a known generator $\rho (G)$ with positive entries described as follows. Recall that an i-treeT of a digraph is a spanning tree where the i-th vertex is its unique sink (equivalently, the i-th is the only vertex of the tree with no edges leaving from it) and we call $k^{T}$ the product of the labels of all the edges of T. Then, the i-th coordinate of $\rho (G)$ equals

$$\begin{aligned} \rho _i(G)=\underset{T\; an \; i-tree}{\sum }k^{T}. \end{aligned}$$

(36)

We refer the reader to Mirzaev and Gunawardena (2013) and Tutte (1948) for a detailed account.

Proof of Theorem 5.3

Recall that in a MESSI network there are two types of species: intermediate and core. Our proof proceeds by performing (invertible) linear operations on the steady-state equations, which in the end yield (equivalent) binomial equations.

We begin by operating on the steady-state equations of intermediate species. Given a core complex y, we consider the following set of intermediate complexes: $I_{y}=\{y' \text { intermediate} : y\rightarrow _\circ y'\}$. Following the reasoning in Feliu and Wiuf (2013a), we build a labeled directed graph denoted by $G_{y}$, with node set $I_{y}\cup \{y\}$, and labeled directed edges as in G, except that any reaction of the form $y'\overset{\kappa }{\rightarrow } y''$, where $y'\in I_y$ and $y''$ is any core complex, is replaced by $y'\overset{\kappa }{\rightarrow } y$, with the same rate constant (if there are several core complexes to which $y'$ reacts, the edges are collapsed and the label equals the sum of the labels of the corresponding collapsed edges). Note that, as all intermediate complexes in MESSI networks react via intermediates to some core complex, the graph $G_y$ is strongly connected.

Number the species in $I_y$ and denote by $x_1, \dots , x_{n_y}$ the corresponding concentrations. Then, the mass-action ODEs corresponding to them in the given network, are given by $f_\ell =\dot{x}_\ell =({\mathcal {L}}(G_{y}))_\ell {{\mathbf {x}}}^{\top }$, where $({\mathcal {L}}(G_y))_\ell$ is the $\ell$th row of the Laplacian ${\mathcal {L}}(G_{y})$ of $G_{y}$, $\mathbf{x}=(x_1, \dots , x_{n_y},m(y))$, and m(y) is the monomial associated with the complex y. Call $\rho _\ell =\rho (G_{y})_\ell$ for $1\le \ell \le n_y+1$. From the matrix-tree theorem we know that, up to sign, the determinant of the first $n_y\times n_y$ principal minor of ${\mathcal {L}}(G_{y})$ equals $\rho _{n_y+1}\ne 0$. Call A the $(n_y+1)\times (n_y+1)$ block matrix

$$\begin{aligned} A=\left( \begin{array}{c|c}A_1 &{} 0 \\ \hline 0 &{} 1 \end{array}\right) , \end{aligned}$$

where $A_1$ is the $n_y\times n_y$ is such that $A\cdot {\mathcal {L}}(G_{y})$ has the form:

$$\begin{aligned} A\cdot {\mathcal {L}}(G_{y})=\left( \begin{array}{ccc|c}1 &{} &{} 0 &{}- \alpha _1\\ &{} \ddots &{} &{} \vdots \\ 0 &{} &{}1&{} - \alpha _{n_y}\\ \hline * &{} \dots &{} * &{} * \end{array}\right) . \end{aligned}$$

Such a matrix A exists since the $n_y \times n_y$ first principal minor of ${\mathcal {L}}(G_{y})$ is invertible. Observe that $\ker ({\mathcal {L}}(G_{y}))$ is generated by $(\rho _1,\rho _2,\dots ,\rho _{n_y},\rho _{n_y+1})$, so $\rho _\ell - \alpha _\ell \rho _{n_y+1}=0$, for $1\le \ell \le n_y$, and then $\alpha _\ell =\frac{\rho _\ell }{\rho _{n_y+1}}$ (and $\alpha _\ell \ne 0$). By multiplying ${\mathcal {L}}(G_{y})$ on the left by A, which is equivalent to operating linearly on the original equations $f_\ell =0$, we deduce the binomial equations $x_\ell -\alpha _\ell \, m(y)=0$, for the intermediate species $x_\ell$ at steady state, for $1\le \ell \le n_y$. Note that, as $\rho _{n_y+1}\ne 0$ for all $\kappa \in {\mathbb {R}}^m_{>0}$, these operations are well defined for all $\kappa \in {\mathbb {R}}^m_{>0}$.

We can afterward substitute the steady-state value of the intermediate species $x_\ell \in I_y$ into the original steady-state equations, for core species, of G. We moreover show that this substitution can be achieved via linear operations. Indeed, for a core species $X_k$, write the corresponding ODE as:

$$\begin{aligned} f_k~=~\dot{x}_k~=~p_k+\sum _{\ell =1}^{n_y} \kappa _\ell x_{\ell }~, \end{aligned}$$

where $p_k$ is a polynomial that does not depend on $x_1,\dots , x_{n_y}$ and $\kappa _\ell \ge 0$ is positive precisely when $X_{\ell }$ reacts with rate constant $\kappa _\ell$ to a core complex that involves $X_k$. We now subtract a linear combination of the intermediate-species binomials $(x_\ell -\alpha _\ell \, m(y))$:

$$\begin{aligned} f_k-\sum _{\ell =1}^{n_y} \kappa _\ell (x_{\ell }-\alpha _{\ell } m(y)) ~=~ p_k+ \sum _{\ell =1}^{n_y} (\kappa _\ell \alpha _{\ell }) m(y)~, \end{aligned}$$

and so we replace $f_k=0$ by $p_k+ \sum _{\ell =1}^{n_y} (\kappa _\ell \alpha _{\ell }) m(y)=0$, where all the intermediates in $I_y$ have been eliminated by performing linear operations on the original steady-state equations (for core species) and the new binomials (for the intermediates).

A key observation pertaining to how we obtained binomial equations for all intermediate species is that, by condition $({\mathcal {C}})$, the set of intermediate complexes can be written as the disjoint union of sets $I_y$ for a certain (finite) number of core complexes y. By the natural bijection between intermediate complexes and intermediate species, we can then obtain the corresponding binomials by operating linearly on the original equations. We also, as described above, eliminated all the intermediate species from the core-species equations by linear operations. We will denote this procedure as follows. First assume, without loss of generality, that the intermediate species are the last $s-n$ species. Then we can assert that there exists an invertible matrix $M_1=M_1(\kappa )\in {\mathbb {Q}}(\kappa )^{s\times s}$ which is well defined for all $\kappa \in {\mathbb {R}}^m_{>0}$ such that $M_1(f_1,\dots ,f_s)^{\top }=(\tilde{f}_1,\dots ,\tilde{f}_n,\tilde{h}_{n+1},\dots ,\tilde{h}_s)^{\top }$, where $\tilde{f}_1,\dots ,\tilde{f}_n$ do not depend on the intermediate-species concentrations $x_{n+1},\dots , x_{s}$, and $\tilde{h}_{n+\ell }$ is the binomial for the intermediate species $x_{n+\ell }$ ($1\le \ell \le s-n$) and its form is $x_{n+\ell }-\alpha _{n+\ell } \, m(y)=0$.

Before we continue operating on the core-species equations (the $\tilde{f}_i$’s are not binomials), we describe the map $\varvec{\tau }=\varvec{\tau }(\kappa )$ mentioned in (34). For each $X_i+X_j\rightarrow _\circ X_\ell +X_m$ in G, the reaction constant $\tau$ in $G_1$ which gives the label $X_i+X_j\overset{\tau }{\longrightarrow } X_\ell +X_m$ has the form

$$\begin{aligned} \tau =\kappa +\overset{s-n}{\underset{k=1}{\sum }}\kappa _k\alpha _k, \end{aligned}$$

where $\kappa \ge 0$ is positive when $X_i+X_j\overset{\kappa }{\longrightarrow } X_\ell +X_m$ in G (and $\kappa =0$ otherwise), and $\kappa _k\ge 0$ is positive if there is a reaction from the intermediate species $X_{n+k}\overset{\kappa _k}{\longrightarrow } X_\ell +X_m$ and $X_i+X_j\rightarrow _\circ X_{n+k}$ in G (and $\kappa _k=0$ otherwise). As we pointed out in Remark 5.2, the steady states of G are in one-to-one correspondence with those of $G_1$ and, in fact, the polynomials $\tilde{f}_i$ can be read from the digraph $G_1$ (see Theorem 3.2 in Feliu and Wiuf 2013a).

What we show now is that we can operate linearly on the core-species equations $\tilde{f}_i=0$ (for $1\le i\le n$) in order to obtain equivalent binomial equations. To avoid unnecessary notation, we will assume in what follows that the partition of is minimal. Recall that a vertex in a directed graph has outdegree zero if it is not the tail of any directed edge. Let us define subsets of indices based on the graph $G_E$:

As is finite and there are no directed cycles in $G_E$, there must exist a subset with $1\le \beta \le m$ such that its outdegree in $G_E$ is zero. This means that $L_0\ne \emptyset$.

Consider $\alpha \in L_0$. By the assumption of minimality of the partition, there is a connected component of $G^\circ _2$, which we denote by $H_\alpha$, with vertices the species in . Let $\widetilde{H}_\alpha$ be the corresponding underlying undirected graph. As $\widetilde{H}_\alpha$ is a tree, consider $X_i$, a leaf of the tree (this is, a vertex of degree one). $X_i$ is only connected to one vertex $X_j$, so $\tilde{f}_i$ is already a binomial of the form

$$\begin{aligned} \tilde{f}_i=\tau _{ji}x_jx_\ell -\tau _{ij}x_ix_h, \end{aligned}$$

for some species , $\beta , \gamma$ in levels strictly greater than 0. Moreover, $\tilde{f}_j$ is

$$\begin{aligned} \tilde{f}_j=p_j+\tau _{ij}x_ix_h-\tau _{ji} x_jx_\ell =p_j-\tilde{f}_i, \end{aligned}$$

with $p_j$ a polynomial that does not depend on $x_i$. Then, $\tilde{f}_j+\tilde{f}_i=p_j$. And we can replace $\tilde{f}_j$ with $p_j$. As the associated digraph $G_E$ has no directed cycles, all the reactions are enzymatic. This means that the reactions that correspond to $\tau _{ij}$ and $\tau _{ji}$ in $G_1$ are $X_i+X_h\overset{\tau _{ij}}{\longrightarrow }X_j+X_h$ and $X_j+X_\ell \overset{\tau _{ji}}{\longrightarrow }X_i+X_\ell$, respectively, and none of these reactions affect either $\tilde{f}_h$ or $\tilde{f}_\ell$. Moreover, as with $\alpha \in L_0$ and has outdegree zero in $G_E$, $x_i$ only appears in $\tilde{f}_i$ and $\tilde{f}_j$. We have then eliminated by linear operations the variable $x_i$ from all the equations other than $\tilde{f}_i$. And this can be done for all the species whose vertices are leaves of $\widetilde{H}_\alpha$. We can then erase all the leaves from $\widetilde{H}_\alpha$, and by an inductive argument, we see that we can operate linearly, with integer coefficients, on $\tilde{f}_1,\dots ,\tilde{f}_n$ to obtain binomials for the species in . This argument holds for any $\alpha \in L_0$. As the species in the ’s for $\alpha \in L_0$ do not appear in any label of the $L_i$’s for $i\ge 1$ and all the reactions in all the $H_\alpha$’s with $\alpha \in L_0$ do not affect the equations of those species that appear on its labels, by an inductive argument we can complete the proof to obtain binomial equations by operating linearly on the equations $\tilde{f}_j=0$ for the species in for $\alpha \in L_i$, $i \ge 1$.

We have then proved that there exists a matrix with integer entries $\widetilde{M}_2\in {\mathbb {Q}}^{n\times n}$, and an invertible block matrix $M_2\in {\mathbb {Q}}^{s\times s}$ of the form

$$\begin{aligned} M_2=\left( \begin{array}{c|c} \widetilde{M}_2 &{} 0\\ \hline 0 &{} Id_{s-n} \end{array}\right) , \end{aligned}$$

such that $M_2~(\tilde{f}_1,\dots ,\tilde{f}_n,\tilde{h}_{n+1},\dots ,\tilde{h}_s)^{\top }= (\tilde{h}_1,\dots ,\tilde{h}_s)^{\top }$, where $\tilde{h}_1,\dots , \tilde{h}_s$ are binomials.

We have so far that there are invertible matrices $M_1\in {\mathbb {Q}}(\kappa )^{s\times s}$ and $M_2\in {\mathbb {Q}}^{s\times s}$, with $M_1$ well defined for all $\kappa \in {\mathbb {R}}^m_{>0}$, such that $M_2M_1(f_1,\dots ,f_s)^{\top }=(\tilde{h}_1,\dots ,\tilde{h}_s)^{\top }$, with $\det (M_2M_1)\ne 0$. If $f_{j_1},\dots ,f_{j_{s-d}}$ is a basis of the ${\mathbb {Q}}(\kappa )$-linear subspace generated by $f_1,\dots ,f_s$ there must exist a set of $s-d$ binomials $\{\bar{h}_{j_1},\dots ,\bar{h}_{j_{s-d}}\}\subseteq \{\tilde{h}_1,\dots ,\tilde{h}_s\}$ and an invertible matrix $M\in {\mathbb {Q}}(\kappa )^{(s-d)\times (s-d)}$, which is well defined for all $\kappa \in {\mathbb {R}}^m_{>0}$, such that $M(f_{j_1},\dots ,f_{j_{s-d}})^{\top }=(\bar{h}_{j_1},\dots ,\bar{h}_{j_{s-d}})^{\top }$, as we wanted to prove. $\square$

Example A.1

Consider the following network:

which has $s=6$ species:

There are 2 conservation laws:

$$\begin{aligned} x_1+x_2+x_3+x_4+x_6&=c_1\\ x_5+x_6&=c_2. \end{aligned}$$

The equations $f_{c,\kappa }(x)$ are

$$\begin{aligned} f_{c,\kappa ,1}= & {} x_1+x_2+x_3+x_4+x_6-c_1,\\ f_{c,\kappa ,2}= & {} \kappa _3x_6-\,\kappa _4x_2x_5+\kappa _7x_3-\,\kappa _8x_2,\\ f_{c,\kappa ,3}= & {} \kappa _4x_2x_5-\,\kappa _5x_3x_5+\kappa _6x_4-\,\kappa _7x_3,\\ f_{c,\kappa ,4}= & {} \kappa _5x_3x_5-\,\kappa _6x_4,\\ f_{c,\kappa ,5}= & {} x_5+x_6-c_2,\\ f_{c,\kappa ,6}= & {} \kappa _1x_1x_5-(\kappa _2+\kappa _3)x_6. \end{aligned}$$

Matrix $M_1(\kappa )$ is the product of two matrices: The first one multiplied by $(f_1,\dots ,f_6)^{\top }$ gives binomials for the intermediate-species equations; the second one eliminates the intermediate species from the core-species equations.

$$\begin{aligned} M_1=M_1(\kappa )=\left( \begin{array}{ccc|c} &{} &{} &{} -\,\kappa _2\\ &{} &{} &{} -\,\kappa _3\\ &{} Id_5 &{} &{} 0 \\ &{} &{} &{} 0 \\ &{} &{} &{} -\,\kappa _2-\,\kappa _3\\ \hline 0 &{} \dots &{} 0 &{} 1 \end{array}\right) \cdot \left( \begin{array}{ccc|c} &{} &{} &{} 0\\ &{} &{} &{} \\ &{} Id_5 &{} &{} \vdots \\ &{} &{} &{} \\ &{} &{} &{} 0\\ \hline 0 &{} \dots &{} 0 &{} -\dfrac{\phantom {A^A}1}{\kappa _2+\kappa _3} \end{array}\right) , \end{aligned}$$

where $Id_5$ is the $5\times 5$ identity matrix. This leads to:

$$\begin{aligned} M_1(f_1,\dots ,f_6)^{\top }&=\left( \kappa _8x_2-\frac{\kappa _1\kappa _3}{\kappa _2+\kappa _3}x_1x_5,~ -\,\kappa _4x_2x_5+\kappa _7x_3-\,\kappa _8x_2+\frac{\kappa _1\kappa _3}{\kappa _2+\kappa _3}x_1x_5, \right. \nonumber \\&\quad \left. f_3,~f_4,~0,~-\frac{\kappa _1}{\kappa _2+\kappa _3}x_1x_5+x_6 \right) ^{\top } =(\tilde{f}_1,\dots ,\tilde{f}_5,\tilde{h}_6). \end{aligned}$$

(37)

The corresponding digraph $G_2^\circ$ defined in Sect. 5.2 is:

and the underlying undirected graph is a tree with leaves and . Then $\tilde{f}_1$ and $\tilde{f}_4$ are already binomials, and $p_2=\tilde{f}_2-\tilde{f}_1=-\,\kappa _4x_2x_5+\kappa _7x_3$ (from (37)), and $p_3=\tilde{f}_3+\tilde{f}_4=(\kappa _4x_2x_5-\,\kappa _5x_3x_5+\kappa _6x_4-\,\kappa _7x_3)+(\kappa _5x_3x_5-\,\kappa _6x_4)=\kappa _4x_2x_5-\,\kappa _7x_3$, which are binomials. Matrix $M_2$ is

$$\begin{aligned} M_2=\left( \begin{array}{cccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{array}\right) , \end{aligned}$$

and $M_2~(\tilde{f}_1,\dots ,\tilde{f}_5,\tilde{h}_6)^{\top }=(\tilde{h}_1,\dots ,\tilde{h}_6)^{\top }$.

In order to obtain $h_{c,\kappa ,2}$, $h_{c,\kappa ,3}$, $h_{c,\kappa ,4}$, $h_{c,\kappa ,6}$ we need to build $f_1$ and $f_5$ from $f_{c,\kappa ,2}$, $f_{c,\kappa ,3}$, $f_{c,\kappa ,4}$, $f_{c,\kappa ,6}$, multiply by the product $M_2M_1$, and then pick 4 linearly independent binomials:

$$\begin{aligned} M'= & {} \left( \begin{array}{cccccc} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{array}\right) M_2M_1 \left( \begin{array}{rrrr} -\,1 &{}\quad -\,1 &{}\quad -\,1 &{}\quad -\,1 \\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{array}\right) \\ {}= & {} \left( \begin{array}{rrrr} -1 &{}\quad -1 &{}\quad -1 &{}\quad -\frac{\kappa _3}{\kappa _2+\kappa _3} \\ 0 &{}\quad -1 &{}\quad -1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -\frac{1}{\kappa _2+\kappa _3} \end{array}\right) , \end{aligned}$$

which leads to $M'(f_{c,\kappa ,2},f_{c,\kappa ,3},f_{c,\kappa ,4},f_{c,\kappa ,6})^{\top }= (\kappa _8x_2-\frac{\kappa _1\kappa _3}{\kappa _2+\kappa _3}x_1x_5,-\,\kappa _4x_2x_5+\kappa _7x_3, \kappa _5x_3x_5-\,\kappa _6x_4,-\frac{\kappa _1}{\kappa _2+\kappa _3}x_1x_5+x_6)^{\top }$. Notice that, as $M_1(\kappa )$ is well defined for all $\kappa \in {\mathbb {R}}_{>0}^8$, then $M'$ is also well defined for all $\kappa \in {\mathbb {R}}_{>0}^8$.

We now choose matrix $M=M(\kappa )$:

$$\begin{aligned} M=\left( \begin{array}{rrrr} \frac{1}{\kappa _8} &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad \frac{1}{\kappa _7} &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -\frac{1}{\kappa _6} &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{array}\right) M', \end{aligned}$$

and the effective parameters $\bar{a}_1=\frac{\kappa _1\kappa _3}{\kappa _8(\kappa _2+\kappa _3)}$, $\bar{a}_2=\frac{\kappa _4}{\kappa _7}$, $\bar{a}_3=\frac{\kappa _5}{\kappa _6}$, and $\bar{a}_4=\frac{\kappa _1}{\kappa _2+\kappa _3}$. Then, the reparametrization map $\bar{a}$ is surjective, $M(\kappa )$ is well defined for all $\kappa \in {\mathbb {R}}_{>0}^8$, and $\det (M)=(-\frac{1}{\kappa _6\kappa _7\kappa _8})\cdot (-\frac{1}{\kappa _2+\kappa _3})>0$.

Appendix B: Proof of Lemma 6.4

In this appendix, we prove Lemma 6.4.

Proof of Lemma 6.4

Let $\varepsilon >0$. By hypothesis, f(a, z) and $(a^*,z^*) \in {\mathbb {R}}^{n+1}$ satisfy:

• (I) $f(a^*,z^*)=0$,

• (II) $\frac{\partial f}{\partial z}(a^*,z^*)=0$,

• (III) $\frac{\partial ^2 f}{\partial z^2}(a^*,z^*) \ne 0$, and

• (IV) $\frac{\partial f}{\partial a_\ell }(a^*,z^*)\ne 0$.

By assumptions (I) and (IV), the implicit function theorem applies. So, there exists a function

$$\begin{aligned} {\tilde{\beta }}:B_{\varepsilon '}(a_1,\dots ,a_{\ell -1},a_{\ell +1},\dots ,a_n,z)\rightarrow {\mathbb {R}} \end{aligned}$$

defined on a ball of some radius $\varepsilon ' \le \varepsilon$ in ${\mathbb {R}}^n$ such that ${\tilde{\beta }}(a_1^*,\dots ,a_{\ell -1}^*,a_{\ell +1}^*,\dots ,a_n^*,z^*)=a_\ell ^*$ and, near the point of interest $(a^*, z^*)$, the $f=0$ locus is the graph of ${\tilde{\beta }}$.

Call $\beta : (z^*-\epsilon ', z^*+\epsilon ') \rightarrow {\mathbb {R}}$ the restriction $\beta (z) ={\tilde{\beta }}(a_1^*,\dots ,a_{\ell -1}^*,a_{\ell +1}^*,\dots ,a_n^*,z)$, and call ${\hat{\beta }}(z)=(a_1^*,\dots ,a_{l-1}^*,\beta (z),a_{\ell +1}^*,\dots , a_n^*)$; then near $(a^*, z^*)$ we have

$$\begin{aligned} f({\hat{\beta }}(z),z)~=~0 \quad \mathrm{for~} z \in (z^*-\varepsilon ', z^*+\varepsilon ')~. \end{aligned}$$

(38)

Take the derivative of equation (38), via the chain rule:

$$\begin{aligned} \frac{ \partial f }{\partial a_\ell } \left( {\hat{\beta }}(z),z \right) \beta '(z) + \frac{ \partial f }{\partial z} ({\hat{\beta }}(z),z) ~=~0~. \end{aligned}$$

(39)

Evaluating this equation at $z=z^*$, and recalling that $\beta (z^*)=a_\ell ^*$ and ${\hat{\beta }}(z^*)=a^*$, we obtain:

$$\begin{aligned} \frac{ \partial f }{\partial a_\ell } (a^*,z^*) \beta '(z^*) + \frac{ \partial f }{\partial z} (a^*,z^*) ~=~0~. \end{aligned}$$

Recall that $\frac{ \partial f }{\partial a_\ell } (a^*,z^*)\ne 0$, by hypothesis (IV), and $\frac{ \partial f }{\partial z} (a^*,z^*)=0$, by (II). Thus, $\beta '(z^*)=0$. Next, take another derivative, applying the chain rule to equation (39), and then evaluate at $z=z^*$:

$$\begin{aligned}&\frac{ \partial f }{\partial a_\ell } (a^*,z^*) \beta ''(z^*) + \frac{ \partial ^2 f }{\partial a_\ell ^2} (a^*,z^*) \left( \beta '(z^*) \right) ^2 + 2 \frac{ \partial ^2 f }{\partial z \partial a_\ell } (a^*,z^*) \beta '(z^*)\\&\quad + \frac{ \partial ^2 f }{\partial z^2} (a^*,z^*) ~=~0~. \end{aligned}$$

Thus, we deduce from (III) and (IV) that $\beta ''(z^*) \ne 0$. It follows that the univariate function $\beta$ has a maximum or a minimum at $z^*$ (depending on the sign of the second derivative). Hence, there exists a sufficiently small $\delta >0$ such that for all $\delta ' \in (0,\delta )$, either $b_\ell ^{**}=a_\ell ^*-\delta '$ (if $a_\ell ^*$ is a local maximum) or $a_\ell ^{**}=a_\ell ^*+\delta '$ (if $a_\ell ^*$ is a local minimum) yields $f(a^{**}, z)=0$, for $a^{**}=(a_1^*,\dots ,a_{\ell -1}^*,a_\ell ^{**},a_{\ell +1}^*,\dots ,a_n^*)$, with two distinct real solutions within distance $\varepsilon$ of $z^*$. $\square$

Appendix C: Proof of Theorem 6.11

The goal of this appendix is to prove Theorem 6.11. We first recall some definitions and a result of Kapur, Sun, and Wang from the theory of comprehensive Gröbner bases (Kapur et al. 2010). For basic concepts from computational algebraic geometry, see the books (Cox et al. 2005, 2007).

Let $h\in {{\mathbb {C}}}[a, x] := {{\mathbb {C}}}[a_1, a_2, \ldots , a_n, x_1, x_2, \ldots , x_s]$. We denote by

$$\begin{aligned} {\mathrm{lpp}}_x(h) \quad \quad \mathrm{and} \quad \quad \mathrm{lc}_x(h)~, \end{aligned}$$

the leading monomial (or “leading power product”) and leading coefficient of h, respectively, when h is viewed in ${\mathbb {C}}(a)[ x]$ taken with the lexicographic order $x_{s}<\cdots<x_2<x_{1}$. For instance, if $h=a_1^2x_1 + x_2$, then ${\mathrm{lpp}}_x(h)=x_1$ and $\mathrm{lc}_x(h)=a_1^2$.

Definition C.1

(Kapur et al. 2010, Definition 4.1) Given $H\subseteq {{\mathbb {C}}}[a, x]$, a subset $H'$ of H is a noncomparable subset ofH if

• (1) for every $h\in H$, there exists $g\in H'$ such that ${\mathrm{lpp}}_x(h)$ is a multiple of ${\mathrm{lpp}}_x(g)$, and

• (2) for every $g_1, g_2\in H'$, with $g_1 \ne g_2$, the leading monomial ${\mathrm{lpp}}_x(g_1)$ is not a multiple of ${\mathrm{lpp}}_x(g_2)$, and ${\mathrm{lpp}}_x(g_2)$ is not a multiple of ${\mathrm{lpp}}_x(g_1)$.

Example C.2

Consider $H=\{a_2x_2^2-1, a_1x_1-1, (a_1+1)x_1-x_2, (a_1+1)x_2-a_1\}$. Let $H'=\{a_1x_1-1, (a_1+1)x_2-a_1\}~ (\subseteq H)$. We verify that $H'$ is a noncomparable subset of H:

• (1) Note that $\{{\mathrm{lpp}}_x(h) \mid h \in H\}=\{x_2^2, x_2, x_1\}$ and $\{{\mathrm{lpp}}_x(g) \mid g \in H'\}=\{x_2, x_1\}$. So, every monomial in the first set is a multiple of some monomial in the second set.

• (2) For the two polynomials in $H'$, their leading monomials are, respectively, $x_1$ and $x_2$. We see that $x_1$ is not a multiple of $x_2$, and $x_2$ is not a multiple of $x_1$.

The following straightforward lemma shows that noncomparable subsets always exist and explains how (in theory) to effectively find one.

Lemma C.3

(Existence of noncomparable subsets). Let H be a finite, nonempty subset of ${{\mathbb {C}}}[a,x]$. The following procedure yields a noncomparable subset of H:

• (1) Let $M=\{{\mathrm{lpp}}_x(h) \mid h\in H\}$, and let $D=\emptyset$.

• (2) Pick a monomial from M, say $m_1$. Let $d:=m_1$. Search M. If there exists $m \in M$ such that m|d and $m\ne d$, then set $d := m$. Continue searching until there is no m in M such that m|d and $m\ne d$. Then add d into D.

• (3) Let $M' := \{ m \in M: d|m\}$, and let $M := M \backslash M'$.

• (4) Repeat steps 2–3 until M is empty.

• (5) For each $d \in D$, pick some $f_d \in F$ for which ${\mathrm{lpp}}_x(f)=d$. Output $F_D:=\{f_d \mid d \in D \}$.

Now consider a set of s polynomials $H=\{h_1,h_2, \ldots , h_s\}$ in ${{\mathbb {C}}}[a,x]$. Denote by ${{\mathcal {I}}}(H)$ the ideal generated by H in the polynomial ring. Denote by V(H) the variety generated by H (or, equivalently generated by ${{\mathcal {I}}}(H)$) in ${{\mathbb {C}}}^{n+s}$:

$$\begin{aligned} V(H) ~:=~ \{(a;x)=(a_1, a_2, \ldots a_n, x_1, x_2, \ldots , x_s)\in {{\mathbb {C}}}^{n+s} \mid h(a;x)=0 ~\mathrm{for~all}~h\in {{\mathcal {I}}}(H)\}~. \end{aligned}$$

Below, we show that if the system H is a general zero-dimensional system (Definition 6.7), then H admits a triangular form (see the proof of Theorem 6.11). More specifically, we prove that any noncomparable subset of a Gröbner basis of ${{\mathcal {I}}}(H)$ with respect to the lexicographic order $a_n<\cdots<a_2<a_1<x_{s}<\cdots<x_2<x_{1}$ has the desired triangular form. The proof requires the following result, due to Kapur, Sun, and Wang (Kapur et al. 2010, Theorem 4.3), which relates Gröbner bases of ${{\mathcal {I}}}(H)$ to those of the specialized ideal ${{\mathcal {I}}}(H|_{b=b^*})$:

Proposition C.4

(Specialization of Gröbner bases Kapur et al. 2010). Consider $H \subseteq {{\mathbb {C}}}[a, x]$, and let ${{\mathcal {G}}}$ be a Gröbner basis of the ideal ${{\mathcal {I}}}(H) \subseteq {{\mathbb {C}}}[a,x]$ with respect to the lexicographic order $a_n<\cdots<a_2<a_1<x_{s}<\cdots<x_2<x_{1}$. Let ${{\mathcal {G}}}_{\cap }={{\mathcal {G}}}\cap {{\mathbb {C}}}[a]$, let ${{\mathcal {G}}}_m$ be a noncomparable subset of ${{\mathcal {G}}}\backslash {{\mathcal {G}}}_{\cap }$, and let $h=\Pi _{g \in {{\mathcal {G}}}_m}\mathrm{lc}_{x}(g)$. For any $a^*=(a_1^*, a_1^*, \ldots , a_n^*)\in {{\mathbb {C}}}^n$, if ${{\mathcal {G}}}_{\cap }|_{a=a^*} \subseteq \{0\}$ and $h|_{a=a^*}\ne 0$, then ${{\mathcal {G}}}_m|_{a=a^*}$ is a Gröbner basis of the ideal ${{\mathcal {I}}}(H|_{a=a^*}) ~\subseteq ~ {{\mathbb {C}}}[x]$ with respect to the lexicographic order $x_{s}<\cdots<x_2<x_{1}$.

To prove Theorem 6.11, we also need the following lemma:

Lemma C.5

For a general zero-dimensional system $H=\{h_1,h_2, \ldots , h_s\}\subseteq {{\mathbb {C}}}[a,x]$, we have ${{\mathcal {I}}}(H)\cap {{\mathbb {C}}}[a] = \{0\}.$

Proof

Since H is a general zero-dimensional system (Definition 6.7), there exists a proper variety ${{\mathcal {W}}} \subsetneq {\mathbb {C}}^n$ such that for every $a^* \in {\mathbb {C}}^n \setminus {{\mathcal {W}}}$ the specialized system $H|_{a=a^*}$ has at least one complex solution. It follows that ${{\mathbb {C}}}^n\backslash {{\mathcal {W}}}\subseteq \pi (V(H))$, where $\pi : {\mathbb {C}}^{n+s} \rightarrow {\mathbb {C}}^n$ denotes the standard projection given by $(a,x) \mapsto a$. Thus,

$$\begin{aligned} \overline{{{\mathbb {C}}}^n\backslash {{\mathcal {W}}}} ~\subseteq ~ \overline{\pi (V(H))} ~\subseteq ~ {\mathbb {C}}^n~. \end{aligned}$$

Note that $\overline{{{\mathbb {C}}}^n\backslash {{\mathcal {W}}}}={{\mathbb {C}}}^n$ (as ${{\mathcal {W}}} \subsetneq {\mathbb {C}}^n$), so $\overline{\pi (V(H)}={{\mathbb {C}}}^n$. By Cox et al. (2007), p. 193, Theorem 3, we know that $\overline{\pi (V(H))} = V({{\mathcal {I}}}(H)\cap {{\mathbb {C}}}[a])$. So, $V({{\mathcal {I}}}(H)\cap {{\mathbb {C}}}[a])={{\mathbb {C}}}^n$. The only ideal that generates the variety ${{\mathbb {C}}}^n$ is the zero ideal. So, ${{\mathcal {I}}}(H)\cap {{\mathbb {C}}}[a] = \{0\}$. $\square$

Proof of Theorem 6.11

Let $H=\{h_1,h_2, \ldots , h_s\}$. Let ${{\mathcal {G}}}_m$ be a noncomparable subset of ${{\mathcal {G}}}$ (which exists by Lemma C.3). By Lemma C.5, we have:

$$\begin{aligned} {{\mathcal {G}}}_m\cap {{\mathbb {C}}}[a] ~\subseteq ~ {{\mathcal {I}}}(H)\cap {{\mathbb {C}}}[a] ~=~\{0\}~. \end{aligned}$$

In fact, $0 \notin {{\mathcal {G}}}_m$ (by the definition of noncomparable subset and because ${\mathcal {G}} \ne \{0\}$), so ${{\mathcal {G}}}_m\cap {{\mathbb {C}}}[a] =\emptyset$. So, by Proposition C.4, for every $a^*\in {{\mathbb {C}}}^n\backslash V(h)$, where $h=\Pi _{g \in {{\mathcal {G}}}_m}\mathrm{lc}_{x}(g)$, the set ${{\mathcal {G}}}_m|_{a=a^*}$ is a Gröbner basis of ${{\mathcal {I}}}(H|_{a=a^*}) \subseteq {{\mathbb {C}}}[x]$ with respect to $x_{s}<\cdots <x_{1}$.

(1) We show that a subset $\{g_1, g_2, \ldots , g_s\}$ of ${{\mathcal {G}}}_m$ has the required triangular form. As H is a general zero-dimensional system, let ${\mathcal {W}}$ be the variety in ${{\mathbb {C}}}^n$ such that H satisfies hypotheses (A1)–(A3) in Definition 6.7. Let $c^*\in {{\mathbb {C}}}^n\backslash \left( {{\mathcal {W}}}\cup V(h)\right)$. Then by (A1) in Definition 6.7, we know that $V({{\mathcal {G}}}_m|_{a=c^*})=V(H|_{a=c^*})$ is a nonempty finite set in ${{\mathbb {C}}}^s$. Hence, by Cox et al. (2007), p. 234, Theorem 6 (i) and (iii), for $i\in \{1, 2, \ldots , s\}$, there exists $g_i \in {{\mathcal {G}}}_m$ such that the leading monomial of $g_i|_{a=c^*}$ has the form $x_i^{N_i}$, where $N_i$ is a nonnegative integer. In particular, $g_s|_{a=c^*}\in {\mathbb {C}}[x_{s}]$.

It follows that ${\mathrm{lpp}}_x(g_i) = x_i^{N_i}$, because $c^*\notin V(h)$ and so $\mathrm{lc}_x(g_i)|_{a=c^*} \ne 0$. Hence, if we show that $N_2=N_3=\dots =N_s=1$, then, by the definition of the lexicographic order, $g_1, g_2, \ldots , g_s$ have the forms shown in Theorem 6.11 (1).

Hence, to finish proving (1), we need only show that $N_2=N_3=\dots =N_s=1$. Let $N:= |V(H|_{a=c^*}) |$. Then every $x^* \in V(H|_{a=c^*})$ has a distinct $x_s$-coordinate (by (A2) in Definition 6.7), and every such coordinate is a root of $g_s|_{a=c^*}\in {\mathbb {C}}[x_{s}]$. Hence,

$$\begin{aligned} N_s ~=~ \mathrm{deg}(g_s|_{a=c^*} ) ~\ge ~ N~. \end{aligned}$$

(40)

Next, by (A3) in Definition 6.7 and Cox et al. (2007), p. 235, Proposition 8(ii), we know that

$$\begin{aligned} N~=~N_1N_2 \cdots N_s~. \end{aligned}$$

(41)

So, by (40) and (41), we have $N_s=N$ and $N_2 = N_3 =\cdots =N_{s-1}=1$.

(2) Consider the following claim:

Claim:${{\mathcal {G}}}_m= \{g_1, ~g_2, ~\ldots , ~g_s\}$.

This claim implies that $h=Q_{s, N} \, Q_1 \, Q_2 \cdots Q_{s-1}$ and so, by what we saw earlier, for every $a^*\in {{\mathbb {C}}}^n\backslash V(Q_{s, N} \, Q_1\, Q_2 \cdots Q_{s-1})$, the set $\{g_1|_{a=a^*},~ g_2|_{a=a^*},~ \ldots ,~ g_s|_{a=a^*} \}$ is a Gröbner basis of the ideal ${{\mathcal {I}}}(H|_{a=a^*})$ with respect to $x_{s}<\cdots <x_{1}$. So, to complete the proof, we need only prove the Claim.

First, the containment ${{\mathcal {G}}}_m\supseteq \{g_1, g_2, \ldots g_s\}$ follows from the fact that the $g_i$’s were selected from ${{\mathcal {G}}}_m$. Next, we show the containment ${{\mathcal {G}}}_m\subseteq \{g_1, g_2, \ldots , g_s\}$ by proving the following equality by induction on i:

$$\begin{aligned} {{\mathcal {G}}}_m\cap {{\mathbb {C}}}[a, x_{i},x_{i+1} \ldots , x_{s}] ~\subseteq ~ \{g_{i},g_{i+1}, \ldots , g_s\}~, \quad \mathrm{for~all}~ i\in \{1, 2, \ldots , s\}~. \end{aligned}$$

(42)

For $i=s$, assume that $g \in {{\mathcal {G}}}_m\cap {{\mathbb {C}}}[a, x_{s}]$. Then ${\mathrm{lpp}}_x(g)= x_s^{\tilde{N}}$ for some $\tilde{N} \ge 0$, and also recall that ${\mathrm{lpp}}_x(g_s)= x_s^N$ for some $N >0$. However, ${{\mathcal {G}}}_m$ is noncomparable, so $g=g_s$.

For the inductive step, assume that containment (42) holds for all $i \in \{j, j+1, \dots , s \}$ (for some $j\le s$). For $i=j-1$, let $g\in {{\mathcal {G}}}_m\cap {{\mathbb {C}}}[a_1, a_2, \ldots , a_n, x_{j-1}, x_j,\ldots , x_{s}]$, and write ${\mathrm{lpp}}_x(g) = x_s^{a_s}\cdots x_j^{a_j}x_{j-1}^{a_{j-1}}$. If $a_{j-1}=0$, then $g\in {{\mathcal {G}}}_m\cap {{\mathbb {C}}}[a_1, a_2, \ldots , a_n, x_j, x_{j+1},\ldots , x_s]$, so by the induction hypothesis, $g\in \{g_j, g_{j+1},\ldots , g_s\}$. If $a_{j-1}>0$, then ${\mathrm{lpp}}_x(g_{j-1})=x_{j-1} | {\mathrm{lpp}}_x(g)$. Hence, $g=g_{j-1}$ (because ${\mathcal {G}}_m$ is noncomparable). $\square$