\label{thm1}
	Let $u\in C_{loc}(\rn)$ be a convex viscosity solution of \eqref{eqma}, where $f:\rn\rightarrow{\mathbb R}$ satisfies
	{\textnormal{(H)}}. If $n\geq3$, then there exist $b\in\rn$, $A\in\mathcal{A}$ and $u_p\in C^{2,\alpha}(\rn)$ such that
	\begin{equation*}
		\begin{aligned}
		\left|u(x)-\left(\frac{1}{2}x'Ax+b\cdot x+u_p(x)\right)\right|\leq C_0(1+|x|)^{2-\min\{n,\beta\}},\ \ \ \forall\ x\in\rn,
		\end{aligned}
	\end{equation*}
	where $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\cdots,n$,
    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\beta$, ${\alpha}$ and $a_1,\cdots,a_n$.

\begin{theorem}\label{thm1}
	Let $u\in C_{loc}(\rn)$ be a convex viscosity solution of \eqref{eqma}, where $f:\rn\rightarrow\rr$ satisfies
	{\textnormal{(H)}}. If $n\geq3$, then there exist $b\in\rn$, $A\in\A$ and $u_p\in C^{2,\alpha}(\rn)$ such that
	\begin{equation*}
		\begin{aligned}
		\left|u(x)-\left(\frac{1}{2}x'Ax+b\cdot x+u_p(x)\right)\right|\leq C_0(1+|x|)^{2-\min\{n,\beta\}},\ \ \ \forall\ x\in\rn,
		\end{aligned}
	\end{equation*}
	where $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\cdots,n$,
    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\be$, $\az$ and $a_1,\cdots,a_n$.
\end{theorem}

Monge-Amp\`{e}re equations are a class of fully nonlinear equations and can be found in several contexts of analysis and geometry.
The asymptotic behavior of solution to the Monge-Amp\`{e}re equation
\begin{equation}\label{eqma}
 \det(D^2u)=f,\ \ \ \ \ \text{in}\ \mathbb{R}^n,
\end{equation}
as an extension of Liouville's theorem, has been investigated extensively in the last years, starting with the pioneering works by
J\"{o}rgens \cite{j}, Calabi \cite{ce} and Pogorelov \cite{p}. They showed that any convex classical solution of \eqref{eqma} with $f\equiv1$
must be a quadratic polynomial.
Caffarelli \cite{ca} extended the result from classical solution to viscosity solution.
Caffarelli and Li \cite{cl}
considered the case $f$ is equal to $1$ outside a compact subset of $\rn$ and Bao, Li and Zhang \cite{blz} focused on a more general situation
that $f$ is a perturbation of $1$ near infinity. They all proved that any convex viscosity solution $u$ is close to a
quadratic polynomial at infinity for $n\geq3$ (there is an additional logarithmic term for $n=2$). In \cite{lb}, Bao and Liu concentrated on
the regularity of $f$ and weakened it from $f\in C^3$ in \cite{blz} to $f\in C^2$ by the spherical harmonic expansion of the solutions of
nonhomogeneous linearized equations. Bao and Qi \cite{qb} developed a nonlocal method to obtain the asymptotic behavior of $u$ under
the hypothesis on regularity that $f$ is just H\"{o}lder continuous. In \cite{jtx}, Jin, Tu and Xiong considered the case that the right-hand side of the equation is a measure.

On the other hand, $f\equiv1$ is also a special periodic function. In the well known work \cite{cl2}, Caffarelli and Li investigated the case
that $f\in C^{\alpha}$ is a positive periodic function and showed that the difference between $u$ and a quadratic polynomial is periodic. Li and Lu considered the
positive periodic function $f\in L^\infty$ in a following work \cite{ll}. As the counterpart of \cite{blz} in the aperiodic case, Teixeira and Zhang \cite{tz} studied the
the asymptotic behavior of solution to \eqref{eqma}, in which the right hand side $f\in C^{1,{\alpha}}$ is asymptotically close to periodic function.
They showed that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.

In this paper we discuss the asymptotic behavior of solution $u$ to \eqref{eqma} with a H\"{o}lder continuous term $f$, compared to the condition
$f\in C^{1,{\alpha}}$ in \cite{tz}. More precisely, our assumptions on $f$ are as follows:

Let $f_p$ be a positive periodic function in $\rn$ such that, for some constants $d_0>0$, $\alpha\in(0,1)$ and $a_1,\cdots,a_n>0$ there hold
\begin{equation*}
 \begin{array}{rl}
 &d_0^{-1}\leq f_p\leq d_0,\ \ \ [f_p]_{C^\alpha(\rn)}\leq d_0,\\
 &f_p(x+a_ie_i)=f_p(x),\ \ \ \forall x\in\rn,
 \end{array}
\end{equation*}
where $e_i=(0,\cdots,0,1,0,\cdots,0)$ denotes the $i$-th standard coordinate vector in $\rn$, $i=1,\cdots,n$.

We suppose that $f$ is asymptotically close to $f_p$ in the following sense:
\vspace{-0.2cm}
\begin{itemize}
 \item[(H)] The function $f\in C(\rn)\cap C_{loc}^{\alpha}({\rn\setminus\overline{\mathcal{O}}})$ for
 some bounded open subset $\mathcal{O}\subset\rn$ and satisfies, for $x\in\rn\setminus\{0\}$,
 \begin{equation*}\label{eqfcdinfi}
 \begin{aligned}
 |x|^{\beta}\left|\left(f-f_p\right)(x)\right|+
 |x|^{\beta+\alpha}\left[f-f_p\right]_{C^{\alpha}(\overline{B_{\frac{|x|}{2}}(x)})}\leq d_1
 \end{aligned}
 \end{equation*}
 for some constants $\beta>2$ and $d_1>0$.
\end{itemize}
\vspace{-0.2cm}
Let ${\mathbb M}^{n\times n}$ be the set of the real valued, $n\times n$ matrices and
\begin{equation*}
 \mathcal{A}:=\left\{A\in{\mathbb M}^{n\times n}:\ A\text{ is symmetric, positive definite and }\det(A)=
 \int\hspace{-1.05em}-_{\raisebox{-5pt}{\tiny$\prod_{1\leq i\leq n}[0,a_i]$}}fdx\right\}.
\end{equation*}

Our main result is

\begin{equation}\label{eqma}
	\det(D^2u)=f,\ \ \ \ \ \text{in}\ \mathbb{R}^n,
\end{equation}

\begin{remark}\label{remcf}
 It is clear that $f\in C^\az(\rn\setminus\O)$. Moreover, the viscosity solution $u$ in Theorem \ref{thm1}
 is really $C^{2,\alpha}$ near infinity, see Section 3.
\end{remark}

For the case in $A_1$, we first write
\begin{gather*}
 \Gamma_1:=\int_{|y|\leq\frac{|x|}{2}}(-\De)^sg(y)\left(\frac{1}{|x+e-y|^{n-2s}}-\frac{1}{|x-y|^{n-2s}}\right)dy,\\
 \Gamma_2:=\int_{|y|\leq\frac{|x|}{2}}(-\De)^sg(y)\left(\frac{1}{|x-y|^{n-2s}}-\frac{1}{|x-e-y|^{n-2s}}\right)dy.
\end{gather*}
Note that $|x+e|\leq\frac{|x|}{4}$ in $\Omega_2$ and $(-\De)^sg$ is bounded, we have, for $\rho>0$ small,
\begin{equation*}
 \begin{aligned}
 \Gamma_1=&\left(\int_{B_{\frac{|x|}{2}}\setminus B_\rho(x+e)}+\int_{B_\rho(x+e)}\right)
 (-\De)^sg(y)\left(\frac{1}{|x+e-y|^{n-2s}}-\frac{1}{|x-y|^{n-2s}}\right)dy\\
 \leq&C_{n,s}\int_{B_{\frac{|x|}{2}}\setminus B_\rho(x+e)}|(-\De)^sg(y)|\frac{|e|}{|\xi-y|^{n+1-2s}}dy+C_{n,s}(\rho^{2s}+\rho^n),
 \end{aligned}
\end{equation*}
where $\xi$ is obtained by the Lagrange mean value theorem. In fact, $\xi$ can be chosen such that $\xi=x+\theta e$ for $\theta\in[\frac{1}{8},\frac{3}{8}]$,
which implies $|\xi|\geq|x|-\frac{3}{8}|e|\geq\frac{17}{32}|x|$ since $|e|\leq\frac{5}{4}|x|$.
Moreover, $|\xi-y|\geq\frac{1}{32}|x|$ for $|y|\leq\frac{|x|}{2}$. Passing to the limit $\rho\rightarrow0$ one gets
\begin{equation*}
 |\Gamma_1|\leq\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\int_{|y|\leq\frac{|x|}{2}}|(-\De)^sg(y)|dy.
\end{equation*}
Using the same argument we also get
\begin{equation*}
 |\Gamma_2|\leq\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\int_{|y|\leq\frac{|x|}{2}}|(-\De)^sg(y)|dy.
\end{equation*}
Thus, noticing that $\frac{|x|}{2}>1$, we have
\begin{equation}\label{eqg}
\begin{aligned}
 \int_{A_1}|(-\Delta)^sg(y)||\Psi|dy
 &\leq\frac{C_{n,s}|e|}{|x|^{n+1-2s}}\left(\int_{B_{1}}+\int_{B_{\frac{|x|}{2}}\setminus B_{1}}\right)|(-\Delta)^sg(y)|dy.
\end{aligned}
\end{equation}
The boundedness of $(-\Delta)^sg$ implies
\begin{equation*}
 \int_{B_{1}}|(-\Delta)^sg(y)|dy\leq C'
\end{equation*}
for some $C'>0$. On the other hand, equation \eqref{eqlemFgd} yields that
\begin{align*}
 \int_{B_{\frac{|x|}{2}}\setminus B_{1}}|(-\Delta)^sg(y)|dy&\leq C\int_{B_{\frac{|x|}{2}}\setminus B_{1}}|y|^{-\min\{\be,n\}-2s}dy\\
 &\leq
 \left\{
 \begin{array}{ll}
 C|x|^{n-\min\{\be,n\}-2s},&\ \ \ \min\{\be,n\}+2s\neq n,\\
 C\ln|x|,&\ \ \ \min\{\be,n\}+2s=n.
 \end{array}
 \right.
\end{align*}
It follows that
\begin{align*}
 \int_{B_{\frac{|x|}{2}}}|(-\Delta)^sg(y)|dy
 &\leq\left\{
 \begin{array}{ll}
 C(|x|^{n-\min\{\be,n\}-2s}+1),&\ \ \ \be+2s\neq n,\\
 C(\ln|x|+1),&\ \ \ \be+2s=n.
 \end{array}
 \right.
\end{align*}
Since
\begin{align*}
 (|x|^{n-\min\{\be,n\}-2s}+1)\cdot{|x|^{-n-1+2s}}&\leq|x|^{-\min\{\be,n-2s\}-1},
\end{align*}
we use \eqref{eqg} and \eqref{eqex} to obtain
\begin{align*}
 \frac{C}{|e|^{2s}}\int_{A_1}|(-\Delta)^sg(y)||\Psi|dy
 &\leq\left\{
 \begin{array}{ll}
 C|e|^{1-2s}|x|^{-\min\{\be,n-2s\}-1},\\
 C|e|^{1-2s}|x|^{-n+2s-1}(\ln|x|),
 \end{array}
 \right.\\
 &\leq\left\{
 \begin{array}{ll}
 C|x|^{-\min\{\be+2s,n\}},&\ \ \ \be+2s\neq n,\\
 C|x|^{-\beta-2s}(\ln|x|),&\ \ \ \be+2s=n,
 \end{array}
 \right.\ \ \ \text{for}\ |x|>2.
\end{align*}

By \cite[Theorem 0.2]{cl2} (see also \cite{l}), there exists $v_p\in C^{2,\az}(\rn)$ such that
\begin{equation}\label{eqvp}
 \left\{
 \begin{array}{ll}
 \det(I+D^2v_p)=(f_p)_T,\ \ \ \text{in}\ \rn,\\
 I+D^2v_p>0.
 \end{array}
 \right.
\end{equation}
Here, the function $v_p$ is periodic and has the same period as that of $(f_p)_T$. Let
\begin{equation*}
 w(x):=v(x)-\frac{1}{2}|x|^2-v_p(x).
\end{equation*}
Applying Lemma \ref{lemvh} and the facts that $\De^s_e\left(\frac{1}{2}|x|^2\right)=|e|^{2-2s}$ and $\De^s_ev_p(x)=0$, one has
\begin{equation*}
 \De^s_ew(x)-h(x)=\De^s_ev(x)-|e|^{2-2s}-h(x)\leq0,\ \ \ \ \forall x\in\rn,\ e\in E.
\end{equation*}
Combined with the equations \eqref{eqv} and \eqref{eqvp} we have
\begin{equation*}
 \det(D^2v(x))-\det(D^2(\frac{1}{2}|x|^2+v_p(x)))=f_T(x)-(f_p)_T(x),\ \ \ \ \forall x\in\rn,
\end{equation*}
from which we get the equation for $w$:
\begin{equation*}
 \tilde{a}_{ij}(x)\pat_{ij}w(x)=f_T(x)-(f_p)_T(x),\ \ \ \ \forall x\in\rn,
\end{equation*}
where
\begin{equation*}
 \tilde{a}_{ij}(x):=\int_0^1cof_{ij}\left[tD^2v(x)+(1-t)D^2\left(\frac{1}{2}|x|^2+v_p(x)\right)\right]dt.
\end{equation*}
In fact, we can use Proposition \ref{pro2} and \eqref{eqvp} to get that $\tilde{a}_{ij}(x)$ is uniformly elliptic. On the other hand, we can
use the fundamental solution to construct (as that of $h$) a function $h_1$ which satisfies
\begin{equation*}
 \tilde{a}_{ij}(x)\pat_{ij}h_1(x)=(f_p)_T(x)-f_T(x),\ \ \ \ \forall x\in\rn.
\end{equation*}
Therefore,
\begin{equation}\label{eqwh1}
 \tilde{a}_{ij}(x)\pat_{ij}\left(w+h_1\right)(x)=0,\ \ \ \ \forall x\in\rn.
\end{equation}
By \cite[Lemma 3.2]{lb} and assumption (H) we conclude
\begin{equation*}
 |h_1(x)|\leq
 \left\{
 \begin{array}{ll}
 C(1+|x|)^{2-\min\{\be,n\}},&\ \ \ \ \be\neq n,\\
 C(1+|x|)^{2-n}\ln(2+|x|),&\ \ \ \ \be=n.
 \end{array}
 \right.
\end{equation*}

By \eqref{eqglim} we have $\max_{B_1}g\geq\frac{3}{4}$, thus the linear function $l=a\cdot x$ for $|a|\geq\frac{3}{4}$, where we have using the fact that
$g(0)=l(0)=0$. It follows
\begin{equation}\label{eqwro}
 \frac{w(ry)}{M_r}-a\cdot y\rightarrow0\ \ \ \ \text{as}\ r\rightarrow\infty\ \ \ \ \text{for }y\in B_{\frac{3}{4}},
\end{equation}
from which we obtain for $r$ large that
\begin{equation}\label{eqcon}
 w(2^Ne)>\frac{M_r}{2r}2^N|e|,
\end{equation}
where $e\in E$ satisfies $a\cdot e>\frac{2}{3}$ and integer $N$ is chosen so that $2^N|e|<\frac{3}{4}r\leq2^{N+1}|e|$. Let $\bar{l}$ be a linear function
that agrees with $w$ at $0$ and $e$. Then the function $\overline{w}:=w-\bar{l}$ satisfies $\overline{w}(0)=\overline{w}(e)=0$ and
\begin{equation*}
 \overline{w}(2^Ne)>\frac{M_r}{4r}2^N|e|
\end{equation*}
for $r$ large. Since, by Remark \ref{remhss},
\begin{equation*}
 \De^s_e\overline{w}(x)=\De^s_ew(x)\leq C|x|^{-2s-\sigma},\ \ \ \ \forall |x|\geq1
\end{equation*}
for some $\sigma>0$ small, we have
\begin{equation*}
 \overline{w}(2e)\leq2\overline{w}(e)-\overline{w}(0)+C|e|^{2s}|e|^{-2s-\sigma}\leq C|e|^{-\sigma}.
\end{equation*}
Similarly,
\begin{align*}
 \overline{w}(4e)&\leq2\overline{w}(2e)-\overline{w}(0)+C|2e|^{2s}|2e|^{-2s-\sigma}\\
 &\leq 2C|e|^{-\sigma}+C|2e|^{-\sigma}.
\end{align*}
By induction,
\begin{align*}
 \overline{w}(2^Ne)&\leq C|e|^{-\sigma}\left(2^{N-1}+2^{N-2-\sigma}+2^{N-3-2\sigma}+\cdots+2^{-\sigma(N-1)}\right)\\
 &\leq 2^{N-1}C|e|^{-\sigma}\frac{1-2^{-N(1+\sigma)}}{1-2^{-(1+\sigma)}}\leq C2^N|e|,
\end{align*}
a contradiction to \eqref{eqcon}. Thus \eqref{eqtmr} is proved.

Now we rewrite \eqref{eqwro} as
\begin{equation}\label{eqwo}
 \left|w(x)-\frac{M_r}{r}a\cdot x\right|=o(1)M_r,\ \ \ \ \ \text{for }x\in B_{\frac{3}{4}r}.
\end{equation}
Let
\begin{equation*}
 b:=\lim_{r\rightarrow\infty}\frac{M_r}{r}a\ \ \ \ \text{and }\ \ \ w_1(x):=w(x)-b\cdot x.
\end{equation*}
Then we get from \eqref{eqwo} that $|w_1(x)|=o(|x|)$ for $|x|$ large, which gives
\begin{equation*}
 |w_1(x)|\leq C+o(|x|),\ \ \ \ \forall x\in\rn.
\end{equation*}
In other words, $w_1$ is bounded in $\rn$ and so is $w_1+h_1$. We also have
\begin{equation*}
 \tilde{a}_{ij}(x)\pat_{ij}\left(w+h_1\right)(x)=0,\ \ \ \ \forall x\in\rn.
\end{equation*}
Thus, by the Harnack inequality we see that $w_1+h_1\equiv C$. Theorem \ref{thm1} is proved.
\qed

\begin{remark}\label{remcf}
	It is clear that $f\in C^{\alpha}(\rn\setminus\mathcal{O})$. Moreover, the viscosity solution $u$ in Theorem \ref{thm1}
	is really $C^{2,\alpha}$ near infinity, see Section 3.
\end{remark}

\begin{remark}\label{rembeta}
	The assumption $\beta>2$ is essentially optimal, one can see the example in \cite{blz,tz}.
\end{remark}

The major difficulties in this paper are caused by the lower regularity and the aperiodicity of $f$. To get the asymptotic behavior near infinity,
the authors both in \cite{cl2} and \cite{tz} paid much attention on the second-order increment of $u$. Their vital step is deducing a suitable
equation of the increment. More specifically, in \cite{cl2}, although $f$ is just H\"{o}lder continuous they obtained a homogeneous equation of the increment
since $f$ is periodic, while in \cite{tz} the authors got a nonhomogeneous equation in which the nonhomogeneous term can be controlled since the higher
regularity of $f$.
In the case $f\in L^\infty$ is periodic \cite{ll}, the authors established the asymptotic results by following the main ideas in
\cite{cl2} with additional approximation arguments.
It is not the case in our paper and the second-order increment is not an appropriate object for us. We will propose a novel
fractional order increment and deduce a nonhomogeneous equation of it. Then we employ the nonlocal ideas in \cite{qb} and refine the
nonlocal method to measure the decay of the nonhomogeneous term. The asymptotic behavior is finally established with the aid of the fractional order increment.

\subsection*{Notations and the structure of the article}

Throughout the paper, we use the following notations:

The point $x\in\rn$ will also be written as $x=(x_1,x')=(x_1,\cdots,x_n)$. The notation $|x|$ denotes the Euclidean norm of $x$.
$B_r(x)\subset\rn$ denotes the ball centered at $x$ with radius $r$.
We drop the center if it coincides with the origin, i.e., $B_r=B_r(0)$.

The identity matrix is denoted by $I$.
For an $n\times n$ matrix $B=(b_{ij})_{n\times n}$, $1\leq i,j\leq n$, $cof_{ij}B$ stands for the algebraic cofactor of
element $b_{ij}$. We will always assume that $i,j\in\mathbb{N}_+$ and $i,j\leq n$ in this paper
except for additional explanation.

Given a function $u:\rn\rightarrow{\mathbb R}$ and a point $x\in\rn$, we denote by $Du(x)$ and $\Delta u(x)$ the gradient  vector and
the  Laplacian respectively.

Let $k\in\mathbb{N}$, $\kappa\in(0,1)$ and $\Omega\subset\rn$ be open. For multi-indices
$\gamma=(\gamma_1,\cdots,\gamma_n)\in\mathbb{N}^n$, we let $|\gamma|$ denote the sum of its components.
The space $C^k(\overline{\Omega})$ consists of functions $u: \overline{\Omega}\rightarrow{\mathbb R}$, which admit derivatives
up to order $k$, such that \begin{equation*}
	\|u\|_{C^{k}(\overline{\Omega})}:=\sup_{\substack{ \gamma\in\mathbb{N}^n\\|\gamma|\leq k}}
	\sup_{x\in{\Omega}}\left|D^\gamma u(x)\right|<+\infty.\end{equation*}
The H\"{o}lder space $C^{k,\kappa}(\overline{\Omega})$ consists of function $u\in C^k(\overline{\Omega})$ satisfying
\begin{gather*}
	\|u\|_{C^{k,\kappa}(\overline{\Omega})}:=\|u\|_{C^{k}(\overline{\Omega})}+[u]_{C^{k,\kappa}(\overline{\Omega})}<+\infty,
\end{gather*}
where the seminorm
\begin{equation*}
	[u]_{C^{k,\kappa}(\overline{\Omega})}:=\sup_{\substack{ \gamma\in\mathbb{N}^n\\|\gamma|=k}}
	\sup_{\substack{x,y\in{\Omega} \\ x\neq y}}\frac{|D^\gamma u(x)-D^\gamma u(y)|}{|x-y|^\kappa}.
\end{equation*}
A function $u\in C^{k,\kappa}_{loc}({\Omega})$ if $u\in C^{k,\kappa}(K)$ for all compact
$K\subset\Omega$. For convenience, we may write $C^{0}(\overline{\Omega})$, $C^{k,\kappa}(\overline{\Omega})$ as $C(\overline{\Omega})$,
$C^{k+\kappa}(\overline{\Omega})$,  respectively.

The paper is organized as follows: In Section 2, we introduce the fractional Laplacian and a series of relevant properties,
which will be applied in the sequel. In Section 3, we deduce the equation for the fractional order increment (Lemma \ref{lemeqv}) and then
show the crucial estimate (Lemma \ref{lemeF}) by a nonlocal method. Finally, Theorem \ref{thm1} is proved in Section 4.

The post-theorem introduction explains that “the major difficulties … are caused by the lower regularity and the aperiodicity of $f$,” so the usual approach via “the second-order increment of $u$” (as in \cite{cl2,tz,ll}) “is not … appropriate” here. Instead, the authors “propose a novel fractional order increment and deduce a nonhomogeneous equation of it.” Then they “employ the nonlocal ideas in \cite{qb} and refine the nonlocal method to measure the decay of the nonhomogeneous term.” With this decay control, “the asymptotic behavior is finally established with the aid of the fractional order increment.”

They also indicate the proof’s placement/structure: in Section 3 they “deduce the equation for the fractional order increment (Lemma \ref{lemeqv}) and then show the crucial estimate (Lemma \ref{lemeF}) by a nonlocal method,” and “finally, Theorem \ref{thm1} is proved in Section 4.”

The post-theorem introduction explains that “the major difficulties … are caused by the lower regularity and the aperiodicity of $f$,” so the usual approach via “the second-order increment of $u$” (as in \cite{cl2,tz,ll}) “is not … appropriate” here. Instead, the authors “propose a novel fractional order increment and deduce a nonhomogeneous equation of it.” Then they “employ the nonlocal ideas in \cite{qb} and refine the nonlocal method to measure the decay of the nonhomogeneous term.” With this decay control, “the asymptotic behavior is finally established with the aid of the fractional order increment.”

They also indicate the proof’s placement/structure: next they deduce the equation for the fractional order increment. \begin{lemma}\label{lemeqv}
	Let $E$, $v$ and $f_T$ be as above. Then, for $s\in(0,\frac{\az}{2})$,
	\begin{equation*}
		a_{ij}(x)\pat_{ij}(\De_e^sv(x))\geq F_s(x),\ \ \ \ \forall x\in\rn,\ e\in E\setminus\{0\},
	\end{equation*}
	where $a_{ij}(x)=cof_{ij}(D^2v(x))$ and
	\begin{equation*}
		F_s(x):=n\left(\det(D^2v(x))^{\frac{n-1}{n}}\right)\left(\frac{f_T^{\frac{1}{n}}(x+e)+f_T^{\frac{1}{n}}(x-e)-2f_T^{\frac{1}{n}}(x)}{|e|^{2s}}\right).
	\end{equation*}
\end{lemma}
They then show the crucial estimate by a nonlocal method. \begin{lemma}\label{lemeF}
	Let $F_s$ be the one obtained in Lemma \ref{lemeqv}. Then we have
	\begin{equation}\label{eqlemeF1}
		|F_s(x)|\leq
		\left\{
		\begin{array}{ll}
			C_0(1+|x|)^{-\min\{\be+2s,n\}},&\ \ \ \be+2s\neq n,\ \be\neq n,\\
			C_0(1+|x|)^{-\beta-2s}(\ln(2+|x|)),&\ \ \ \be+2s=n,\\
			C_0(1+|x|)^{-\beta-2s}(\ln(2+|x|)),&\ \ \ \be=n,
		\end{array}
		\right.\ \ \ \ \
	\end{equation}
	for all $e\in E\setminus\{0\},\ x\in\rn.$ The constant $C_0>0$ depends only $d_0,\ d_1,\ \be,\ n$ and $s$.
\end{lemma}
Finally, in establishing the main theorem, they complete the argument in the subsequent section.

\label{thm1}
	Let $u\in C_{loc}(\rn)$ be a convex viscosity solution of
\begin{equation}\label{eqma}
	\det(D^2u)=f,\ \ \ \ \ \text{in}\ \mathbb{R}^n,
\end{equation}
, where $f:\rn\rightarrow{\mathbb R}$ satisfies
	{\textnormal{(H)}}. If $n\geq3$, then there exist $b\in\rn$, $A\in\mathcal{A}$ and $u_p\in C^{2,\alpha}(\rn)$ such that
	\begin{equation*}
		\begin{aligned}
		\left|u(x)-\left(\frac{1}{2}x'Ax+b\cdot x+u_p(x)\right)\right|\leq C_0(1+|x|)^{2-\min\{n,\beta\}},\ \ \ \forall\ x\in\rn,
		\end{aligned}
	\end{equation*}
	where $u_p$ is $a_i$-periodic in the $i$-th variable, $i=1,\cdots,n$,
    and the constant $C_0>0$ depends on $d_0$, $d_1$, $n$, $\beta$, ${\alpha}$ and $a_1,\cdots,a_n$.,

Existential–Universal
Inequality or Bound

Question:
Let $n\ge 3$, and let $u\in C_{\mathrm{loc}}(\mathbb{R}^n)$ be a convex viscosity solution of
\[
\det(D^2u)=f\quad \text{in }\mathbb{R}^n.
\]
Assume there exist constants $d_0>0$, $d_1>0$, $\alpha\in(0,1)$, $\beta>2$, and $a_1,\dots,a_n>0$, together with a positive periodic function $f_p:\mathbb{R}^n\to\mathbb{R}$ such that
\[
d_0^{-1}\le f_p\le d_0,\qquad [f_p]_{C^\alpha(\mathbb{R}^n)}\le d_0,
\]
and
\[
f_p(x+a_i e_i)=f_p(x)\quad \text{for all }x\in\mathbb{R}^n,\ i=1,\dots,n.
\]
Also assume $f\in C(\mathbb{R}^n)\cap C^{\alpha}_{\mathrm{loc}}(\mathbb{R}^n\setminus \overline{\mathcal O})$ for some bounded open set $\mathcal O\subset \mathbb{R}^n$, and that for every $x\in\mathbb{R}^n\setminus\{0
\}$,
\[
|x|^{\beta}|(f-f_p)(x)|+|x|^{\beta+\alpha}[f-f_p]_{C^\alpha(\overline{B_{|x|/2}(x)})}\le d_1.
\]
Define
\[
\mathcal A:=\left\{A\in \mathbb{R}^{n\times n}: A\text{ is symmetric, positive definite, and }\det(A)=\dashint_{\prod_{i=1}^n[0,a_i]} f(x)\,dx\right\},
\]
where $\dashint$ denotes the average over the rectangle $\prod_{i=1}^n[0,a_i]$. Under these assumptions, which quantitative estimate holds?

There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta\}}\qquad \forall x\in\mathbb{R}^n,
\]
where $C_0>0$ depends only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$.

Choices:
B.
There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta+\alpha\}}\qquad \forall x\in\mathbb{R}^n,
\]
where $C_0>0$ depends only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$.
C.
There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p:\mathbb{R}^n\to\mathbb{R}$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta\}}\qquad \forall x\in\mathbb{R}^n,
\]
for some constant $C_0>0$.
D.
There exists a constant $C_0>0$, depending only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$, such that for every $A\in\mathcal A$ there exist $b\in\mathbb{R}^n$ and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, and
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\min\{n,\beta\}}\qquad \forall x\in\mathbb{R}^n.
\]
E.
There exist $b\in\mathbb{R}^n$, $A\in\mathcal A$, and a function $u_p\in C^{2,\alpha}(\mathbb{R}^n)$ that is $a_i$-periodic in the $i$-th variable for each $i=1,\dots,n$, such that
\[
\left|u(x)-\left(\frac12 x^T A x+b\cdot x+u_p(x)\right)\right|\le C_0(1+|x|)^{2-\beta}\qquad \forall x\in\mathbb{R}^n,
\]
where $C_0>0$ depends only on $d_0,d_1,n,\beta,\alpha,$ and $a_1,\dots,a_n$.

decay exponent tied to nonlocal estimate

dropped the stated $C^{2,\alpha}$ regularity and parameter dependence details for $u_p$

existential choice of the asymptotic matrix $A$

case split at $\beta\ge n$ producing the $\min\{n,\beta\}$ rate

The stem does not explicitly reveal the correct option. It presents hypotheses and asks for the guaranteed asymptotic conclusion, without embedding the exact answer.

This is very close to theorem-statement recall: the correct choice is essentially the sharp conclusion of the cited result under the stated hypotheses. The alternatives create some comparison, but the item is still largely a near-restatement.

There is some reasoning pressure in distinguishing the sharp rate, the role of min{n, β}, the uniform-for-all-x bound versus asymptotic little-o, and the determinant constraint on A. However, the task mostly tests recognition of the exact theorem statement rather than substantial derivation.

Several distractors are mathematically meaningful and target common errors: B overstates the decay rate, D alters the quantifier/strength of the conclusion, and E corrupts the determinant condition on A. But C is a weaker statement that appears to be true if A is true, so it weakens the single-correct-answer structure.

A technically sophisticated but theorem-recall-heavy MCQ. It avoids answer leakage and uses mostly plausible distractors, but it is close to a direct theorem restatement and is weakened by including a distractor that is also true in a weaker form.


\setcounter{equation}{0}

We consider the following diffusive prey-predator model with nonlocal dispersal
\begin{equation}\label{pp}
\begin{cases}
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-V](x,t), \; x\in\bR,\, t>0,\\
V_t(x,t)=d \bN_2[V(\cdot,t)](x)+[bV(1-V/U)](x,t), \; x\in\bR,\, t>0,
\end{cases}
\end{equation}
where the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by
\beaa
\bN_i[u(\cdot,t)](x):=\int_{\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\; u=U,V,
\eeaa
where $J_i$ is a probability density function satisfying the following conditions:
\begin{enumerate}
\item[(H1)]\,  $J_i$ is a nonnegative continuous function defined in $\bR$;
\item[(H2)]\, $\int_{\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\in \mathbb{R}$;
\item[(H3)] there exists $\hat{\lambda}_i\in(0,\infty]$ such that  $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy < \infty$ for any $\lambda \in (0, \hat{\lambda_i})$ and $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy\to\infty$ as $\lambda \uparrow \hat{\lambda}_i$.
\end{enumerate}
Note that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\in(0,1)$, when $a>1$.

When the nonlocal dispersal in system \eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \cite{CGS21}. In fact, system \eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \eqref{pp}.

The main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \eqref{pp} is called a traveling wave solution of \eqref{pp}, if there exist a constant $s\in\R$ ({\it the wave speed}) and a function $(\phi,\psi)$ ({\it the wave profile}) of class $C^1(\R)$ such that
$$(U,V)(x,t)=(\phi,\psi)( z),\ \ z:=x-st.$$
We are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\{s,\phi,\psi\}$ satisfies the following system of equations:
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
together with the following asymptotic boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}

For convenience, for $i=1,2$, we set
\beaa
I_i(\lambda):=\int_\bR J_i(y)e^{\lambda y}dy,\ \ \lambda\in\R,
\eeaa
with $I_i(\lambda)\in(0,\infty)$ if $0\le\lambda<\hat{\lambda}_i$ and $I_i(\lambda)=\infty$ if $\lambda\ge\hat{\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\hat{\lambda}_i,\hat{\lambda}_i)$. Also, we introduce the quantity
\beaa
s^*:=\inf_{\lambda\in(0,\hat{\lambda}_2)}\frac{d[I_2(\lambda)-1]+b}{\lambda}.
\eeaa
Note that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.

We now state our main theorem as follows.

We consider the following diffusive prey-predator model with nonlocal dispersal
\begin{equation}\label{pp}
\begin{cases}
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-V](x,t), \; x\in\bR,\, t>0,\\
V_t(x,t)=d \bN_2[V(\cdot,t)](x)+[bV(1-V/U)](x,t), \; x\in\bR,\, t>0,
\end{cases}
\end{equation}
where the unknown functions $U,V$ stand for the population densities of prey and predator species at position $x$ and time $t$, respectively, $d,a,b$ are positive constants such that $1,d$ are diffusion coefficients and $a,b$ are intrinsic growth rates of species $U,V$, respectively. The functional response of predator to prey is normalized to be $1$. The prey obeys the logistic growth and its carrying capacity is normalized to be 1. However, the density of predator follows a logistic dynamics with a varying carrying capacity proportional to the density of prey. Moreover, for $i=1,2$, $\bN_i$ formulates the spatial nonlocal dispersal of individuals and is defined by
\beaa
\bN_i[u(\cdot,t)](x):=\int_{\mathbb{R}}J_i(y)u(x-y,t)dy-u(x,t),\; u=U,V,
\eeaa
where $J_i$ is a probability density function satisfying the following conditions:
\begin{enumerate}
\item[(H1)]\, $J_i$ is a nonnegative continuous function defined in $\bR$;
\item[(H2)]\, $\int_{\mathbb{R}}J_i(y)dy=1$ and $J_i(y)=J_i(-y)$ for all $y\in \mathbb{R}$;
\item[(H3)] there exists $\hat{\lambda}_i\in(0,\infty]$ such that $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy < \infty$ for any $\lambda \in (0, \hat{\lambda_i})$ and $\int_{\mathbb{R}}J_i(y)e^{\lambda y} dy\to\infty$ as $\lambda \uparrow \hat{\lambda}_i$.
\end{enumerate}
Note that there is always the predator-free state $(1,0)$, and there is the unique constant co-existence state $(a^*,a^*)$, $a^*:=1-1/a\in(0,1)$, when $a>1$.

When the nonlocal dispersal in system \eqref{pp} is replaced by the classical diffusion, the dynamical behaviors of the corresponding system was investigated in the survey paper \cite{CGS21}. In fact, system \eqref{pp} without diffusion (the ODE system) arises in the control of introduced rabbits to protect native birds from introduced cat predation in an island (cf. \cite{CLS}), when we consider the case without rabbits and control. For the detailed biological background of the full ODE system including the rabbits and control, we refer the reader to \cite{CLS,CS}. However, it is reasonable and more realistic to take into account the influence of spatial movements of birds and cats, namely, the effect of diffusion. On the other hand, to model the long range movements and nonadjacent interactions of individuals it is more realistic to consider the nonlocal dispersal instead of the random movement with classical diffusion. This motivates us to study system \eqref{pp}.

The main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \eqref{pp} is called a traveling wave solution of \eqref{pp}, if there exist a constant $s\in\R$ ({\it the wave speed}) and a function $(\phi,\psi)$ ({\it the wave profile}) of class $C^1(\R)$ such that
$$(U,V)(x,t)=(\phi,\psi)( z),\ \ z:=x-st.$$
We are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\{s,\phi,\psi\}$ satisfies the following system of equations:
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
& \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z) =0, &\; z\in \mathbb{R},\\
& d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
together with the following asymptotic boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}

For convenience, for $i=1,2$, we set
\beaa
I_i(\lambda):=\int_\bR J_i(y)e^{\lambda y}dy,\ \ \lambda\in\R,
\eeaa
with $I_i(\lambda)\in(0,\infty)$ if $0\le\lambda<\hat{\lambda}_i$ and $I_i(\lambda)=\infty$ if $\lambda\ge\hat{\lambda}_i$. Note that, due to the symmetry of $J_i$, the function $I_i$ is even, and it is also strictly convex in $(-\hat{\lambda}_i,\hat{\lambda}_i)$. Also, we introduce the quantity
\beaa
s^*:=\inf_{\lambda\in(0,\hat{\lambda}_2)}\frac{d[I_2(\lambda)-1]+b}{\lambda}.
\eeaa
Note that $s^*$ is well-defined, the infimum is reached, and $s^*>0$.

We now state our main theorem as follows.

\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}

The main purpose of this paper is to study the existence and nonexistence of traveling wave solutions to \eqref{pp} connecting the predator-free state and the co-existence state. Here a solution $(U,V)$ to \eqref{pp} is called a traveling wave solution of \eqref{pp}, if there exist a constant $s\in\R$ ({\it the wave speed}) and a function $(\phi,\psi)$ ({\it the wave profile}) of class $C^1(\R)$ such that
$$(U,V)(x,t)=(\phi,\psi)( z),\ \ z:=x-st.$$
We are interested in the traveling waves connecting the predator-free state $(1,0)$ to the co-existence state $(a^*,a^*)$. Therefore, $\{s,\phi,\psi\}$ satisfies the following system of equations:
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
& \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z) =0, &\; z\in \mathbb{R},\\
& d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
together with the following asymptotic boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}

We now state our main theorem as follows.

Hereafter, a function $(\phi,\psi)$ is positive if $\phi,\psi>0$ in $\bR$. A corresponding theorem to Theorem~\ref{th:main}, under the condition $a\ge 4$, on the traveling waves in the classical diffusion case was derived in \cite{CGS21}. Theorem~\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\eqref{TWS}.

The existence of wave profiles $(\phi,\psi)$ to \eqref{TWS}-\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\phi,\psi)(-\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\ref{sec2} is devoted to the existence part in Theorem~\ref{th:main}, while the nonexistence is shown in Section~\ref{sec3}.

\begin{proposition}\label{le6}
Let $s>0$ be given. Let $(\overline{\phi },\overline{\psi })$ and $(\underline{\phi},\underline{\psi})$ be a pair of bounded upper and lower solutions of \eqref {TWS}. Then system \eqref {TWS} admits a positive solution $(\phi,\psi)$ of class $C^1(\R)$ such that
\begin{equation*}
\underline{\phi }( z )\leq \phi(z)\leq \overline{\phi}( z ),\; \upsi(z)\leq \psi(z)\leq\overline{\psi}( z ),\; z \in \mathbb{R}.
\end{equation*}
\end{proposition}

For a given $s>s^*$, we consider the quantity
\beaa
A(\lambda)=A(\lambda;s):=d[I_2(\lambda)-1]-s\lambda+b,\; \lambda>0.
\eeaa
It follows from the definition of $s^*$ and the strict convexity of $A(\lambda)$ that there are two positive constants $\lambda_1<\lambda_2<\hat{\ld}_2$ such that
$$A(\lambda_1)=A(\ld_2)=0$$
and $A(\lambda)<0$ for all $\lambda\in(\lambda_1,\lambda_2)$. Also, set
$$B(\ld)=B(\ld;s):=[I_1(\ld)-1]-s\ld.$$
Since $B(0)=0$ and $B'(0)=-s<0$, we can choose a constant $\ld_0\in(0,\min\{\ld_1,\hat{\ld}_1\})$ small enough such that
$$B(\ld_0)<0.$$
Now, for a fixed constant $\mu\in(1,\min{\{ \lambda_2/ \lambda_1,2\}})$, we choose
\be\label{q0}
q>\max\Big\{1,\frac{2b}{-A(\mu\ld_1)}\Big\}.
\ee
Then, the function $f(z) := e^{ -\lambda_1z}-qe^{-\mu \lambda_1z}$ has exactly one zero $z_0>0$ and exactly one maximum point $z_M\in(z_0,\infty)$, and there holds $f(z_M)>0$. Thus, using $a\ge4>1$ and $d<b$, we can choose $\delta$ such that
$$0<\delta<\min\Big\{f(z_M),\underbrace{1-\frac{1}{a}}_{=a^*},\frac12\Big(1-\frac{d}{b}\Big)\Big\},$$
and a point $z_1\in(z_0,z_M)$ such that $f(z_1)=\delta$. Note that
\be\label{delta}
d<b(1-2\delta).
\ee
Furthermore, the inequalities $z_1>z_0>0$, $\ld_1>0$ and $\mu>1$ imply
\be\label{q}
{e^{(\mu-1)\ld_1z_1}>e^{(\mu-1)\ld_1z_0}=q.}
\ee
With these choices of $\mu,q,\delta$, we finally choose $\e$ such that
\be\label{ep1}
0<\e<\min{\left\{\frac{\delta}{1+s\ld_1+a},\frac{e^{(\mu-1)\lambda_1z_1}-q}{(1+s\lambda_1+a)e^{(\mu-1)\lambda_1z_1}} \right\}}.
\ee
Note that the constant $\e$ is admissible, due to \eqref{q}, and that $0<\e<\delta<1/2$.

\begin{lemma}\label{la:upper-lower2}
Assume that $a\ge 4$, $d<b$, and that $J_2$ has a compact support such that its support is contained in $[-S,S]$ for some $S\in(0,\infty)$. For $s=s^{*}$, the functions $(\overline{\phi},\overline{\psi})$ and~$(\underline{\phi},\underline{\psi})$ defined in \eqref{uplosol2} are a pair of upper and lower solutions of \eqref{TWS}.
\end{lemma}

First, for any $s\ge s^*$, the existence of a $C^1(\R)$ solution $(\phi,\psi)$ to~\eqref{TWS} such that $1/2\le\underline{\phi}\le\phi\le\overline{\phi}<1$ and $0<\underline{\psi}\le\psi\le\overline{\psi}\le1$ in $\R$ follows from Lemmas~\ref{la:upper-lower1}-\ref{la:upper-lower2} and Proposition~\ref{le6}. Moreover, by the construction of upper and lower solutions, it is clear that $(\phi,\psi)(\infty)=(1,0)$.

\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}

\begin{equation}\label{pp}
\begin{cases}
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-V](x,t), \; x\in\bR,\, t>0,\\
V_t(x,t)=d \bN_2[V(\cdot,t)](x)+[bV(1-V/U)](x,t), \; x\in\bR,\, t>0,
\end{cases}
\end{equation}

\begin{theorem}\label{th:main}
Let $a$, $b$ and $d$ be given positive constants such that $a\ge 4$ and $d<b$. Then there exists a positive solution $(\phi,\psi)$ of \eqref{TWS}-\eqref{BC} for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of \eqref{TWS}-\eqref{BC} for $s < s^{*}$.
\end{theorem}

Hereafter, a function $(\phi,\psi)$ is positive if $\phi,\psi>0$ in $\bR$. A corresponding theorem to Theorem~\ref{th:main}, under the condition $a\ge 4$, on the traveling waves in the classical diffusion case was derived in \cite{CGS21}. Theorem~\ref{th:main} characterizes the minimal speed $s^*$ in the nonlocal dispersal case, but it needs the extra condition $d<b$ in the construction of lower and upper solutions to~\eqref{TWS}.

It is worth to note that there are some difficulties in the study of system \eqref{pp}. It is a non-monotone system which is lack of comparison principle and a singularity may occur in the $V$-equation when $U$ reaches zero in a finite time which has been confirmed numerically in \cite{CLS}. Thus system \eqref{pp} is named as a {\it singular} prey-predator model due to the latter fact. This is in contrast to the system with $U$-equation in \eqref{pp} replaced by
\beaa
U_t(x,t)= \bN_1[U(\cdot,t)](x)+[aU(1-U)-kUV](x,t), \; x\in\bR,\, t>0,
\eeaa
in which the functional response of predation is a linear function of prey. In this case, the strong maximum principle for the scalar equation gives that $U>0$ for all $t>0$. Hence no singularity can occur for the nonlinear term $V/U$ in the $V$-equation.

The existence of wave profiles $(\phi,\psi)$ to \eqref{TWS}-\eqref{BC} for all speeds $s>s^*$ is based on the construction of lower and upper solutions, which is new for this singular nonlocal system. The case $s=s^*$ is more involved, it requires a special care and uses the boundedness of the support of $J_2$. The characterization of the limiting state behind the front, that is, $(\phi,\psi)(-\infty)=(a^*,a^*)$ is carried out with a squeezing method and an argument by contradiction. The proof of the nonexistence of wave profiles for all speeds $s<s^*$ is also done by contradiction. Section~\ref{sec2} is devoted to the existence part in Theorem~\ref{th:main}, while the nonexistence is shown in Section~\ref{sec3}.

The post-theorem introduction outlines the proof of Theorem~\ref{th:main} as follows:

- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to~\eqref{TWS}.”

- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).

- **Limiting state behind the front:** the claim “$(\phi,\psi)(-\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”

- **Nonexistence for $s<s^*$:** proved “by contradiction.”

- **Organization:** “Section~\ref{sec2} is devoted to the existence part in Theorem~\ref{th:main}, while the nonexistence is shown in Section~\ref{sec3}.”

The post-theorem introduction outlines the proof strategy used to establish the main theorem as follows:

- **Existence for $s>s^*$:** “based on the construction of lower and upper solutions,” which the authors note “is new for this singular nonlocal system,” and the extra condition $d<b$ is needed “in the construction of lower and upper solutions to
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}.”

- **Existence at $s=s^*$:** “more involved,” requiring “special care” and using “the boundedness of the support of $J_2$” (i.e., compact support).

- **Limiting state behind the front:** the claim “$(\phi,\psi)(-\infty)=(a^*,a^*)$” is proved “with a squeezing method and an argument by contradiction.”

- **Nonexistence for $s<s^*$:** proved “by contradiction.”

- **Organization:** Next they prove the existence part in the main theorem, while the nonexistence is shown later.

\label{th:main}
Let $a$, $b$ and $d$ be given positive constants such that $a\ge 4$ and $d<b$. Then there exists a positive solution $(\phi,\psi)$ of
\begin{equation}\label{TWS}
\left\{
\begin{aligned}
&   \bN_1[\phi](z)+ s\phi'(z) +a\phi(z)\left[1-\phi(z)\right]-\psi(z)    =0, &\; z\in \mathbb{R},\\
&  d \bN_2[\psi](z)+ s\psi'(z) +b\psi(z)\left[1-\frac{\psi(z)}{\phi(z)}\right] =0, & \; z\in \mathbb{R},
\end{aligned}
\right.
\end{equation}
with boundary conditions
\begin{equation}\label{BC}
(\phi,\psi)(\infty) = (1,0),\quad (\phi,\psi)(-\infty) = (a^{*},a^{*}).
\end{equation}
for each $s > s^{*}$. This existence also holds for $s=s^*$, if we further assume that $J_2$ has a compact support. Moreover, there exist no positive solutions of the system \eqref{TWS} with boundary conditions \eqref{BC} for $s < s^{*}$.

Question:
Let \(a,b,d>0\) with \(a\ge 4\) and \(d<b\), and set \(a^*=1-\frac1a\). For \(i=1,2\), define the nonlocal dispersal operator
\[
\mathcal N_i[u](z)=\int_{\mathbb R}J_i(y)u(z-y)\,dy-u(z),
\]
where each kernel \(J_i\) is a nonnegative continuous symmetric probability density on \(\mathbb R\) satisfying the exponential-moment condition: there exists \(\hat\lambda_i\in(0,\infty]\) such that \(\int_{\mathbb R}J_i(y)e^{\lambda y}\,dy<\infty\) for \(0<\lambda<\hat\lambda_i\), and this integral tends to \(\infty\) as \(\lambda\uparrow \hat\lambda_i\). Let
\[
I_2(\lambda)=\int_{\mathbb R}J_2(y)e^{\lambda y}\,dy,
\qquad
s^*=\inf_{\lambda\in(0,\hat\lambda_2)}\frac{d\,[I_2(\lambda)-1]+b}{\lambda}.
\]
Consider the traveling-wave profile system
\[
\begin{cases}
\mathcal N_1[\phi](z)+s\phi'(z)+a\phi(z)(1-\phi(z))-\psi(z)=0, & z\in\mathbb R,\\
 d\,\mathcal N_2[\psi](z)+s\psi'(z)+b\psi(z)\left(1-\dfrac{\psi(z)}{\phi(z)}\right)=0, & z\in\mathbb R,
\end{cases}
\]
with boundary conditions
\[
(\phi,\psi)(+\infty)=(1,0),\qquad (\phi,\psi)(-\infty)=(a^*,a^*).
\]
Which statement holds for positive solutions \((\phi,\psi)\) of this boundary-value problem?

A positive solution exists for every wave speed \(s>s^*\). It also exists for the critical speed \(s=s^*\) provided that \(J_2\) has compact support. For every \(s<s^*\), no positive solution exists.

Choices:
B. A positive solution exists for every wave speed \(s\ge s^*\). Moreover, for every \(s<s^*\), no positive solution exists.
C. For every \(s>s^*\), there exists a positive solution. In addition, for every \(s<s^*\), no positive solution exists.
D. A positive solution exists for every wave speed \(s>s^*\), and also for \(s=s^*\) without any additional assumption on \(J_2\). For every \(s<s^*\), no positive solution exists.
E. A positive solution exists for every wave speed \(s>s^*\). It also exists for the critical speed \(s=s^*\) provided that both \(J_1\) and \(J_2\) have compact support. For every \(s<s^*\), no positive solution exists.

critical-speed compact-support hypothesis at s=s^*

dropped the critical-speed existence clause

dependence of critical-speed existence on compact support of J_2

which kernel requires compact support at the critical speed

The stem does not state the conclusion outright. It defines the setting and the threshold speed $s^*$, but the exact existence/nonexistence regime—especially the subtle critical-speed compact-support exception—is not leaked.

The item is very close to asking for the theorem’s exact statement in this setting, so it is largely a recall/restatement question. Still, the options differ in meaningful ways at the critical speed, so it is not a pure tautology.

Some reasoning is needed to compare the nuanced alternatives ($s>s^*$, $s=s^*$, compact support, and universal nonexistence for $s<s^*$). However, it mainly tests precise theorem recall rather than generating a conclusion from the equations.

The distractors are plausible and target natural failure modes: overclaiming existence at $s=s^*$, omitting the compact-support hypothesis, making the threshold too strict, or weakening the nonexistence quantifier. They are distinct and mathematically relevant.

A solid theorem-discrimination MCQ with strong distractors and little answer leakage, but it leans more toward exact-result recall than genuine generative reasoning.


Question:
Let \(a,b,d>0\) with \(a\ge 4\) and \(d<b\), and set \(a^*=1-\frac1a\). For \(i=1,2\), define the nonlocal dispersal operator
\[
\mathcal N_i[u](z)=\int_{\mathbb R}J_i(y)u(z-y)\,dy-u(z),
\]
where each kernel \(J_i\) is a nonnegative continuous symmetric probability density on \(\mathbb R\) satisfying the exponential-moment condition: there exists \(\hat\lambda_i\in(0,\infty]\) such that \(\int_{\mathbb R}J_i(y)e^{\lambda y}\,dy<\infty\) for \(0<\lambda<\hat\lambda_i\), and this integral tends to \(\infty\) as \(\lambda\uparrow \hat\lambda_i\). Let
\[
I_2(\lambda)=\int_{\mathbb R}J_2(y)e^{\lambda y}\,dy,
\qquad
s^*=\inf_{\lambda\in(0,\hat\lambda_2)}\frac{d\,[I_2(\lambda)-1]+b}{\lambda}.
\]
Consider the traveling-wave profile system
\[
\begin{cases}
\mathcal N_1[\phi](z)+s\phi'(z)+a\phi(z)(1-\phi(z))-\psi(z)=0, & z\in\mathbb R,\\
 d\,\mathcal N_2[\psi](z)+s\psi'(z)+b\psi(z)\left(1-\dfrac{\psi(z)}{\phi(z)}\right)=0, & z\in\mathbb R,
\end{cases}
\]
with boundary conditions
\[
(\phi,\psi)(+\infty)=(1,0),\qquad (\phi,\psi)(-\infty)=(a^*,a^*).
\]
Which statement holds for positive solutions \((\phi,\psi)\) of this boundary-value problem?

Choices:
B. A positive solution exists for every wave speed \(s\ge s^*\). Moreover, for every \(s<s^*\), no positive solution exists.
C. For every \(s>s^*\), there exists a positive solution. In addition, for every \(s<s^*\), no positive solution exists.
D. A positive solution exists for every wave speed \(s>s^*\), and also for \(s=s^*\) without any additional assumption on \(J_2\). For every \(s<s^*\), no positive solution exists.
E. A positive solution exists for every wave speed \(s>s^*\). It also exists for the critical speed \(s=s^*\) provided that both \(J_1\) and \(J_2\) have compact support. For every \(s<s^*\), no positive solution exists.

The stem gives definitions and hypotheses but does not directly reveal which existence/nonexistence statement is correct. The correct choice is not implied by wording cues alone.

This is close to a theorem-recall item: the stem presents the full hypotheses and asks for the exact conclusion. However, it is not a pure verbatim restatement because the options introduce competing variants about the critical speed and compact-support assumptions.

Some discrimination is required among subtle alternatives, especially the role of compact support of J2 at s=s*. But the task mainly tests recall of the theorem’s precise statement rather than substantial derivation or generative mathematical reasoning.

The distractors are plausible and well targeted: they vary in logically meaningful ways (dropping the critical-speed clause, overgeneralizing to all s>=s*, removing the compact-support condition, or assigning it to the wrong kernel). These reflect realistic confusion points.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise theorem recall than genuine generative reasoning.


[\cref{thm:main1,thm:mainsimplification}]
\label{thmfirst:intro}
The augmented vertically parametrized system $(C(\kappa \circ x^M), Lx - b)$ admits multiple positive zeros if and only if  there exist an orientation $\sigma \in \{-1,0,1\}^{2 \times \ell}$ and $\rho \in \mathcal{G}^{\sigma}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$
has a solution.

\begin{theorem}[Preclusion of multiple positive zeros]\label{thm:mono}
 Let $F=( C (\k \circ x^M), Lx-b) \in \R[\k,b,x^\pm]^n$ be an augmented vertical system with $C\in \R^{\bs\times \bar{m}}$ principal and $L$ of full row rank.  Let  $(\tau,r,\gamma')$ and $(\alpha,c,\gamma)$ be  row and column partitions, and consider the  associated simplified reduced matrix $P\in \R^{s\times \ell}$ from \eqref{def:simpif_reduced_matrix}.

 If $\mathcal{C}_{\mathcal{S}}^\sigma = \varnothing$ for all orientations $\sigma \in \{-1,0,1\}^{2 \times \ell}$   compatible with $\alpha$ and  all  $\mathcal{S}\in \{-1,0,1\}^{s\times \ell}$
 that are feasible with respect to $(P,\sigma)$, then $F$ does not admit multiple positive zeros for any choice of parameter values.
\end{theorem}

\begin{theorem}[Characterization of multiple positive zeros when $P$ induces a forest]\label{thm:full}
Let $F=( C (\k \circ x^M), Lx-b) \in \R[\k,b,x^\pm]^n$ be an augmented vertical system with $C\in \R^{\bs\times \bar{m}}$ principal and $L$ of full row rank.
Let  $(\tau,r,\gamma')$ and $(\alpha,c,\gamma)$ be  row and column partitions, and assume that the associated simplified reduced matrix $P\in \R^{s\times \ell}$ from \eqref{def:simpif_reduced_matrix} induces a forest.

 Then, $F$ admits multiple positive zeros for some  parameter values $\k\in \R^{\bar{m}}_{>0}$, $b\in \R^{n-\bs}$  if and only if there exist an orientation $\sigma \in \{-1,0,1\}^{2 \times \ell}$ compatible with $\alpha$ and a sign matrix  $\mathcal{S}\in \{-1,0,1\}^{s \times \ell}$ that is feasible with respect to $(P,\sigma)$  such that
$ \mathcal{C}_{\mS}^\sigma \neq \varnothing$.
\end{theorem}

Polynomial systems that arise in applications have often fixed support and their coefficients are functions of some parameters. \bfemph{Vertically parametrized} polynomial systems \cite{FELIU2025630, genericrootcounts} constitute an example of this, and appear in the study of chemical reaction networks \cite{dickenstein2016biochemical, feinberg-book,feliu2025genericgeometrysteadystate}, polynomial optimization \cite{gkz1994} or geometric modelling \cite{craciunsottile}. These are systems defined by a set of polynomials of the form
\begin{equation*}
 C \big( \mathbf{\kappa} \circ x^M \big) \ \in \mathbb{R}[\kappa,x^{\pm}]^{\bar{s}}
\end{equation*}
with parameters $\kappa=(\kappa_1,\dots,\kappa_{\bar{m}})$ and variables $x=(x_1,\dots,x_n)$. Here $C \in \mathbb{R}^{\bar{s} \times \bar{m}}$ is a full rank matrix, the columns of $M \in \mathbb{Z}^{n \times \bar{m}}$ give the exponents of the monomials of the system, and $\kappa \circ x^M$ indicates that
the $i$-th monomial is scaled by $\kappa_i$. A particular feature of vertically parametrized systems is that each parameter accompanies the same monomial in all entries.
For instance,
\begin{align*}
 \big( \k_1x_1- 2 \k_2x_1x_2^2 + \k_3 x_2^2, \k_2x_1x_2^2 - (\k_3+\k_5)x_2^2 , -\k_4 x_1 + 3\k_5 x_2^2\big)
\end{align*}
is a vertically parametrized system with
\[ C=\begin{pmatrix}
 1 & -2 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0& -1 \\ 0 & 0 & 0& -1 & 3
\end{pmatrix}\quad \text{ and } \quad
M=\begin{pmatrix}
 1 & 1 & 0 & 1 & 0 \\ 0 & 2 & 2 & 0 & 2
\end{pmatrix}\, . \]
Sparse (or freely parametrized) polynomial systems, in which each monomial gives rise to a different parameter, are a particular instance of this type of systems.

Motivated by their origin in chemical reaction network theory, we include
additional $n - \bar{s}$ linear entries and consider \bfemph{augmented vertically parametrized} polynomial systems,
\[
\big( C\big( \kappa \circ x^M\big) , Lx - b \big) \in \mathbb{R}[\kappa,b,x^{\pm}]^{n} \, ,
\]
where $b=(b_1,\dots,b_{n-\bar{s}})$ are parameters.

The goal of this work is to decide whether an augmented vertically parametrized system
\bfemph{admits multiple positive zeros}, that is, whether there
exists a choice of parameter values $\kappa\in \mathbb{R}^{\bar{m}}_{>0} $, $b\in \mathbb{R}^{n-\bar{s}}$ and distinct $x,y \in \mathbb{R}_{>0}^n$ such that
\[ C(\kappa \circ x^M) = C(\kappa \circ y^M) =0 \qquad Lx = Ly = 0\, .\]

Our approach builds on ideas from chemical reaction network theory (mainly the higher deficiency algorithm \cite{Ji2011UniquenessOE}, but also \cite{Feinberg1988,advancedellison}) and extends to the general framework of augmented vertically parametrized polynomial systems. Similar approaches also appeared in \cite{ conradiflockerzimulti,hernandez2020fundamentaldecompositionsmultistationaritypowerlaw}.
The main underlying idea is that, under certain hypotheses, deciding upon the existence of multiple positive zeros can be reduced to checking the feasibility of a linear system of equalities and inequalities, or equivalently, deciding whether the relative interiors of a family of polyhedral cones are empty. This reformulation is crucial because it allows the problem to be solved with algorithms that have a more favorable computational complexity than the standard Cylindrical Algebraic Decomposition approach, which relies on Gr\"obner basis computations.

To understand the main ideas behind this system of linear equalities and inequalities, we assume that, in row reduced echelon form, $C$ is of the form
\[
C = \begin{pmatrix} \id_{\bar{s}} & -\bar{P}
\end{pmatrix}\, ,
\]
and let
$P \in \mathbb{R}^{s \times \ell}$ be obtained from $\bar{P}$ by selecting one representative
for each set of pairwise proportional rows and columns (see \eqref{def:simpif_reduced_matrix}).
We consider orientations $\sigma \in \{-1,0,1\}^{2 \times \ell}$ after imposing certain conditions depending on $\bar{P}$, see \cref{section:simplifications}, and for any $\rho \in \mathbb{R}^{m}$, where $m = s + \ell$, we consider matrices $P^{\sigma}$ and $A^\sigma_{\rho}$ defined by
\[
 (P^{\sigma})_{ik} = \sigma_{1k}P_{ik}\, , \qquad (A^\sigma_{\rho})_{ik} = P_{ik}
 (\sigma_{1k}\,e^{\rho_i} - \sigma_{2k}\,e^{\rho_{s+ k}}), \qquad i \in \{1,\dots,s\}, \ k \in \{1,\dots,\ell\}\, .
\]
 With these objects in place, the \bfemph{oriented characteristic system} associated with $(P,\sigma,\rho)$ is the linear system
\[
A_{\rho}^{\sigma}\mu = 0\,, \qquad P^{\sigma}\mu > 0\,, \qquad \mu \in \R_{>0}^{\ell}\, .
\]
Finally, we consider the oriented ground set $\mathcal{G}^{\sigma}\subseteq \mathbb{R}^m$ defined in \cref{section:simplifications}.

\begin{equation}
\label{def:simpif_reduced_matrix}
P_{ik} :=  \oP_{r(i), c(k)} \,, \qquad i\in [s], \ k\in [\ell] \, . \end{equation}

\begin{thmc}[\cref{thm:full}]
Assume that $P\in \R^{s\times \ell}$ induces a forest.
 Then the system $( C (\k \circ x^M), Lx-b)$ admits multiple positive zeros if and only if there exist an orientation $\sigma \in \{-1,0,1\}^{2 \times \ell}$ and feasible $\mathcal{S}\in \{-1,0,1\}^{s \times \ell}$ such that
$ \mathcal{C}_{\mS}^\sigma \neq \varnothing$.
\end{thmc}

\begin{theorem}
\label{thm:mainsimplification}
 Let $F=( C (\k \circ x^M), Lx-b) \in \R[\k,b,x^\pm]^n$ be an augmented vertical system with $C\in \R^{\bs\times \bar{m}}$ principal with reduced matrix $\oP\in \R^{\bs\times \bl}$, and $L\in \R^{(n-\bs)\times n}$ has full rank.
 Let $(\tau,r,\gamma')$ and $(\alpha,c,\gamma)$ be row and column partitions, and consider the associated simplified reduced matrix $P\in \R^{s\times \ell}$ from \eqref{def:simpif_reduced_matrix}.
The following statements are equivalent:
\begin{enumerate}[label=(\roman*)]
 \item There exists $\bar{\rho} \in M^\top(\mO_L)$ such that the characteristic system
\[
A_{\bar{\rho}}\, \bar{\mu} = 0\,, \qquad \oP\, \bar{\mu} > 0\,, \qquad \bar{\mu}\in \R^{\bl}_{>0}
\]
has a solution.
 \item There exist an orientation $\sigma \in \{-1,0,1\}^{2 \times \ell}$ compatible with $\alpha$, and $\rho \in \mathcal{G}^{\sigma}$ such that the oriented characteristic system
\[ A^\sigma_{\rho} \mu = 0\,, \qquad P^{\sigma}\mu > 0\,, \qquad \mu \in \mathbb{R}^{\ell}_{>0}\]
has a solution.
\end{enumerate}
\end{theorem}

\begin{lemma}\label[lemma]{mainlemma}
Let $P\in \R^{s\times \ell}$, $\sigma \in \{-1,0,1\}^{2 \times \ell},\mathcal{S} \in \{-1,0,1\}^{s \times \ell}$ such that
$\mS$ is feasible with respect to $(P,\sigma)$ and consider their associated $\Lambda$-sets.
Assume $P^\sigma\mu^*>0$ for some $\mu^*\in \R^\ell_{>0}$ and let $\rho \in \R^m\setminus \mathcal{D}_{\mS}^{\sigma} $ such that $\sign(A_{\rho}^{\sigma}) = \mathcal{S}$. Then, there exists
 $Q \in \mathbb{R}^{s \times \ell}$ with $\sign(Q) = \mathcal{S}$ such that the system
\[ Q \mu = 0,\qquad P^\sigma \mu>0, \qquad \mu\in \R^\ell_{>0} \]
has a solution and, additionally, for all $i \in [s]$, it holds
 \begin{equation}\label{eq2MainLemma}
 \sum_{j \in \Lambda_i^{\neq}}\frac{Q_{ij} \, \mu_j}{e^{\rho_i} - \frac{\sigma_{2j}}{\sigma_{1j}}\, e^{\rho_{s + j}}} + \sum_{j \notin \Lambda_i}P^{\sigma}_{ij} \mu_j > 0 \, .
 \end{equation}
\end{lemma}
\begin{proof}
As $\rho \notin \mathcal{D}_{\mS}^\sigma$, if $I_{\rho}^+ \cup I_{\rho}^{-}\neq \varnothing$, then there exists $\mu \in \bigcap_{i \in I_{\rho}^+ \cup I_{\rho}^{-}}\Gamma_{\Lambda,i} \neq \varnothing$. If $I_{\rho}^+ \cup I_{\rho}^{-}= \varnothing$, then we let $\mu=\mu^*$.
In both cases $P^\sigma \mu>0$, and hence, by \cref{prop:feasible}, there exists $\widetilde{Q}\in \mathbb{R}^{s \times \ell} $ with $\sign(\wQ) = \mathcal{S}$ such that $\wQ \mu = 0$.
We construct now a parametric matrix $Q(\omega)$ for $\omega = (\omega_1,\dots,\omega_s) \in \mathbb{R}_{>0}^s$ such that, for every $i\in [s]$, $\omega_i$ only modifies the $i$-th row and it holds that
\begin{equation}\label{eq:row_mod}
\sign(\wQ_{ij})=\sign(Q(\omega)_{ij}) \quad\text{for all } j\in [\ell]\, , \qquad (Q(\omega)\mu)_i=0 \, ,
\end{equation}
and \eqref{eq2MainLemma} holds after choosing $\omega_i$ either small or large enough.

Assume that $P$ induces a forest. Let
 $\sigma \in \{-1,0,1\}^{2 \times \ell}$ be an orientation compatible with $\alpha$, and $\mathcal{S}\in \{-1,0,1\}^{s \times \ell}$ be feasible with respect to $(P,\sigma)$. Then, for any $\rho \in\mC_{\mS}^{\sigma}$, the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution.
 \end{theorem}
\begin{proof}
As $\rho \in \mathcal{C}_{\mathcal{S}}^\sigma$, we have in particular that $\sign(A_{\rho}^\sigma) = \mS$ and $\rho \notin \mathcal{D}_{\mS}^\sigma$. Hence by \cref{mainlemma}, there exist $\bar{\mu}\in \R^\ell_{>0}$ and $Q \in \mathbb{R}^{s \times \ell}$ with $\sign(Q) = \mathcal{S}$ such that $Q \bar{\mu} = 0$, $P^\sigma\bar{\mu} > 0$, and
\[
 \sum_{j \in \Lambda_i^{\neq}}\frac{Q_{ij} \, \bar{\mu}_j}{e^{\rho_i} - \frac{\sigma_{2j}}{\sigma_{1j}}\,e^{\rho_{s + j}}} + \sum_{j \notin \Lambda_i}P^{\sigma}_{ij} \bar{\mu}_j > 0
\]
 for all $i \in [s]$. We will now modify $\bar{\mu}$ to obtain $\mu\in \R^\ell_{>0}$ satisfying $A_\rho^\sigma \mu=0$ and $P^\sigma\mu>0$.

Then, $F$ admits multiple positive zeros for some parameter values $\k\in \R^{\bar{m}}_{>0}$, $b\in \R^{n-\bs}$ if and only if there exist an orientation $\sigma \in \{-1,0,1\}^{2 \times \ell}$ compatible with $\alpha$ and a sign matrix $\mathcal{S}\in \{-1,0,1\}^{s \times \ell}$ that is feasible with respect to $(P,\sigma)$ such that
$ \mathcal{C}_{\mS}^\sigma \neq \varnothing$.
\end{theorem}
\begin{proof}
If $F$ admits multiple positive zeros, \Cref{thm:mono} tells us that there exist $\sigma$ and $\mathcal{S}$ as in the statement such that
$ \mathcal{C}_{\mS}^\sigma \neq \varnothing$. Reciprocally, if $\mathcal{C}_{\mS}^\sigma \neq \varnothing$ for some $\sigma$ and $\mS$, then \Cref{thm:multiSimplified} tells us that the oriented characteristic system at any $\rho\in \mathcal{C}_{\mS}^\sigma$ has a solution.
By \Cref{thm:mainsimplification}, this implies that \Cref{thm:main1}(iii) holds and hence $F$ admits multiple positive zeros.
\end{proof}

\begin{example}
\label[example]{example:hybrid_histidine_kinase}
We illustrate \Cref{thm:full} with an augmented vertical system that arises from the study of the steady states of a reaction network named
\emph{hybrid histidine kinase}. This network is known to admit multiple positive zeros \cite{hhk, plos:identifying}. The stoichiometric matrix, the matrix of reactants, and a matrix defining the stoichiometric compatibility classes are:
{\small \[ N = \begin{pmatrix}
 -1 & 0 & 0 & 1 & 0 & 0 \\
 1 & -1 & 0 & 0 & 1 & 0 \\
 0 & 1 & -1 & -1 & 0 & 0 \\
 0 & 0 & 1 & 0 & -1 & 0 \\
 0 & 0 & 0 & -1 & -1 & 1 \\
 0 & 0 & 0 & 1 & 1 & -1 \\
\end{pmatrix}, \quad M = \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}, \quad L = \begin{pmatrix}
1 & 1 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 1\end{pmatrix}.\] }After performing Gaussian elimination to $N$, the steady states of the network are the zeros of the augmented vertical system
$F = (C(\kappa \circ x^M), Lx - b) \in \R[\k,b,x]^6$
with
 \[ C = \begin{pmatrix}
1 & 0 & 0 & 0 & 1 & -1 \\
0 & 1 & 0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 1 & 1 & -1 \end{pmatrix}\, .\]
The last two columns of $C$ give rise to the reduced matrix $\oP \in \R^{4 \times 2}$ from \eqref{eqP}. In this example we have $\bs=4, \bar{m}=6$. We consider the row and column partitions to be $[\bs] = \{1,4\} \sqcup \{2\} \sqcup \{3\}$ and $[\bl] = \{1\} \sqcup \{2\}$, so $s = 3,m=5$. We choose the orientation $\sigma_+$. This gives rise to the following simplified reduced matrix $P$, oriented characteristic matrix $A^{\sigma_+}_\rho$ for $\rho\in \R^5$, and associated graph $G_{P}$:

It turns out that  the feasibility of the oriented characteristic system depends  on the signs of \( A^{\sigma}_{\rho} \). Then, an important observation is that  the strictly increasing nature of the exponential function, implies that the sign of each entry of   \( A^{\sigma}_{\rho} \) is either fixed (if $\sigma_{1k}\sigma_{2k} = -1$) or depends solely on the sign of \( P_{ik}(\sigma_{1k}\rho_i - \sigma_{2k}\rho_{s+k}) \), and can be expressed as a linear inequality in the entries of $\rho$.
Formally,
we first derive simple conditions on the signs of the entries of \( A^{\sigma}_{\rho} \) that preclude the oriented characteristic system from having solutions. Any sign matrix $\mathcal{S} \in \{-1,0,1\}^{s \times \ell}$ not satisfying the conditions is called feasible.
Then, for any $\sigma\in \mathbb{R}^{2\times \ell}$ and feasible $\mathcal{S} \in \{-1,0,1\}^{s \times \ell}$, we introduce a set  $\mathcal{C}_{\mathcal{S}}^\sigma$, which we name the feasible ground set, and for which  deciding whether it is nonempty is equivalent to checking the feasibility of a linear system of equalities and inequalities.

\begin{thmb}[\cref{thm:mono}]
 If $\mathcal{C}_{\mathcal{S}}^\sigma = \varnothing$ for all orientations $\sigma \in \{-1,0,1\}^{2 \times \ell}$ and feasible $\mathcal{S}\in \{-1,0,1\}^{s\times \ell}$, then $(C(\kappa \circ x^M), Lx - b)$ does not admit multiple positive zeros.
\end{thmb}

The converse of  Theorem B holds when   $P$ has a particular structure of zero entries. Namely, we consider the bipartite graph  whose nodes are the sets of rows and columns of $P$, and edges correspond to nonzero entries of $P$. If  the graph is a forest, we say that $P$ \bfemph{induces a forest} (see \cref{forest}).

\begin{thmc}[\cref{thm:full}]
Assume that  $P\in \mathbb{R}^{s\times \ell}$ induces a forest.
 Then the system $( C (\kappa \circ x^M), Lx-b)$ admits multiple positive zeros if and only if there exist an orientation $\sigma \in \{-1,0,1\}^{2 \times \ell}$ and  feasible $\mathcal{S}\in \{-1,0,1\}^{s \times \ell}$  such that
$ \mathcal{C}_{\mathcal{S}}^\sigma \neq \varnothing$.
\end{thmc}

We conclude this introduction by briefly highlighting two applications where augmented vertically parametrized systems arise. As we have already hinted at, they arise when studying
 \bfemph{reaction networks}, which are given by a collection of
 reactions
\[ \sum_{i = 1}^n\alpha_{ij}X_i \ce{->[\k_j]} \sum_{i = 1}^n\beta_{ij}X_i, \qquad j=1,\dots,m \]
 where $\alpha_{ij},\beta_{ij}\in \Z_{\geq 0}$ and $\k_j\in \R_{>0}$ is called the reaction rate constant.
The net production of the species along each reaction is encoded by the  stoichiometric matrix $N=(\beta_{ij} - \alpha_{ij})_ \in\mathbb{Z}^{n\times m}$ and the reactants define the matrix $M  = (\alpha_{ij}) \in \mathbb{Z}^{n \times m}$.
Under the assumption  of mass-action kinetics, the evolution of the concentration of each species in time is  modelled by a system of ordinary differential equations of the form
\[ \frac{d  x }{dt} =  N \big( \kappa \circ x^M \big)\, .\]
Moreover, the trajectories are constrained in the stoichiometric compatibility classes, whose defining equations are of the form $Lx-b=0$ where $Lx=0$ are equations of $\im(N)$ and $b$ depends on the initial condition of the system.
By letting $C$ be a full rank matrix with $\ker(N)=\ker(C)$, the positive steady states  within stoichiometric compatibility classes are the positive zeros of the
augmented vertically parametrized  system $(C(\kappa\circ x^M),  Lx-b)$.

Admitting more than one positive steady state is  necessary  for the network to display multistability, which has been linked to several properties of biological systems \cite{laurentkellershohn, ozbudak}. Thus, deciding upon the existence of multiple positive steady states for some parameter choice   has been an active topic of research in the field. To name a few of the many references, see for example \cite{BanajiPantea2016injectivity,craciun-feinbergI,Conradi_2019, dickenstein_giaroli_perez,feinberg-def0,Feliu_2015, signconditions, wiuf2013power} and the references therein.

The second application we would like to highlight arises when studying \bfemph{singular points of hypersurfaces}. Namely, let $f \in \mathbb{R}[x]$ be an $n$-variate polynomial with a prescribed support  $\mathcal{A} \subset \mathbb{Z}^n$   and signs $\varepsilon\colon \mathcal{A} \mapsto \{-,+\}$:
\[ f = \sum_{a \in \mathcal{A}}\varepsilon(a)\kappa_{a}x^{a} \, , \qquad \kappa \in \mathbb{R}_{>0}^{\mathcal{A}} \, .\]
The singular positive zeros of $f$
are the points of $\mathbb{R}^n_{>0}$ satisfying $f = x_1\frac{d}{dx_1} f = \dots = x_n\frac{d}{dx_n} f = 0$. By letting $A$ have as columns the elements of $\mathcal{A}$ in some order, and $\overline{A}$ be obtained by adding a top row of $1's$ to $A$ and multiplying the $i$-th column by $\epsilon(a_i)$,
the singular positive zeros of $f$ are precisely the positive zeros of the  vertically parametrized system
\[ \overline{A}( \kappa \circ x^A) \, .  \]
The existence of signed coefficients of $f$ giving rise to a polynomial with multiple singular positive  points depends on the combinatorial properties of the signed support and can be studied with the methods in this work. This question becomes relevant when studying $\mathcal{A}$-discriminants \cite{gkz1994}, the number of connected components of fewnomial hypersurfaces, e.g. \cite{bihan2024bounds, feliutelek, telekdescartesrule}, or copositivity of polynomials and SONC decompositions \cite{Ferrer:SONC}.

\subsection*{Structure of the Paper}
In \Cref{section:characterization}, we present the characterization of multiple positive zeros for augmented vertically parametrized polynomial systems via the characteristic system. \Cref{section:simplifications} introduces orientations, showing that the question of multiple zeros can be   reduced to the feasibility of the oriented characteristic system. In \Cref{section:feasibility}, we analyze the feasibility of oriented characteristic systems. First, we establish sufficient conditions for the existence of multiple positive zeros by reducing the feasibility of the characteristic system to checking whether the relative interiors over a family of polyhedral cones are empty. Secondly, we assume that the simplified reduced matrix $P$ induces a forest and characterize the existence of multiple positive zeros using the same family of polyhedral cones.

\subsection*{Acknowledgments}
This project has been funded by the European Union under the Grant Agreement number 101044561, POSALG. Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or European Research Council (ERC). We thank Joan Ferrer, Oskar Henriksson and Nidhi Kaihnsa for useful discussions.

\medskip
\noindent
\textbf{Notation. }
For two vectors $\alpha,\beta \in \mathbb{R}^{m}$,  $\alpha \circ \beta$  denotes its component-wise multiplication, i.e.
$(\alpha \circ \beta)_{i} = \alpha_i\beta_i$ for $i = 1,\dots,m.$
Similarly, the operations $\frac{\alpha}{\beta}, e^{\alpha}, \ln{\alpha}$ are taken component-wise.

For a  vector $v\in \mathbb{R}^n$, we let $\sign(v)\in \{-1,0,1\}^n$ be obtained by taking the sign entry-wise.
For a set $V\subseteq \mathbb{R}^n$, we let $\sign(V)=\{\sign(v) \colon v\in V\}$. For a vector $v\in \mathbb{R}^n$,   $v>0$ is shorthand notation for   $v\in \mathbb{R}^n_{>0}$.

For an integer $n$ we let $[n]:=\{1,\dots,n\}$. We  write $b \in [a,c]^\circ$ if $b$ is in the relative interior of the interval $[a,b]$, that is if either
\[ a < b < c\quad  \text{  or  } \quad a = b = c\, .\]

To analyze Theorem~\ref{thmfirst:intro}, the introduction explains that the feasibility of the oriented characteristic system “depends on the signs of \( A^{\sigma}_{\rho} \)”. Using that “the strictly increasing nature of the exponential function” fixes these signs, it states that each entry’s sign is either “fixed (if $\sigma_{1k}\sigma_{2k} = -1$) or depends solely on the sign of \( P_{ik}(\sigma_{1k}\rho_i - \sigma_{2k}\rho_{s+k}) \)”, and hence “can be expressed as a linear inequality in the entries of $\rho$.”

Formally, it sketches the following steps:
\begin{itemize}
\item “we first derive simple conditions on the signs of the entries of \( A^{\sigma}_{\rho} \) that preclude the oriented characteristic system from having solutions.”
\item Any sign matrix $\mathcal{S}\in\{-1,0,1\}^{s\times\ell}$ not excluded by these conditions is called “feasible”.
\item For any orientation $\sigma\in\mathbb{R}^{2\times\ell}$ and feasible $\mathcal{S}$, one “introduce[s] a set $\mathcal{C}_{\mathcal{S}}^\sigma$, which [is] name[d] the feasible ground set,” such that deciding nonemptiness is “equivalent to checking the feasibility of a linear system of equalities and inequalities.”
\end{itemize}
(Additionally, the text notes a sufficient nonexistence criterion via Theorem B and, under the extra assumption that $P$ “induces a forest,” an equivalence in terms of $\mathcal{C}_{\mathcal{S}}^\sigma\neq\varnothing$ in Theorem C.)

Biconditional or Equivalence
Existential–Universal

Question:
Consider an augmented vertically parametrized polynomial system
\[
F(x;\kappa,b)=\big(C(\kappa\circ x^M),\,Lx-b\big),
\]
where $C\in\mathbb{R}^{\bar s\times \bar m}$, $M\in\mathbb{Z}^{n\times \bar m}$, $L$ has $n-\bar s$ rows, $\kappa\in\mathbb{R}_{>0}^{\bar m}$, $b\in\mathbb{R}^{n-\bar s}$, and $x\in\mathbb{R}_{>0}^n$. Say that $F$ admits multiple positive zeros if there exist parameter values $\kappa\in\mathbb{R}_{>0}^{\bar m}$ and $b\in\mathbb{R}^{n-\bar s}$ and distinct points $x,y\in\mathbb{R}_{>0}^n$ such that $F(x;\kappa,b)=F(y;\kappa,b)=0$, equivalently,
\[
C(\kappa\circ x^M)=C(\kappa\circ y^M)=0,
\qquad Lx=Ly=b.
\]
Let $P\in\mathbb{R}^{s\times \ell}$ be the reduced matrix associated with the system, let $m=s+\ell$, and for an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and $\rho\in\mathbb{R}^m$ define
\[
(P^{\sigma})_{ik}=\sigma_{1k}P_{ik},
\qquad
(A^{\sigma}_{\rho})_{ik}=P_{ik}\big(\sigma_{1k}e^{\rho_i}-\sigma_{2k}e^{\rho_{s+k}}\big)
\quad (i\in[s],\ k\in[\ell]).
\]
The oriented characteristic system associated with $(P,\sigma,\rho)$ is
\[
A^{\sigma}_{\rho}\mu=0,\qquad P^{\sigma}\mu>0,\qquad \mu\in\mathbb{R}_{>0}^{\ell},
\]
and $\mathcal{G}^{\sigma}\subseteq\mathbb{R}^m$ is the oriented ground set associated with $\sigma$. Which statement holds?

The augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros if and only if there exist an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and a vector $\rho\in\mathcal{G}^{\sigma}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution $\mu\in\mathbb{R}_{>0}^{\ell}$.

Choices:
B. The augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros if and only if for every orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ there exists a vector $\rho\in\mathcal{G}^{\sigma}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution $\mu\in\mathbb{R}_{>0}^{\ell}$.
C. If there exist an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and a vector $\rho\in\mathcal{G}^{\sigma}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution $\mu\in\mathbb{R}_{>0}^{\ell}$, then the augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros.
D. The augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros if and only if there exist an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and a vector $\rho\in\mathbb{R}^{m}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution $\mu\in\mathbb{R}_{>0}^{\ell}$.
E. The augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros if and only if there exist an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and a vector $\rho\in\mathcal{G}^{\sigma}$ such that the sign pattern of $A^{\sigma}_{\rho}$ is feasible and $P^{\sigma}\mu>0$ for some $\mu\in\mathbb{R}_{>0}^{\ell}$.

dropped the converse direction of the iff statement

replaced solvability of $A^{\sigma}_{\rho}\mu=0$ by mere feasible sign-pattern data

The stem does not explicitly reveal choice A or state the exact equivalent condition outright. It sets up definitions and asks for the equivalent statement, so there is no direct answer leakage.

This is essentially a theorem-recall item: the correct option is a near-verbatim statement of the equivalence characterizing multiple positive zeros. It mainly asks the student to recognize the exact formal statement.

Some reasoning is needed to track subtle changes in quantifiers, positivity, and ground-set membership across the options, but the task is still primarily recognition of the canonical theorem statement rather than genuine derivation or synthesis.

The distractors are strong: they vary by plausible mathematical failure modes such as dropping a constraint, weakening strict inequalities, removing ground-set membership, or replacing existence by universality. They are distinct and nontrivial.

A technically well-designed recall question with high-quality distractors, but it is strongly theorem-statement recognition and therefore weak on tautology avoidance and only moderate on generative reasoning.


Question:
Consider an augmented vertically parametrized polynomial system
\[
F(x;\kappa,b)=\big(C(\kappa\circ x^M),\,Lx-b\big),
\]
where $C\in\mathbb{R}^{\bar s\times \bar m}$, $M\in\mathbb{Z}^{n\times \bar m}$, $L$ has $n-\bar s$ rows, $\kappa\in\mathbb{R}_{>0}^{\bar m}$, $b\in\mathbb{R}^{n-\bar s}$, and $x\in\mathbb{R}_{>0}^n$. Say that $F$ admits multiple positive zeros if there exist parameter values $\kappa\in\mathbb{R}_{>0}^{\bar m}$ and $b\in\mathbb{R}^{n-\bar s}$ and distinct points $x,y\in\mathbb{R}_{>0}^n$ such that $F(x;\kappa,b)=F(y;\kappa,b)=0$, equivalently,
\[
C(\kappa\circ x^M)=C(\kappa\circ y^M)=0,
\qquad Lx=Ly=b.
\]
Let $P\in\mathbb{R}^{s\times \ell}$ be the reduced matrix associated with the system, let $m=s+\ell$, and for an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and $\rho\in\mathbb{R}^m$ define
\[
(P^{\sigma})_{ik}=\sigma_{1k}P_{ik},
\qquad
(A^{\sigma}_{\rho})_{ik}=P_{ik}\big(\sigma_{1k}e^{\rho_i}-\sigma_{2k}e^{\rho_{s+k}}\big)
\quad (i\in[s],\ k\in[\ell]).
\]
The oriented characteristic system associated with $(P,\sigma,\rho)$ is
\[
A^{\sigma}_{\rho}\mu=0,\qquad P^{\sigma}\mu>0,\qquad \mu\in\mathbb{R}_{>0}^{\ell},
\]
and $\mathcal{G}^{\sigma}\subseteq\mathbb{R}^m$ is the oriented ground set associated with $\sigma$. Which statement holds?

Choices:
B. The augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros if and only if for every orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ there exists a vector $\rho\in\mathcal{G}^{\sigma}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution $\mu\in\mathbb{R}_{>0}^{\ell}$.
C. If there exist an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and a vector $\rho\in\mathcal{G}^{\sigma}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution $\mu\in\mathbb{R}_{>0}^{\ell}$, then the augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros.
D. The augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros if and only if there exist an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and a vector $\rho\in\mathbb{R}^{m}$ such that the oriented characteristic system associated with $(P,\sigma,\rho)$ has a solution $\mu\in\mathbb{R}_{>0}^{\ell}$.
E. The augmented vertically parametrized system $\big(C(\kappa\circ x^M),Lx-b\big)$ admits multiple positive zeros if and only if there exist an orientation $\sigma\in\{-1,0,1\}^{2\times \ell}$ and a vector $\rho\in\mathcal{G}^{\sigma}$ such that the sign pattern of $A^{\sigma}_{\rho}$ is feasible and $P^{\sigma}\mu>0$ for some $\mu\in\mathbb{R}_{>0}^{\ell}$.

The stem does not explicitly reveal the correct option. It sets up notation and definitions, but does not state the theorem-level equivalence itself.

This is essentially a direct theorem-identification item: the correct answer is the exact iff characterization, with distractors formed by small logical perturbations. It mainly tests recognition of the stated result rather than application.

Some reasoning is needed to compare existential vs universal quantifiers, iff vs one-way implication, and the role of the ground set condition. However, the item still leans more toward recall/recognition of a theorem statement than genuine generative problem solving.

The distractors are mathematically plausible and target common failure modes: quantifier strengthening (B), weaker true implication (C), dropping a necessary domain condition (D), and replacing exact solvability by weaker sign-feasibility data (E).

Technically well-constructed with strong distractors, but it is mostly a theorem-recall question rather than a non-tautological test of generative mathematical reasoning.