\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


Question:
Let $a,b,d>0$ with $a\ge 4$ and $d<b$. For $i=1,2$, define the nonlocal dispersal operators
\[
\mathcal N_i[u](z)=\int_{\mathbb R}J_i(y)u(z-y)\,dy-u(z),
\]
where $J_i$ are the dispersal kernels from the model, and set
\[
a^*=1-\frac1a,
\qquad
I_2(\lambda)=\int_{\mathbb R}J_2(y)e^{\lambda y}\,dy,
\qquad
s^*=\inf_{\lambda\in(0,\hat\lambda_2)}\frac{d\,[I_2(\lambda)-1]+b}{\lambda}.
\]
Consider positive profile pairs $(\phi,\psi)$, meaning $\phi(z)>0$ and $\psi(z)>0$ for all $z\in\mathbb R$, satisfying
\[
\begin{cases}
\mathcal N_1[\phi](z)+s\phi'(z)+a\phi(z)(1-\phi(z))-\psi(z)=0, & z\in\mathbb R,\\
 d\,\mathcal N_2[\psi](z)+s\psi'(z)+b\psi(z)\left(1-\dfrac{\psi(z)}{\phi(z)}\right)=0, & z\in\mathbb R,
\end{cases}
\]
with asymptotic boundary conditions
\[
(\phi,\psi)(\infty)=(1,0),\qquad (\phi,\psi)(-\infty)=(a^*,a^*).
\]
Which statement holds for the existence and nonexistence of such positive solutions as the wave speed $s$ varies?

For every $s>s^*$, there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions. The same existence conclusion also holds for $s=s^*$ if, in addition, $J_2$ has compact support. For every $s<s^*$, there is no positive solution of the system with those boundary conditions.

Choices:
B. For every $s\ge s^*$, there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions, without any additional assumption on $J_2$. For every $s<s^*$, there is no positive solution of the system with those boundary conditions.
C. For every $s>s^*$, there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions. For every $s<s^*$, there is no positive solution of the system with those boundary conditions.
D. There exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions for every $s>s^*$, and there is no positive solution for $s\le s^*$.
E. If $J_2$ has compact support, then there exists a positive solution $(\phi,\psi)$ of the above system with the stated boundary conditions for every $s\ge s^*$. Moreover, for every $s<s^*$ there is no positive solution only for some choice of the parameters $a,b,d$ with $a\ge 4$ and $d<b$.

compact-support requirement at the critical speed

exceptional critical case $s=s^*$ with compact support

universal quantifier in the nonexistence statement


[\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


Question: For the augmented vertically parametrized system \((C(\kappa\circ x^M),Lx-b)\), say that it **admits multiple positive zeros** if there exist parameter values \(\kappa\in \mathbb{R}_{>0}^{\bar m}\) and \(b\in \mathbb{R}^{n-\bar s}\) such that the system \(C(\kappa\circ x^M)=0\), \(Lx-b=0\) has two distinct solutions \(x,y\in \mathbb{R}_{>0}^n\). For an orientation \(\sigma\in\{-1,0,1\}^{2\times \ell}\), define \((P^{\sigma})_{ik}=\sigma_{1k}P_{ik}\) and, for \(\rho\in\mathbb{R}^m\) with \(m=s+\ell\), define \((A_{\rho}^{\sigma})_{ik}=P_{ik}(\sigma_{1k}e^{\rho_i}-\sigma_{2k}e^{\rho_{s+k}})\). Let \(\mathcal{G}^{\sigma}\subseteq \mathbb{R}^m\) be the oriented ground set associated with \(\sigma\). Which of the following statements is equivalent to \((C(\kappa\circ x^M),Lx-b)\) admitting multiple positive zeros?

There exist an orientation \(\sigma\in\{-1,0,1\}^{2\times \ell}\), a vector \(\rho\in\mathcal{G}^{\sigma}\), and a vector \(\mu\in\mathbb{R}_{>0}^{\ell}\) such that \(A_{\rho}^{\sigma}\mu=0\) and \(P^{\sigma}\mu>0\); equivalently, the oriented characteristic system associated with \((P,\sigma,\rho)\) has a solution.

Choices:
B. There exist an orientation \(\sigma\in\{-1,0,1\}^{2\times \ell}\), a vector \(\rho\in\mathbb{R}^m\), and a vector \(\mu\in\mathbb{R}_{>0}^{\ell}\) such that \(A_{\rho}^{\sigma}\mu=0\) and \(P^{\sigma}\mu>0\); equivalently, the oriented characteristic system associated with \((P,\sigma,\rho)\) has a solution.
C. There exist an orientation \(\sigma\in\{-1,0,1\}^{2\times \ell}\) and a vector \(\rho\in\mathcal{G}^{\sigma}\) such that there is some \(\mu\in\mathbb{R}_{>0}^{\ell}\) with \(A_{\rho}^{\sigma}\mu=0\).
D. For every orientation \(\sigma\in\{-1,0,1\}^{2\times \ell}\) there exist a vector \(\rho\in\mathcal{G}^{\sigma}\) and a vector \(\mu\in\mathbb{R}_{>0}^{\ell}\) such that \(A_{\rho}^{\sigma}\mu=0\) and \(P^{\sigma}\mu>0\); equivalently, each oriented characteristic system associated with \((P,\sigma,\rho)\) has a solution.
E. There exist an orientation \(\sigma\in\{-1,0,1\}^{2\times \ell}\), a vector \(\rho\in\mathcal{G}^{\sigma}\), and a vector \(\mu\in\mathbb{R}^{\ell}\) such that \(A_{\rho}^{\sigma}\mu=0\) and \(P^{\sigma}\mu\ge 0\); equivalently, the oriented characteristic system associated with \((P,\sigma,\rho)\) has a solution.


\begin{thm} \label{thm:main}
	Let $1 \le \ell \le k-1$ be such that $k \ge 3$ and $(k,\ell) \neq (3,1)$, let $t = \floor{\frac{k}{k-\ell}}(k - \ell)$, and let $n$ be sufficiently large and divisible by $k-\ell$.
	Then every $k$-uniform $n$-vertex hypergraph without isolated vertices and having minimum supported co-degree at least $(1 - 1/t)n-(k-3)$ will contain a Hamilton $\ell$-cycle.
\end{thm}

\label{sec:intro}
    A widespread research theme in extremal graph theory is to determine sufficient conditions that ensure the existence of specified spanning structures in graphs and hypergraphs. A classic result in this vein, due to Dirac~\cite{dirac1952some}, states that any graph on $n\geq 3$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle, and it is not too difficult to see that the degree condition is the best possible. Dirac's theorem has been generalised to various distinct settings over the years (see the surveys \cite{gould2014recent,kuhn2014hamilton,simonovits2020embedding,kuhn2012survey,frieze2019hamilton,rodl2010dirac}) and this has contributed to the development of several powerful techniques, such as regularity, absorption, and rotation-extension methods. In this paper, we will focus on hypergraph extensions of Dirac's theorem.

	A \emph{$k$-uniform hypergraph} or simply a \emph{$k$-graph} $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)$, where each edge consists of exactly $k$ vertices. The most common generalisation of a minimum degree condition from graphs to hypergraphs is accomplished via the notion of a \emph{minimum co-degree} -- which is the minimum $d$ such that every set of $k-1$ vertices in $G$ is contained in at least $d$ edges -- and we denote it here by $\delta(G)$. The larger uniformity $k$ allows for multiple distinct ways to define a cycle in a $k$-graph. The cyclic structures we search for are \emph{$\ell$-cycles}, which are perhaps the most common extensions of graph cycles. Intuitively, we think of an $\ell$-cycle as a spanning path formed by a cyclic set of edges such that each edge has exactly $\ell$ vertices in common with the preceding edge (it could intersect other edges as well). More formally, given an $n$-vertex $k$-graph $G$ and any $\ell \in [k-1]$ such that $k - \ell$ divides $n$, we define a \emph{Hamilton $\ell$-cycle} in $G$ to be an ordering of the vertices of $G$, say $v_1 \dots v_n$, such that the vertices of the subsequence $v_{i(k - \ell)+1} \dots v_{i(k - \ell) +k}$ form an edge for all $i \ge 0$, where we view the indices cyclically modulo $n$. Since each edge of an $\ell$-cycle contains $k-\ell$ vertices that were not in the previous edge, we trivially require that $k - \ell$ divides $n$ so that when we ``cycle around'' the vertices of $G$, the original sequence of edges gets repeated. We call $(k-1)$-cycles and $1$-cycles \emph{tight} and \emph{loose} cycles respectively.

The question of determining sufficient minimum co-degree conditions to ensure a tight Hamilton cycle was first raised by Katona and Kierstead~\cite{katona1999hamiltonian} who conjectured that if $\delta(G)\ge (n-k+2)/2$ then $G$ has a tight Hamilton cycle, and they showed that this bound is the best possible. R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2006dirac,rodl2008approximate} first proved this asymptotically for $k\geq 3$ and $\delta(G)\geq n/2+o(n)$, and later obtained an exact result for $k=3$~\cite{rodl2011dirac}. Additionally, Liu and Liu \cite{liu2022hamiltonian} made progress towards an exact result for $k = 4$. In fact, for any $\ell \in [k-1]$ such that $k - \ell$ divides $k$, a tight Hamilton cycle will contain such a Hamilton $\ell$-cycle (provided the necessary divisibility condition holds). If we further suppose that $k$ divides $n$, then this Hamilton $\ell$-cycle will imply the existence of a perfect matching which necessitates $\delta(G) \ge n/2 - k$ as shown in \cite{kuhn2010hamilton,rodl2009perfect}, and hence the aforementioned tight cycle bound of $\delta(G) \ge n/2 + o(n)$ is tight. If $k$ does not divide $n$, then R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2009perfect} have determined the minimum codegree threshold for a near-perfect matching of size $\left\lfloor n/k \right\rfloor$ to be roughly $n/k$. It is therefore plausible that the correct minimum codegree threshold for a spanning $\ell$-cycle, when $k-\ell$ divides $k$ but is not $1$ and $n$ is not divisible by $k$, is significantly lower than $n/2$. Conversely, if $k-\ell$ does not divide $k$, a series of works by K\"uhn and Osthus~\cite{kuhn2006loose}, Keevash, K\"uhn, Mycroft and Osthus~\cite{keevash2011loose}, H\`an and Schacht~\cite{han2010dirac} and K\"uhn, Mycroft and Osthus~\cite{kuhn2010hamilton} have established that
\[
\delta(G)\geq \frac{n}{\ceil{\frac{k}{k-\ell}}(k-\ell)}+o(n)
\]
suffices and is optimal up to the $o(n)$ term, yielding a much lower threshold. Exact bounds, however, have proven notoriously difficult to obtain and are known only in a few special cases. To the best of our knowledge, these are $(k,\ell)=(3,2)$ by R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2011dirac}, $(k,\ell)=(3,1)$ by Czygrinow and Molla~\cite{czygrinow2014tight}, $(k,\ell)=(4,2)$ by Garbe and Mycroft~\cite{garbe2018hamilton}, and $k\ge 3$ and $\ell<k/2$ by Han and Zhao~\cite{han2015minimum}. We refer the reader to the surveys~\cite{kuhn2014hamilton,zhao2016recent,rodl2010dirac} for more detailed discussions.

A drawback of a minimum co-degree condition is that it tends to be a fairly strong requirement. For instance, if we start with a complete $k$-graph and remove all edges that contain a fixed pair of vertices, it is immediate that the resulting hypergraph has minimum co-degree equal to zero. However, if $k\ge 3$, it is not hard to see that even fairly small hypergraphs of this form are quite dense, and trivially contain Hamilton cycles. In fact, there are several natural hypergraph classes that are quite dense, but have minimum co-degree equal to zero (multipartite hypergraphs, for instance), and hence the previous theorems are not immediately applicable. This motivates the notion of the \emph{minimum supported co-degree} of a $k$-graph $G$, which we denote $\delta^*(G)$, and define as the maximum integer $d$ such that every $(k-1)$-set that is contained in at least one edge is contained in at least $d$ edges. A minimum supported co-degree requirement immediately handles the previous issues of zero co-degree sets, although it does not rule out isolated vertices, and hence we will usually impose the (fairly weak) requirement of all vertices being non-isolated.

Multiple recent studies have investigated questions that arise from replacing a minimum co-degree with a minimum supported co-degree requirement in classical extremal hypergraph theoretic problems. The notion of supported co-degrees was introduced by Balogh, Lemons and Palmer~\cite{balogh2021maximum} (they use the term ``positive co-degree'' instead) to formulate a generalisation of the Erd\H{o}s-Ko-Rado theorem, providing bounds on the sizes of intersecting families satisfying a minimum supported co-degree condition. Subsequently, Frankl and Wang~\cite{frankl2025intersecting} improved these bounds in almost all cases, and Spiro~\cite{spiro2023t} has extended these studies to the more general case of $t$-intersecting families. Other papers have worked on generalisations of the Andr\'asfai-Erd\H{o}s-S\'os theorem~\cite{liu2024positive} and unique colourability of hypergraphs~\cite{liu2024uniquely}.

Halfpap, Lemons and Palmer~\cite{halfpap2025positive} investigated variants of hypergraph Tur\'an problems, and established the asymptotic supported co-degree threshold for a host $3$-graph to contain a copy of a fixed graph $F$ for several distinct $3$-graphs $F$, and Wu~\cite{wu2025positive} has consequently addressed this question for other examples. Halfpap, Lemons and Palmer~\cite{halfpap2025positive} also study ``jumps'' in this threshold, which has further been tackled by Balogh, Halfpap, Lidick\'y, and Palmer~\cite{balogh2024positive}. Pikhurko~\cite{pikhurko2023limit} recently proved that these problems are well-defined for a generalisation to the ``minimum positive/supported $\ell$-co-degree'' for any $\ell \in [k-1]$, providing analogous results to those Lo and Markstr\"om~\cite{lo2014degree} regarding the usual notion of $\ell$-co-degree.

The question of finding minimum supported co-degree conditions that ensure the existence of spanning structures in hypergraphs was first raised by Halfpap and Magnan~\cite{halfpap2024positive}. They prove an exact optimal lower bound on the minimum supported co-degree required to guarantee a perfect matching in $3$-graphs and a slightly weaker bound for all higher uniformities, which was later improved to an exact tight bound by Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025matching}. Furthermore, Halfpap and Magnan also establish an exact best-possible minimum supported co-degree condition for Hamilton Berge cycles and an asymptotic one for $3$-uniform loose cycles. In a different direction, Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} showed that an $n$-vertex $k$-graph with $\delta^*(G)\ge n/2 + o(n)$ contains a spanning $k$-sphere, asymptotically confirming a conjecture of Georgakopoulos, Haslegrave, Montgomery and Narayanan~\cite{georgakopoulos2022spanning}.

The problem of determining the optimal minimum supported co-degree for $k$-uniform $\ell$-cycles was initially suggested by Halfpap and Magnan in~\cite[Section 5]{halfpap2024positive}, and reiterated by Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} for the special case of tight cycles. This question was recently tackled by Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025positive}, who prove an asymptotically optimal result for all $k\ge 3$ and $\ell\in [k-1]$.

Our main result provides an improved sufficient minimum supported co-degree condition for $k$-uniform $\ell$-cycles for all $k\ge 3$ and $\ell\in [k-1]$ except $(k,\ell)=(3,1)$, which is the best possible for infinitely many values of $k$ and $\ell$, and is off by at most $1$ for all values of $k$ and $\ell$. We discuss the optimality of our result in \Cref{sec:lower bound}. Importantly, we emphasise that our results are (essentially) exact and not asymptotic, a rather stark difference from the minimum co-degree version of the problem where very few cases have been resolved exactly.

\label{sec:intro}
 A widespread research theme in extremal graph theory is to determine sufficient conditions that ensure the existence of specified spanning structures in graphs and hypergraphs. A classic result in this vein, due to Dirac~\cite{dirac1952some}, states that any graph on $n\geq 3$ vertices with minimum degree at least $n/2$ contains a Hamilton cycle, and it is not too difficult to see that the degree condition is the best possible. Dirac's theorem has been generalised to various distinct settings over the years (see the surveys \cite{gould2014recent,kuhn2014hamilton,simonovits2020embedding,kuhn2012survey,frieze2019hamilton,rodl2010dirac}) and this has contributed to the development of several powerful techniques, such as regularity, absorption, and rotation-extension methods. In this paper, we will focus on hypergraph extensions of Dirac's theorem.

The question of determining sufficient minimum co-degree conditions to ensure a tight Hamilton cycle was first raised by Katona and Kierstead~\cite{katona1999hamiltonian} who conjectured that if $\delta(G)\ge (n-k+2)/2$ then $G$ has a tight Hamilton cycle, and they showed that this bound is the best possible. R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2006dirac,rodl2008approximate} first proved this asymptotically for $k\geq 3$ and $\delta(G)\geq n/2+o(n)$, and later obtained an exact result for $k=3$~\cite{rodl2011dirac}. Additionally, Liu and Liu \cite{liu2022hamiltonian} made progress towards an exact result for $k = 4$. In fact, for any $\ell \in [k-1]$ such that $k - \ell$ divides $k$, a tight Hamilton cycle will contain such a Hamilton $\ell$-cycle (provided the necessary divisibility condition holds). If we further suppose that $k$ divides $n$, then this Hamilton $\ell$-cycle will imply the existence of a perfect matching which necessitates $\delta(G) \ge n/2 - k$ as shown in \cite{kuhn2010hamilton,rodl2009perfect}, and hence the aforementioned tight cycle bound of $\delta(G) \ge n/2 + o(n)$ is tight. If $k$ does not divide $n$, then R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2009perfect} have determined the minimum codegree threshold for a near-perfect matching of size $\left\lfloor n/k \right\rfloor$ to be roughly $n/k$. It is therefore plausible that the correct minimum codegree threshold for a spanning $\ell$-cycle, when $k-\ell$ divides $k$ but is not $1$ and $n$ is not divisible by $k$, is significantly lower than $n/2$. Conversely, if $k-\ell$ does not divide $k$, a series of works by K\"uhn and Osthus~\cite{kuhn2006loose}, Keevash, K\"uhn, Mycroft and Osthus~\cite{keevash2011loose}, H\`an and Schacht~\cite{han2010dirac} and K\"uhn, Mycroft and Osthus~\cite{kuhn2010hamilton} have established that
\[
\delta(G)\geq \frac{n}{\ceil{\frac{k}{k-\ell}}(k-\ell)}+o(n)
\]
suffices and is optimal up to the $o(n)$ term, yielding a much lower threshold. Exact bounds, however, have proven notoriously difficult to obtain and are known only in a few special cases. To the best of our knowledge, these are $(k,\ell)=(3,2)$ by R\"{o}dl, Ruci\'nski and Szemer\'edi~\cite{rodl2011dirac}, $(k,\ell)=(3,1)$ by Czygrinow and Molla~\cite{czygrinow2014tight}, $(k,\ell)=(4,2)$ by Garbe and Mycroft~\cite{garbe2018hamilton}, and $k\ge 3$ and $\ell<k/2$ by Han and Zhao~\cite{han2015minimum}. We refer the reader to the surveys~\cite{kuhn2014hamilton,zhao2016recent,rodl2010dirac} for more detailed discussions.

A drawback of a minimum co-degree condition is that it tends to be a fairly strong requirement. For instance, if we start with a complete $k$-graph and remove all edges that contain a fixed pair of vertices, it is immediate that the resulting hypergraph has minimum co-degree equal to zero. However, if $k\ge 3$, it is not hard to see that even fairly small hypergraphs of this form are quite dense, and trivially contain Hamilton cycles. In fact, there are several natural hypergraph classes that are quite dense, but have minimum co-degree equal to zero (multipartite hypergraphs, for instance), and hence the previous theorems are not immediately applicable. This motivates the notion of the \emph{minimum supported co-degree} of a $k$-graph $G$, which we denote $\delta^*(G)$, and define as the maximum integer $d$ such that every $(k-1)$-set that is contained in at least one edge is contained in at least $d$ edges. A minimum supported co-degree requirement immediately handles the previous issues of zero co-degree sets, although it does not rule out isolated vertices, and hence we will usually impose the (fairly weak) requirement of all vertices being non-isolated.

The problem of determining the optimal minimum supported co-degree for $k$-uniform $\ell$-cycles was initially suggested by Halfpap and Magnan in~\cite[Section 5]{halfpap2024positive}, and reiterated by Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} for the special case of tight cycles. This question was recently tackled by Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025positive}, who prove an asymptotically optimal result for all $k\ge 3$ and $\ell\in [k-1]$.

Our main result provides an improved sufficient minimum supported co-degree condition for $k$-uniform $\ell$-cycles for all $k\ge 3$ and $\ell\in [k-1]$ except $(k,\ell)=(3,1)$, which is the best possible for infinitely many values of $k$ and $\ell$, and is off by at most $1$ for all values of $k$ and $\ell$. We discuss the optimality of our result in \Cref{sec:lower bound}. Importantly, we emphasise that our results are (essentially) exact and not asymptotic, a rather stark difference from the minimum co-degree version of the problem where very few cases have been resolved exactly.

\begin{ex}\label{ex:lower bd weak}
 Let $k\ge 3$, $\ell \in [k-1]$ and $t = \floor{\frac{k}{k-\ell}}(k - \ell)$. Let $n$ be chosen such that $k-\ell$ divides $n$. Now, consider an $n$-vertex $k$-graph $G$ with a partition $A\sqcup B$ of its vertex set such that $|A| = \floor{n/t} + 1$ and $E(G)$ consists of all $k$-sets $e \in \binom{V(G)}{k}$ that satisfy $|e \cap A| \le 1$ (and so $A$ is a strong independent set). Suppose $S$ is any supported $(k-1)$-set. If $|S\cap A| = 0$, then $S$ supports all vertices outside itself, and so $d_G^1(S) = n-k+1$. If $|S\cap A| = 1$, then it supports all vertices in $B\setminus S$, meaning that $d_G^1(S) = n - \floor{n/t} - (k-2)$, and this is equal to $\delta^*(G)$ since there is no supported set with $|S\cap A| \ge 2$. The following observation shows that $G$ cannot contains a Hamilton $\ell$-cycle.
 \end{ex}

\begin{ex}\label{ex:lower bd strong}
 Next, we provide a specialised modification of the previous example. Let $k\ge 3$ be odd, $\ell = (k-1)/2$, and consequently $t = \floor{\frac{k}{k-\ell}}(k - \ell) = (k+1)/2 = \ell+1 = k - \ell$. Let $n$ be such that $t$ divides $n$ and $n/t+1$ is even. Yet again, we consider an $n$-vertex $k$-graph $G$ with a partition $A\sqcup B$ of its vertex set with $|A| = n/t + 1$. Write $A = \{a_1, \dots, a_{|A|}\}$ and let $E(G)$ consist of all $e\in \binom{V(G)}{k}$ such that $e\cap A \subseteq \{a_{2i-1}, a_{2i}\}$ for some $i\in [|A|/2]$ (so the graph $\partial^2[A]$ is a perfect matching). Clearly $|e\cap A| \le 2$ for all $e \in E(G)$. If $S$ is any supported $(k-1)$-set, then, similar to the previous example, it is not too hard to see that
 \[d_G^1(S) =
 \left\{
 \begin{array}{ll}
 n - (k-1) & \textnormal{if} \ |e\cap A|=0,\\
 |B| - (k - 2) +1 = |B| - (k-3) & \textnormal{if} \ |e\cap A|=1,\\
 |B| - (k - 3) & \textnormal{if} \ |e\cap A|=2,\\
 \end{array}
 \right.
 \]
 and so $\delta^*(G) = |B| - (k - 3) = n - n/t - (k - 2)$. We now argue that $G$ cannot contain a Hamilton $\ell$-cycle, which shows that the bound in \cref{thm:main} cannot be improved.

\begin{restatable}{thm}{thmHamiltonCycleNonExtremal}
 \label{thm:Hamilton cycle non-extremal}
 Let $k \ge 3$ and $\ell \in [k-1]$ be such that $(k,\ell) \neq (3,1)$, and set $t = \floor{\frac{k}{k-\ell}}(k-\ell)$. Let $1/n \ll \eps \ll \mu \ll 1/k$ such that $n$ is divisible by $k-\ell$. If $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\delta^*(G)\geq (1-1/t-\eps)n$ such that every set of $\floor{n/t}$ vertices of $G$ contains at least $\mu n^2$ supported pairs, then $G$ has a Hamilton $\ell$-cycle.
 \end{restatable}

\begin{restatable}{thm}{thmHamiltonCycleExtremal}
 \label{thm:Hamilton cycle extremal}
 Let $k \ge 3$ and $\ell \in [k-1]$ be such that $(k,\ell) \neq (3,1)$, and set $t = \floor{\frac{k}{k-\ell}}(k-\ell)$.
 Let $1/n\ll \eps \ll 1/k\leq 1/3$ be such that $n$ is divisible by $k-\ell$.
 Suppose $G$ is an $n$-vertex $k$-graph with no isolated vertices and $\delta^*(G)\geq (1-1/t)n-(k-3)$. If there is a subset $A \subseteq V(G)$ with $|A|=\floor{n/t}$ that contains at most $\eps n^2$ supported pairs, then $G$ has a Hamilton $\ell$-cycle.
 \end{restatable}

\begin{prop} \label{prop:supported-l-paths}
 Let $G$ be a $k$-graph, where $k \ge 3$, $\ell \in [k-1]$ and $t = \floor{\frac{k}{k-\ell}}(k-\ell)$.
 \begin{enumerate}[label = \rm(\roman*)]
 \item \label{itm:extended-path-1}
 Let $P$ and $Q$ be two sequences of vertices in $G$ of length at least $t$, both of which support an $\ell$-path, and whose vertices are disjoint. If the sequence consisting of the last $t$ vertices in $P$ followed by the first $t-1$ vertices in $Q$ is a tight path, then $PQ$ supports an $\ell$-path.
 \item \label{itm:extended-path-2}
 Let $P_1, \ldots, P_m$ be sequences of vertices in $G$ of length at least $t$, each of which supports an $\ell$-path, and whose vertices are pairwise disjoint. If $P_iP_{i+1}$ supports an $\ell$-path for every $i \in [m]$ (addition of indices taken modulo $m$) then $P_1 \ldots P_m$ is an $\ell$-cycle.
 \item \label{itm:extended-path-3}
 Let $P = v_0 \ldots v_r$ be a sequence of distinct vertices where $r$ is divisible by $t$. If every subsequence of at most $k$ consecutive vertices that contains at most one vertex $v_i$ with $i$ divisible by $t$ is supported by an edge, then $P$ supports an extended $\ell$-path.
 \item \label{itm:extended-path-4}
 Let $P$ be a sequence of distinct vertices of length $r$ such that $r = -1 \pmod{k-\ell}$ and $r \ge t-1$, and let $u,v$ be two vertices that do not appear in $P$, such that $uP$ and $Pv$ are tight paths (or supported sets if $r+1 \le k$). Then $uPv$ supports an extended $\ell$-path.
 \end{enumerate}
 \end{prop}
 \begin{proof}
 For \ref{itm:extended-path-1}, say that the length of $P$ is $p$ and that of $Q$ is $q$. Let $PQ = v_1 \dots v_{p+q}$, where $P = v_1 \dots v_p$ and $Q = v_{p+1} \dots v_{p+q}$. Since $P$ and $Q$ both support $\ell$-paths, we know that $p$ and $q$ are divisible by $k-\ell$, implying the same is true of $p+q$. Furthermore, as $q\ge t$, we also know that the last $t$ vertices of $PQ$ correspond to the last $t$ vertices of $Q$, and hence are supported. All that is left to show is that $v_{i(k-\ell) + 1} \ldots v_{i(k-\ell)+k}$ is an edge for all $i\le \frac{p + q - t}{k-\ell} - 1$. Since $P$ and $Q$ support $\ell$-paths (and in particular their length is divisible by $k-\ell$), we only need to restrict our attention to $i \in [\frac{p - t}{k - \ell}, \frac{p}{k - \ell}-1]$. For this, it would suffice to argue that the subsequence $v_{i_1} \dots v_{i_2}$ forms a tight path, where $i_1 = p - t + 1$ and $i_2 = p + \ell$. By assumption, we know that $v_{p - t + 1} \dots v_{p+t-1}$ is a tight path, and we know $t-1\ge \ell$ (see \Cref{obs:useful t info}), which completes the proof.

These results contribute to the broader research theme of finding the best possible minimum supported co-degree to ensure a specified spanning substructure, an area which poses some tantalising questions. Our original motivation for working on this problem arose from Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite[Conjecture $1.4$]{illingworth2025spanning}, who conjectured that $\delta^*(G) \ge (1-1/k)n$ guarantees a tight Hamilton cycle. The main focus of their paper is the minimum supported co-degree required to ensure a spanning $(k-1)$-sphere -- a subset of edges in a $k$-graph that induces a homogeneous simplicial complex homeomorphic to the $(k-1)$-sphere $\mathbb{S}^{k-1}$, and whose $0$-skeleton spans the entire vertex set. They tackle the following conjecture of Georgakopoulos, Haslegrave, Montgomery and Narayanan~\cite{georgakopoulos2022spanning} (paraphrased).
 \begin{conj}\label{conj:spheres}
 Every $k$-graph $G$ of order $n>k \ge 2$ with $\delta^*(G) \ge n/2$ and with no isolated vertices contains a spanning copy of $\mathbb{S}^{k-1}$.\footnote{In the original phrasing of the conjecture $G$ is required to be tightly connected -- meaning that any two edges can be joined by a tight walk -- but this follows from the other assumptions.}
 \end{conj}
 The chief result in~\cite{georgakopoulos2022spanning} provides an asymptotically optimal minimum co-degree condition for a $3$-graph to contain a vertex-spanning copy of any \emph{surface}, meaning an arbitrary connected, closed $2$-manifold. In particular, this addresses the minimum co-degree necessary to guarantee a spanning $2$-sphere in a $3$-graph.
 \begin{thm}[{\cite[Theorem 1.4]{georgakopoulos2022spanning}}]\label{thm:spheres-codegree}
 Let $\cS$ be an arbitrary surface and $\eps>0$. Then any sufficiently large $n$-vertex $3$-graph with $\delta(G) \ge n/3 + \eps n$ contains a spanning copy of $\cS$. Moreover, for any $n \in \mathbb{N}$, there exists an $n$-vertex $3$-graph $H$ with $\delta(H) = \floor{\frac{n}{3}} - 1$ such that there are at most $2\ceil{\frac{n}{3}}$ vertices in the $0$-skeleton of a copy of any surface in $H$.
 \end{thm}

This improves the aforementioned asymptotic results of Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025positive}, who showed that $\delta^*(G) \ge (1 - 1/t)n + o(n)$ suffices. Our methods are largely different from theirs. Our analysis splits into two regimes: the extremal and non-extremal (see \Cref{sec:stability} for details). The extremal regime does not feature in \cite{mycroft2025positive} at all, and for this we use ad-hoc structural analysis with lemmas about random matchings in bipartite graphs and about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs. For the non-extremal case, while Mycroft and Z\'arate-Guer\'en rely on a version of the regularity lemma for hypergraphs (called the weak regularity lemma) along with the absorption method, our approach stems from Lang's breakthrough work \cite{lang2023tiling} on hypergraph tilings, which allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method. For the latter part, however, we use a novel idea from \cite{mycroft2025positive} about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\ell$-cycles in certain $k$-partite $k$-graphs, which is one of the steps in our proof.

We point out that Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia~\cite{illingworth2025spanning} conjectured that for all $k\ge 3$, any sufficiently large $k$-graph with $\delta^*(G) \ge (1-1/k)n$ contains a tight Hamilton cycle, which is a special case of \cref{thm:main} (up to the $k-3$ term). Observe that if $k$ divides $n$, then a tight Hamilton cycle in $G$ will contain a perfect matching. Thus, our result implies that $\delta^*(G) \ge (1 - 1/k)n - (k-3)$ ensures a perfect matching, which recovers a result of Mycroft and Z\'arate-Guer\'en~\cite{mycroft2025matching} up to an additive constant of one, who show that a bound of $(1 - 1/k)n - (k - 2)$ suffices.

As alluded to above, the bound in \cref{thm:main} is off by one in most cases. In fact, in many cases (namely whenever $t \ge \ell+2$ or when $\ell = k-1$, that is, we are seeking a tight Hamilton cycle), our techniques can be adapted to prove \cref{thm:main} with the improved bound of $\delta^*(G) \ge (1 - 1/t)n - (k - 2)$ instead (which, for instance, would immediately imply the previously discussed optimal perfect matching bound).
However, since this will require some technical modifications, we choose to present a single unified proof for $\delta^*(G) \ge (1 - 1/t)n - (k - 3)$ that will work for all cases instead. We point out the part of our proof that requires the exact degree condition in \Cref{rem:exact degree}, and briefly discuss how to suitably alter our proof and remove the extra one in the bound in all relevant cases in \Cref{sec:improving the bound}.

Finally, we remark that in a recent personal communication, Mycroft and Z\'arate-Guer\'en informed us that they are preparing a manuscript where they prove that any sufficiently large $3$-graph with no isolated vertices and $\delta^*(G) \ge n/2$ contains a loose Hamilton cycle. This improves the previous asymptotic bound of $n/2 + o(n)$ established by them~\cite{mycroft2025positive} and Halfpap and Magnan~\cite{halfpap2024positive} (which our methods can recover as well) and extends \cref{thm:main} to include $(k,\ell) = (3,1)$, the only case we do not handle.

\paragraph{Organisation of the paper.} In \Cref{sec:overview}, we first provide lower bound constructions to prove the optimality of \cref{thm:main}. We then split the proof of \cref{thm:main} into two complementary cases and handle them separately in \cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}. We also provide brief proof overviews for both these theorems in \Cref{sec:proof-overviews}. Then in \Cref{sec:prelims} we introduce some notation and definitions and provide some basic tools.

\Crefrange{sec:non-extremal}{sec:proof Hamilton paths} are dedicated to proving \cref{thm:Hamilton cycle non-extremal}, and \Crefrange{sec:extremal hypergraphs}{sec:Hamilton cycle extremal} prove \cref{thm:Hamilton cycle extremal}. These two parts are treated largely independently, and can be read as such. We conclude with some open problems in \Cref{sec:conclusion}.

The proof of \cref{thm:main} is split into two regimes: the \emph{extremal} and \emph{non-extremal} cases (see \Cref{sec:stability}). The two cases are then handled separately in \cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}.

In the extremal regime, the authors use “ad-hoc structural analysis” together with “lemmas about random matchings in bipartite graphs” and lemmas “about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs.”

In the non-extremal regime, the approach “stems from Lang's breakthrough work \cite{lang2023tiling} on hypergraph tilings,” which “allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method.” In addition, they use “a novel idea … about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\ell$-cycles in certain $k$-partite $k$-graphs,” described as “one of the steps in our proof.”

They also note that the degree bound is “off by one in most cases,” and indicate they “point out the part of our proof that requires the exact degree condition in \Cref{rem:exact degree},” and “briefly discuss how to suitably alter our proof and remove the extra one … in \Cref{sec:improving the bound}.”

The proof of the main theorem is split into two regimes: the \emph{extremal} and \emph{non-extremal} cases (see \Cref{sec:stability}). The two cases are then handled separately in \cref{thm:Hamilton cycle non-extremal,thm:Hamilton cycle extremal}.

In the extremal regime, the authors use “ad-hoc structural analysis” together with “lemmas about random matchings in bipartite graphs” and lemmas “about finding almost spanning subhypergraphs with large minimum supported codegree in almost complete hypergraphs.”

In the non-extremal regime, the approach “stems from Lang's breakthrough work Lang, \emph{tiling} (2023) on hypergraph tilings,” which “allows us to completely avoid the regularity lemma and greatly simplifies the use of the absorption method.” In addition, they use “a novel idea … about weighted fractional matchings that, coupled with Farkas' linear-algebraic lemma, allows one to find spanning $\ell$-cycles in certain $k$-partite $k$-graphs,” described as “one of the steps in our proof.”

They also note that the degree bound is “off by one in most cases,” and indicate they “point out the part of our proof that requires the exact degree condition in the following remark. \begin{rem}\label{rem:exact degree}
    A critical point to note is that this is the only part of the entire proof that utilises the exact bound $\delta^*(G) \ge n - \floor{\frac{n}{t}} - (k - 3)$ via \cref{obs:min 1-degree,lem:disjoint edges cherries}. As mentioned in \Cref{sec:intro}, we can prove \cref{thm:Hamilton cycle extremal} and consequently \cref{thm:main} with the weaker bound of $\delta^*(G) \ge n - \floor{\frac{n}{t}} - (k - 2)$ instead, for most values of $k$ and $\ell$ (namely when $t \ge \ell+2$ or when $\ell = k-1$) which is achieved by modifying \cref{lem:disjoint edges cherries} to potentially allow for an additional cherry or some edges. We discuss how to suitably adapt the rest of the proof in \Cref{sec:improving the bound}.
\end{rem}” and “briefly discuss how to suitably alter our proof and remove the extra one … in \Cref{sec:improving the bound}.”

\label{thm:main}
	Let $1 \le \ell \le k-1$ be such that $k \ge 3$ and $(k,\ell) \neq (3,1)$, let $t = \floor{\frac{k}{k-\ell}}(k - \ell)$, and let $n$ be sufficiently large and divisible by $k-\ell$.
	Then every $k$-uniform $n$-vertex hypergraph without isolated vertices and having minimum supported co-degree at least $(1 - 1/t)n-(k-3)$ will contain a Hamilton $\ell$-cycle.,

Question:
Let $k \ge 3$ and $1 \le \ell \le k-1$ with $(k,\ell) \ne (3,1)$, and define
\[
t=\left\lfloor \frac{k}{k-\ell}\right\rfloor (k-\ell).
\]
Assume $n$ is sufficiently large and divisible by $k-\ell$. Let $G$ be a $k$-uniform hypergraph on $n$ vertices with no isolated vertices. Its minimum supported co-degree $\delta^*(G)$ is the largest integer $d$ such that every $(k-1)$-subset of vertices that is contained in at least one edge of $G$ is contained in at least $d$ edges of $G$. A Hamilton $\ell$-cycle means a spanning $k$-uniform $\ell$-cycle, i.e. a cyclic ordering of all vertices in which every edge consists of $k$ consecutive vertices and consecutive edges overlap in exactly $\ell$ vertices. Under these assumptions, which existence statement holds?

If $\delta^*(G) \ge \left(1-\frac{1}{t}\right)n-(k-3)$, then $G$ contains a Hamilton $\ell$-cycle.

Choices:
B. If $\delta^*(G) \ge \left(1-\frac{1}{t}\right)n-(k-2)$, then $G$ contains a Hamilton $\ell$-cycle.
C. If $\delta^*(G) \ge \left(1-\frac{1}{t}\right)n-(k-3)$, then $G$ contains an $\ell$-cycle.
D. If for every sufficiently large $n$ divisible by $k-\ell$ there exists a constant $c=c(k,\ell)$ such that every $k$-uniform $n$-vertex hypergraph $G$ without isolated vertices and with $\delta^*(G) \ge \left(1-\frac{1}{t}\right)n-c$ satisfies that $G$ contains a Hamilton $\ell$-cycle, then one may take $c=k-3$ for all such $G$.
E. If $\delta^*(G) > \left(1-\frac{1}{t}\right)n-(k-3)$, then $G$ contains a tight Hamilton cycle.

spanning requirement in Hamilton $\ell$-cycle

quantifier order for the degree slack constant

$\ell$-cycle versus tight Hamilton cycle