\label{THM:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\omega$ given in \eqref{random_data}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1 (\mathbb{T})} > z_0\varepsilon^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noindent
holds if and only if $(c_k) \in \ell^1$.

\begin{proposition}
\label{PROP:linear2}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear2}
\lim_{\eps\to 0^+} \eps \log \PP \big( \|u(t)\|_{L^\infty (\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

\noi
provided $(c_k) \in \l^1$.
\end{proposition}

\begin{theorem}
\label{THM:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\o$ given in \eqref{random_data}.
For any $t = \mathcal O ( \eps^{-1} |\log \eps|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1 (\T)} > z_0\eps^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noi
holds if and only if $(c_k) \in \l^1$.
\end{theorem}

In this paper, we consider the following Cauchy problem of the one-dimensional cubic nonlinear Schr\"odinger equation (NLS) on the torus $\mathbb{T}= \mathbb{R} / (2 \pi \mathbb{Z})$:
\begin{equation}
\label{NLS}
\begin{cases}
i\pa_t u + \partial_x^2 u = \pm \,|u|^2\, u,\\
u(t,x)|_{t=0}=u_0.
\end{cases}
\end{equation}

\noindent
The equation \eqref{NLS} is fundamental in theoretical physics and applied mathematics, for describing the evolution of wave packets in various physical systems, such as nonlinear optics, fluids, and plasmas; see \cite{SS99} for a review.
Bourgain \cite{BO93} proved \eqref{NLS} is deterministically global well-posed in $L^2 (\mathbb{T})$.
The probabilistic study of \eqref{NLS} has emerging importance since the seminal work of Bourgain on invariant measures \cite{BO94,BO96}.

More recently, probabilistic analyses of \eqref{NLS} have also found applications in oceanography, particularly in modelling rare extreme events such as the formation of {\it rogue waves} in deep-sea dynamics~\cite{DGV18,DGV19,GGKS21}. In particular, Garrido et al.~\cite{GGKS21} established a large deviations principle (LDP) describing the probability of observing a wave of unusually large height in the weakly nonlinear regime. Their result relied on a strong exponential decay assumption on the Fourier coefficients of the random initial data, which plays a crucial role in their analysis.
In this paper, we improve the exponential decay condition on coefficients of initial data in \cite{GGKS21} to a $\ell^1$ decay condition.

We consider
random initial data of the form:
\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\mathbb{Z}} c_k g_k e^{ikx},
\end{equation}
where $c_k \in \mathcal{C}$ and $\{g_k\}_{k\in\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\mathbb{E} g_k=0$, $\mathbb{E} g_k g_j=0$ and $\mathbb{E} g_k \overline{g}_j =\delta_{kj}$ for $k,j\in\mathbb{Z}$.
Without loss of any generality, we only consider nonnegative real coefficients $c_k \ge 0$, due to the rotation invariance of complex Gaussian random variables.

The study of the large deviation principle for solutions to \eqref{NLS} was initiated by \cite{DGV18,DGV19}, where the authors conjectured that the existence of an LDP can be used to predict the formation of rogue waves.
In particular, they considered \eqref{NLS} with random initial data $u_0^N$ of the form
\begin{align}
\label{random1}
u_0^N (x) = \sum_{|k| \le N} \theta_k e^{ikx},
\end{align}

\noindent
where the initial data is parametrized by a random vector $\theta = (\theta_k)_{|k| \le N} \in \mathbb C^{2N+1}$.
Here $(\theta_k)$ are independent complex Gaussian random variables with $\mathbb{E} \theta_k = \mathbb{E} \theta_k^2 = 0$ and $\mathbb{E} |\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.
Then, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by
\begin{align}
\label{Dtz}
\mathcal D (t,z) : = \bigg\{ (\theta_k)_{|k| \le N} \in \mathcal{C}^{2N+1} \Big| \sup_x |u (t,x| \theta)| > z\bigg\},
\end{align}

\noindent
where $u$ is the solution to NLS \eqref{NLS} with the truncated initial data \eqref{random1}.
Dematteis et al, then proposed a theoretical framework of a large deviations principle (LDP) to quantify the likelihood of $\mathcal D (t,z)$.
In particular, consider the minimization problem
\begin{align}\label{minp}
\theta^* (z) = {\rm argmin}\,\, \mathcal I (\theta)
\end{align}

\noindent
where
\[
\mathcal I (\theta) = \max_{y \in \mathcal{C}^{2N+1}} [\langle y,\theta \rangle - S(y)] \quad \textup{ and } \quad S(y) = \log \mathbb{E} e^{\langle y, \theta \rangle}.
\]

\noindent
Then Dematteis et al. claim that for $t >0$
\begin{align}
\label{LDP1}
\log \mathbb{P} (\mathcal D(t,z)) = - \mathcal I (\theta^* (z)) + o(1),
\end{align}

\noindent
as $z \to \infty$, provided
the minimization problem \eqref{minp} has a unique solution.

Garrido et al.~\cite{GGKS21} rigorously proved the conjectured LDP~\eqref{LDP1} locally in time for the nonlinear Schr\"odinger equation with weak nonlinearity.
A major difficulty in analysing the LDP problem~\eqref{LDP1} lies in estimating the gradient
$\nabla_{\theta} \sup_{x} |u(t,x|\theta)|$
and in verifying the convexity of the set $\mathcal{D}(t,z)$,
both of which are extremely challenging to establish in practice.
To overcome these obstacles, Garrido et al.~\cite{GGKS21} introduced a new formulation and considered the following LDP problem:
\begin{align}
\label{LDP2}
\log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}
where $I$ denotes the corresponding rate function.
The advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\eqref{minp}.
Within this framework, they established~\eqref{LDP2} for initial data of the form~\eqref{random1} with infinitely many Fourier modes ($N = \infty$) satisfying a \emph{strong decay condition} on the coefficients.

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\mathbb{T} = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

The proof of~\eqref{THM_LDP3} is based on the G\"artner--Ellis theorem combined with resonant approximation techniques.
In their analysis, the rapid decay condition~\eqref{decay} plays a crucial role in both the linear and nonlinear estimates.
They further observed that alternative families of coefficients $(c_k)$ might also yield~\eqref{LDP2}, and posed an \textit{open question} concerning the identification of the optimal decay condition under which~\eqref{LDP2} holds; see~\cite[Remark~1.3]{GGKS21}.
One of the principal aims of the present work is to address this question.

To this purpose, we consider a LDP under Fourier-Lebesgue norm, which is stronger than the $L^\infty (\mathbb{T})$ norm used in \eqref{THM_LDP3}.
Our conventions for the Fourier transform are as follows:
\[
\widehat f (n) = \frac1{{2\pi}} \int_0^{2\pi} f(x) e^{ix n}dx \quad \textup{ and } \quad f(x) = \sum_{n\in \mathbb{Z}} \widehat f(n) e^{ixn}
\]

\noindent
for functions on the circle $\mathbb{T} = \mathbb{R}/(2\pi \mathbb{Z})$.
With the above notations, we define the Fourier-Lebesgue norm by
\[
\begin{split}
\|f\|_{\mathcal{F} L^1 (\mathbb{T})} = \sum_{n \in \mathbb{Z}} |\widehat f(n)|.
\end{split}
\]

\noindent
It is easy to see that
\[
\sup_{x \in \mathbb{T}} |f(x)| \le \|f\|_{\mathcal{F} L^1 (\mathbb{T})}.
\]

\noindent
We note that $L^\infty$ and $\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\mathcal{F} L^1$ share the same rate function.
In particular, for $t \lesssim \varepsilon^{-1}$, we have
\begin{align}\label{obser}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x) | > z_0\varepsilon^{-1/2} \Big) = \lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1} > z_0\varepsilon^{-1/2} \Big),
\end{align}

\noindent
where $u(t,x)$ is the solution to \eqref{weakNLS1} with initial data \eqref{random1} in $\mathcal{F}L^1$.
This motivates us to consider the LDP under the $\mathcal{F} L^1$ norm.
We are ready to state our first main result.

We consider
random initial data of the form:
\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\mathbb{Z}} c_k g_k e^{ikx},
\end{equation}
where $c_k \in \mathcal{C}$ and $\{g_k\}_{k\in\mathbb{Z}}$ are independent, identically distributed, complex Gaussian random variables with $\mathbb{E} g_k=0$, $\mathbb{E} g_k g_j=0$ and $\mathbb{E} g_k \overline{g}_j =\delta_{kj}$ for $k,j\in\mathbb{Z}$.
Without loss of any generality, we only consider nonnegative real coefficients $c_k \ge 0$, due to the rotation invariance of complex Gaussian random variables.

\noindent
where the initial data is parametrized by a random vector $\theta = (\theta_k)_{|k| \le N} \in \mathbb C^{2N+1}$.
Here $(\theta_k)$ are independent complex Gaussian random variables with $\mathbb{E} \theta_k = \mathbb{E} \theta_k^2 = 0$ and $\mathbb{E} |\theta_k|^2 = c_k^2$, for some fast-decaying $c_k > 0$.
Then, the set of initial data that generates a rouge wave of height at least $z>0$ at time $t>0$ is given by
\begin{align}
\label{Dtz}
\mathcal D (t,z) : = \bigg\{ (\theta_k)_{|k| \le N} \in \mathcal{C}^{2N+1} \Big| \sup_x |u (t,x| \theta)| > z\bigg\},
\end{align}

Garrido et al.~\cite{GGKS21} rigorously proved the conjectured LDP~\eqref{LDP1} locally in time for the nonlinear Schr\"odinger equation with weak nonlinearity.
A major difficulty in analysing the LDP problem~\eqref{LDP1} lies in estimating the gradient
$\nabla_{\theta} \sup_{x} |u(t,x|\theta)|$
and in verifying the convexity of the set $\mathcal{D}(t,z)$,
both of which are extremely challenging to establish in practice.
To overcome these obstacles, Garrido et al.~\cite{GGKS21} introduced a new formulation and considered the following LDP problem:
\begin{align}
\label{LDP2}
\log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}
where $I$ denotes the corresponding rate function.
The advantage of this reformulation is that it circumvents the need to solve the minimization problem in~\eqref{minp}.
Within this framework, they established~\eqref{LDP2} for initial data of the form~\eqref{random1} with infinitely many Fourier modes ($N = \infty$) satisfying a \emph{strong decay condition} on the coefficients.

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\mathbb{T} = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

\noindent
We note that $L^\infty$ and $\mathcal{F} L^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\mathcal{F} L^1$ share the same rate function.
In particular, for $t \lesssim \varepsilon^{-1}$, we have
\begin{align}\label{obser}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \sup_{x} |u(t,x) | > z_0\varepsilon^{-1/2} \Big) = \lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1} > z_0\varepsilon^{-1/2} \Big),
\end{align}

\noindent
where $u(t,x)$ is the solution to \eqref{weakNLS1} with initial data \eqref{random1} in $\mathcal{F}L^1$.
This motivates us to consider the LDP under the $\mathcal{F} L^1$ norm.
We are ready to state our first main result.

\begin{align}
\label{LDP1}
\log \PP (\mathcal D(t,z)) = - \mathcal I (\theta^* (z)) + o(1),
\end{align}

\begin{align}
\label{LDP2}
\log \PP \Big( \sup_{x} |u(t,x)| > z \Big) = - I(z) + o(1), \quad z \to \infty,
\end{align}

\begin{align}\label{minp}
\theta^* (z) = {\rm argmin}\,\, \mathcal I (\theta)
\end{align}

\begin{align}
\label{random1}
u_0^N (x) = \sum_{|k| \le N} \theta_k e^{ikx},
\end{align}

\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}

In particular, consider the weakly nonlinear Schr\"odinger equation on the circle $\T = [0,2\pi]$:
\begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align}
It was proved in~\cite{GGKS21} that
\begin{align}
\label{THM_LDP3}
\lim_{\varepsilon \to 0^+}
\varepsilon \log \PP \Big( \sup_{x} |u(t,x)| > z_0 \varepsilon^{-1/2} \Big)
= - \frac{z_0^2}{\sum_k c_k^2},
\end{align}
for $0 < t \lesssim \varepsilon^{-1}$ and $z_0 > 0$,
where $u(t,x)$ denotes the solution to~\eqref{weakNLS1} with random initial data $u_0^\omega$ given by~\eqref{random_data} and coefficients satisfying
\begin{align}\label{decay}
c_k = a e^{-b|k|}
\quad \text{or} \quad
c_k = a e^{-b|k|^2},
\end{align}
for some fixed $a,b > 0$.

\noi
We note that $L^\infty$ and $\FL^1$ scales the same, and furthermore, the LDPs under $L^\infty$ and $\FL^1$ share the same rate function.
In particular, for $t \les \eps^{-1}$, we have
\begin{align}\label{obser}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \sup_{x} |u(t,x) | > z_0\eps^{-1/2} \Big) = \lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1} > z_0\eps^{-1/2} \Big),
\end{align}

\begin{remark}
\rm
We remark that the time scale $t = \mathcal O ( \eps^{-1} |\log \eps|)$ is critical;
see \cite[Remark 1.7]{GGKS21} for more discussion on critical time.
In the limit $\eps \to 0^+$, the large deviation principle \eqref{THM_LDP4} may hold for any fixed time $t > 0$.
\end{remark}

\begin{corollary}\label{COR:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\o$ given in \eqref{random_data}.
For any $t = \mathcal O ( \eps^{-1} |\log \eps|)$ and $z_0 > 0$, we have that \eqref{THM_LDP3} holds provided $(c_k) \in \l^1$.
\end{corollary}

\begin{proposition}
\label{PROP:linear1}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear1}
\lim_{\eps\to 0^+} \eps \log \P \big( \|u(t)\|_{\FL^1(\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

We turn to the summation in \eqref{main} over $k \in B$.
We note that
\begin{align}
\label{main2}
\begin{split}
\eps & \sum_{k \in B} \log \left( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \exp\left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \right) \\
& \le \eps \sum_{k \in B} \log \left( \big( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \big) \exp\left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \right) \\
& \le \eps \sum_{k \in B} \log \big( 1+ \sqrt{\pi}\varepsilon^{-1/2}\, c_k\, \big) + \eps \sum_{k \in B} \left(\frac{\varepsilon^{-1} \, c_k^2}{4}\right) \\
& \le \sqrt{\pi}\varepsilon^{1/2} \sum_{k \in B} \, c_k\, + \frac14 \sum_{k \in B} c_k^2,
\end{split}
\end{align}
which is again uniformly bounded as $(c_k) \in \l^1 \subset \l^2$, where we used $\eps^{1/2} \le c_k$ for $k \in B$.
\end{proof}

\subsection{Linear LDP II - \texorpdfstring{$L^\infty$}{Lg} norm}
In this subsection,
we prove an LDP for the linear solution to \eqref{linear} under $L^\infty$ norm.
In particular, we shall prove
\begin{proposition}
\label{PROP:linear2}
Consider the linear Schr\"odinger equation on the torus $\T = [0,2\pi]$ as in \eqref{linear}, with random initial data $u_0^\o$ given by \eqref{random_data}.
Then,
\begin{align}
\label{LDP_linear2}
\lim_{\eps\to 0^+} \eps \log \PP \big( \|u(t)\|_{L^\infty (\T)} \ge z_0 \eps^{-1/2} \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2},
\end{align}

\noi
where $\mathcal B_\eps$ is defined in \eqref{Beps}.
We first note, see also \cite[Proposition 4.1]{GGKS21},\footnote{Please note that \cite[Proposition 4.1]{GGKS21} only requires that $\|c_k\|_{\l^1_k} < \infty$.} that
\begin{align}
\label{pp400}
\lim_{\eps \to 0^+} \eps \log \PP \big( \| u_{\rm app} (t) \|_{L^\infty (\T)} > z_0 (\eps^{-1/2} + \eps^{-1/2 + \dl}) \big) = - \frac{z_0^2}{\sum_{k \in \Z} c_k^2}.
\end{align}

\begin{align}
\label{THM_LDP4}
\lim_{\eps \to 0^+} \eps \log \PP \Big( \|u(t,x) \|_{\FL^1 (\T)} > z_0\eps^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}

\begin{remark}
\rm
We remark that the time scale $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ is critical;
see \cite[Remark 1.7]{GGKS21} for more discussion on critical time.
In the limit $\varepsilon \to 0^+$, the large deviation principle \eqref{THM_LDP4} may hold for any fixed time $t > 0$.
\end{remark}

\begin{remark}
\rm
To prove Theorem~\ref{THM:main}, we first develop a novel approach to establish
the LDP for the linear Schr\"odinger equation under the sole assumption that
the initial data lies in $\mathcal{F} L^1$. We then carry out the nonlinear analysis,
together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.
\end{remark}

As a consequence of Theorem \ref{THM:main} and the observation \eqref{obser}, we can improve \eqref{THM_LDP3} in terms of the decay of $c_k$.

\begin{corollary}\label{COR:main}
Let $u(t,x)$ be solutions to \eqref{weakNLS1} with initial data $u_0^\omega$ given in \eqref{random_data}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that \eqref{THM_LDP3} holds provided $(c_k) \in \ell^1$.
\end{corollary}

\begin{remark}
\rm
We note that Corollary \ref{COR:main} improves \cite[Theorem 1.1]{GGKS21} in the following sense.
The coefficients $(c_k)$ in \cite[Theorem 1.1]{GGKS21} is required to decay condition \eqref{decay}.
In Proposition \ref{PROP:linear2}, it requires only $(c_k) \in \ell^1$.
It is expected that \eqref{THM_LDP3} holds for initial data $u_0 \in L^2 \cap L^\infty$, where $\mathcal{F} L^1 \subset L^2 \cap L^\infty$.
However, our current argument can only handle $\mathcal{F} L^1$ data.
\end{remark}

\begin{remark}
\rm
Recently, in \cite{BGMS25}, the authors studied rogue waves and large deviations
for two-dimensional pure-gravity deep-water waves, assuming initial data with
exponentially decaying Fourier coefficients; see \cite[(1.14)]{BGMS25}. It is
expected that our argument also applies in their setting and allows one to
relax this assumption to merely $\ell^1$ decay of the Fourier coefficients.
We plan to pursue this in future work.
\end{remark}

To prove Theorem~\ref{THM:main}, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\"odinger equation under the sole assumption that the initial data lies in $\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.”

To prove the main theorem, the authors say they (i) “first develop a novel approach to establish the LDP for the linear Schr\"odinger equation under the sole assumption that the initial data lies in $\mathcal{F} L^1$,” and then (ii) “carry out the nonlinear analysis, together with a perturbative argument, entirely within the $\mathcal{F} L^1$ framework.”,

\label{THM:main}
Let $u(t,x)$ be solutions to \begin{align}
\label{weakNLS1}
i\pa_t u + \Delta u = \varepsilon^{2} |u|^2 u.
\end{align} with initial data $u_0^\omega$ given in \begin{equation}\label{random_data}
u_0 (x)= \sum_{k\in\Z} c_k g_k e^{ikx},
\end{equation}.
For any $t = \mathcal O ( \varepsilon^{-1} |\log \varepsilon|)$ and $z_0 > 0$, we have that
\begin{align}
\label{THM_LDP4}
\lim_{\varepsilon \to 0^+} \varepsilon \log \mathbb{P} \Big( \|u(t,x) \|_{\mathcal{F} L^1 (\mathbb{T})} > z_0\varepsilon^{-1/2} \Big) = - \frac{z_0^2}{\sum_k c_k^2},
\end{align}

\noindent
holds if and only if $(c_k) \in \ell^1.\,

Biconditional or Equivalence
Asymptotic or Limit

Question:
Let $u(t,x)$ solve the weakly nonlinear Schr\"odinger equation on the torus $\mathbb T$,
\[
i\partial_t u+\Delta u=\varepsilon^2|u|^2u,
\]
with random initial data
\[
u_0(x)=\sum_{k\in\mathbb Z} c_k g_k e^{ikx},
\]
where $\{g_k\}_{k\in\mathbb Z}$ are independent identically distributed complex Gaussian random variables and $(c_k)$ is a sequence of coefficients. For any $t=\mathcal O(\varepsilon^{-1}|\log\varepsilon|)$ and any $z_0>0$, suppose one considers the large-deviation asymptotic
\[
\lim_{\varepsilon\to0^+}\varepsilon\log\mathbb P\Big(\|u(t,\cdot)\|_{\mathcal F L^1(\mathbb T)}>z_0\varepsilon^{-1/2}\Big)
= -\frac{z_0^2}{\sum_k c_k^2}.
\]
Which of the following statements is equivalent to this asymptotic formula?

The coefficient sequence satisfies $(c_k)\in \ell^1$.

Choices:
B. The coefficient sequence satisfies $(c_k)\in \ell^2$.
C. The coefficient sequence satisfies $\sum_k c_k^2<\infty$.
D. For each fixed $z_0>0$, there exists a time range $t=\mathcal O(\varepsilon^{-1}|\log\varepsilon|)$ on which the asymptotic formula holds whenever $(c_k)\in \ell^1\cap \ell^2$.
E. The asymptotic formula holds if and only if the coefficient sequence satisfies $(c_k)\in \mathcal F L^1(\mathbb T)$.

sharp \ell^1 threshold replaced by weaker square-summability

dropped the full equivalence and kept only the weaker consequence implied by $(c_k)\in\ell^1$

replaced the iff criterion by a merely sufficient mixed assumption with altered quantifier structure

confused coefficient summability on $(c_k)$ with function-space membership notation


The study of equilateral polygons within lattices combines elements of geometry, number theory, and discrete mathematics. In particular, the problem of characterizing which polygons can be embedded in given planar lattices has drawn sustained interest. A central focus has been on determining for which values of $n$ there exists a convex equilateral $n$-gon whose vertices lie in a specified lattice.

Recent results by Maehara \cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\nu(L)$, satisfies $\nu(L) \equiv 3 \pmod{4}$.

Building on this, Iino and Sakiyama \cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ for square-free integers $m \equiv 3 \pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.

In this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\Lambda(m)$, with $m \equiv 3 \pmod{4}$.

\begin{abstract}
We study the existence of equilateral polygons in planar integer lattices. Maehara showed that it's sufficient to work with rectangular lattices $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ with $m \equiv 3 \pmod{4}$. Building on results of Maehara and of Iino and Sakiyama, we show that for every such $m$ there exists $N$ such that for all $n \geq N$, the lattice $\Lambda(m)$ contains an equilateral $n$-gon. This extends previous classifications of equilateral polygons in planar lattices.
\end{abstract}

Recent results by Maehara \cite{maehara2019planar} established that every planar integral lattice contains convex equilateral $n$-gons for all even $n \geq 4$, and for certain odd values of $n$, depending on a number-theoretic invariant of the lattice. In particular, Maehara showed that a planar integral lattice $L$ contains some equilateral polygon with an odd number of sides if and only if the square-free part of the square of the determinant of $L$, denoted $\nu(L)$, satisfies $\nu(L) \equiv 3 \pmod{4}$.

Building on this, Iino and Sakiyama \cite{iino2025planarlatticesequilateraloddgons} studied rectangular lattices of the form $\Lambda(m) = L[(1,0),(0,\sqrt{m})]$ for square-free integers $m \equiv 3 \pmod{4}$, and provided both necessary and sufficient conditions for the existence of equilateral polygons with a given odd number of sides. Among their key findings is that for such lattices, if a convex equilateral $n$-gon exists, then $n$ must be at least as large as every prime dividing $m$. They also proved that this is a sufficient condition when the largest prime factor of $m$ is less than 29 by a brute-force computer search.

In this paper, we extend the classification of equilateral polygons in rectangular lattices of the form $\Lambda(m)$, with $m \equiv 3 \pmod{4}$.

We will give an existence proof in \Cref{Section: Existence} followed by a explicit construction in \Cref{Section: Construction}.

For any $m \equiv 3 \pmod{4}$ does there exist $N$ such that for all $n \geq N$ there exists a \textbf{convex} equilateral polygon of size n?

\begin{open}
For any $m \equiv 3 \pmod{4}$ does there exist an equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?
\end{open}

For any $m \equiv 3 \pmod{4}$ does there exist a \textbf{convex} equilateral polygon of size $n'$, where $n'$ is the largest prime factor of m?

The post-theorem introduction indicates the proof strategy: it will present "an existence proof" (in \Cref{Section: Existence}) and then an "explicit construction" (in \Cref{Section: Construction}).

\label{thm: new theorem}
For any $m \equiv 3 \pmod{4}$ there exists $N$ such that for all $n \geq N$ there exists an equilateral polygon of size n.,

Question: Let \(\Lambda(m)=L[(1,0),(0,\sqrt m)]=\{(a,b\sqrt m):a,b\in \mathbb Z\}\) be the planar rectangular lattice associated to an integer \(m\equiv 3\pmod 4\). An equilateral polygon of size \(n\) means an \(n\)-gon whose vertices lie in \(\Lambda(m)\) and whose side lengths are all equal. Which statement holds for such lattices?

For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).

Choices:
B. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains a convex equilateral polygon of size \(n\).
C. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(n\) such that the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).
D. There exists an integer \(N\) such that for every integer \(m\equiv 3\pmod 4\) and every integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\).
E. For every integer \(m\equiv 3\pmod 4\), there exists an integer \(N\) such that for every odd integer \(n\ge N\), the lattice \(\Lambda(m)\) contains an equilateral polygon of size \(n\), and one may take \(N\) to be the largest prime factor of \(m\).


\begin{theorem}\label{thm:mainthm}
Let $S$ be a complex algebraic variety admitting a trim resolution. Then
\begin{itemize}
\item
For $s\in S$, the local Euler obstruction $\Eu_S(s)$ equals the Euler characteristic of
the fiber over $s$ in any trim resolution of $S$.
\item
The {\em stringy\/} Chern class of $S$ equals its Chern-Mather class:
$\cstr(S)=\cma(S)$.
\item
The characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.
\end{itemize}
\end{theorem}

\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

There is considerable interest in conditions implying that the characteristic cycle of
the intersection cohomology sheaf is irreducible. Lusztig (\cite[13.7, p.~414]{MR1088333})
expressed the `hope' that this may be the case for Schubert varieties in flag manifolds of
type A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in
$F\ell(8)$ (\cite{MR1458969}, \cite{MR1896039}), while it holds in $F\ell(n)$ for $n\le 7$.
Irreducibility is also known for all Schubert
varieties in the standard Grassmannian (\cite{MR1084458}), and more generally
for all Schubert varieties in cominuscule Grassmannians of
types A, D, and E, see~\cite{MR1451256, MihalceaSingh}.

We hope that Theorem~\ref{thm:mainthm} may help in streamlining such verifications.
For instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve is trim, the irreducibility of the $\IC$ characteristic cycle in this case (first established
in \cite{MR1642745}) follows directly from~Theorem~\ref{thm:mainthm}.

The connection between the irreducibility of the $\IC$ characteristic cycle and
the equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones
in~\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal
index formula of Dubson and Kashiwara.
We freely borrow ideas from~\cite{MR2628830}
in~\S\ref{thmpf}.

A condition on fibers of small resolutions is considered in~\cite{graham}, including a proof
that the condition implies the irreducibility of the IC characteristic cycles.\medskip

{\em Acknowledgments.} This work was supported in part by an award from the Simons
Foundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and
Leonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this
work was carried out.

\section{Lagrangian push-forward}\label{Lagrapf}
Let $X$ be a nonsingular variety and denote by $T^* X$ the cotangent bundle of $X$.
The {\em conormal variety\/} $T^*_W X$ of a closed subvariety $W\subseteq X$ is the
closure in $T^* X$ of the conormal variety to the nonsingular part $W^\circ$ of $V$:
$T^*_W X = \overline{T^*_{W^\circ} X}$. The conormal variety of $X$ itself is the
zero-section $T^*_XX$ of the cotangent bundle. All conormal varieties have dimension
$\dim X$; they determine conormal {\em cycles\/} in $Z_{\dim X} T^* X$.
It will also be convenient to take the projective completion of these constructions:
we will denote by $\Tbb^* X$ the projective completion $\Pbb(T^* X\oplus \one)$
of $T^* X$ (here, $\Pbb$ denotes the projective bundle of lines) and by~$\Tbb^*_WX$
the closure of the conormal variety in $\Tbb^* X$.

For a nonsingular variety $X$, we denote by $\cL (X)$ the free abelian group of
{\em conical Lagrangian cycles\/} in the cotangent bundle $T^* X$.
Conormal cycles are conical Lagrangian, and in fact (cf.~\cite[Lemma~3]{MR1063344})
$\cL (X)$ may be realized as the free abelian group on conormal cycles.
For a closed (and possibly singular) subvariety $V\subseteq X$, we denote by
$\cL(V)$ the subgroup of $\cL(X)$ generated by the conormal cycles $T^*_W X$
with $W\subseteq V$. Thus, elements of~$\cL (V)$ may be viewed as finite integer linear
combinations $\sum_W m_W T^*_WX$ ranging over closed subvarieties $W$ of $V$.
Clearly $\cL(V)$ is isomorphic to the group of algebraic cycles of~$V$, and in particular
it is independent of the ambient nonsingular variety $X$.

Important invariants of $V$ may be expressed directly in terms of Lagrangian cycles
by means of intersection-theoretic operations, after taking the projective completion.
As above, realize $V$ as a closed subvariety of a
nonsingular variety $X$; the results will be independent of the choice of $X$.
Let $\pi: \Tbb^*_V X\to V$ be the natural projection; and let $\cO(1)$ be the
tautological line bundle on $\Tbb^* X=\Pbb(T^* X\oplus \one)$.
\begin{itemize}
\item[---]
The {\em local Euler obstruction\/} $\Eu_V: V\to \Zbb$ is
\[
\Eu_V(p)= (-1)^{\dim X-\dim V} \int c(\pi^* TX|_V) c(\cO(1))^{-1}\cap s(\pi^{-1}(p), \Tbb^*_V X)
\]
where $s(\pi^{-1}(p), \Tbb^*_V X)$ denotes the {\em Segre class\/} in the sense
of~\cite[Chapter~4]{85k:14004};
\item[---]
The {\em Chern-Mather class\/} $\cma(V)\in A_*V$ is
\[
\cma(V)=(-1)^{\dim X-\dim V} c(TX|_V)\cap \pi_* \left(c(\cO(1))^{-1}\cap [\Tbb^*_V X]\right)\saf.
\]
\end{itemize}
Equivalent results were established by C.~Sabbah (\cite[1.2.1, 1.2.2]{MR804052}); we also
address the reader to~\cite{MR1063344} and~\cite{MR2002g:14005} for clear treatments of these
formulas.

Let $\cF(V)$ denote the abelian group of constructible functions $V\to \Zbb$.
Following~\cite[\S1]{MR2002g:14005}, the relation between conormal cycles and local
Euler obstructions is recorded by the homomorphism
\[
\CC: \cF(V) \to \cL(V)
\]
defined by prescribing
\[
\Eu_W \mapsto (-1)^{\dim W} T^*_W X
\]
for all closed subvarieties $W$ of $V$.
In fact, $\CC$ is an {\em isomorphism:\/} it simply matches a basis of $\cF(V)$ with a
basis of $\cL(V)$.

\begin{defin}\label{def:CC}
The {\em characteristic cycle\/} $\CC(\alpha)$ of a constructible function $\alpha$ on $V$
is the image of $\alpha$ in $\cL(V)$ under this isomorphism.
\qede\end{defin}

Likewise, the formula for Chern-Mather classes motivates the introduction of a homomorphism
\[
c_*:\cL(V) \to A_*(V)
\]
defined on generators by
\[
T^*_W X \mapsto (-1)^{\dim W} \cma(W)\saf.
\]
Then the composition $c_*\circ \CC:\cF(V) \to A_*(V)$ agrees with the value at $V$
of MacPherson's natural transformation $\cF \leadsto A_*$, where $\cF$ is taken as
a functor with push-forward defined by Euler characteristics of fibers;
see~\cite{MR0361141} and~\cite[Example 19.1.7]{85k:14004}.
For instance, the {\em Chern-Schwartz-MacPherson class\/} of a (possibly singular)
algebraic variety $V$ is the image $c_*(\CC(\one_V))\in A_*V$ of the characteristic
cycle in~$\cL(V)$ of the constant function $\one_V$.
Sabbah provides an alternative proof of the naturality of this assignment, which is
the main result of~\cite{MR0361141}, by defining a covariant {\em push-forward\/}
\[
\varphi_*: \cL(V') \to \cL(V'')
\]
for every proper map $\varphi: V' \to V''$, making $\cL$ into a functor, in such a way
that the above homomorphisms define natural transformations
\[
\cF \leadsto \cL\quad, \quad \cL \leadsto A_*
\]
whose composition agrees with MacPherson's natural transformation.

We are interested in explicit formulas for this {\em Lagrangian push-forward.\/}
After embedding~$V''$ in a nonsingular variety $X$ and replacing $V'$ by a resolution $Y$,
we can reduce to the case of a proper morphism $f:Y \to X$ of nonsingular varieties.
We are specifically interested in the image $f_*(T^*_YY)$ of the zero-section in this
situation.

Theorem~\ref{thm:mainthm} will be a consequence of the following result. We recall the
definition of `trim' given in the introduction.

\begin{defin}\label{def:contained}
Let $\pi:Y \to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.
For $0\le d\le \dim Y$, let $Y_d$ denote the locus where the rank of the differential
$d\pi$ equals $d$. Then $\pi$ is a {\em trim resolution\/} if $\dim Y_d<d$
for all $0\le d<\dim Y$.
\qede\end{defin}

\begin{prop}\label{prop:Sabpf}
Let $f: Y\to X$ be a proper morphism of nonsingular varieties, such that $Y \to f(Y)$ is
a trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.
\end{prop}

\begin{remark}\label{rem:YtoS}
Let $\pi: Y\to S$ be a proper surjective morphism, with $Y$ nonsingular, and
$\iota:S\hookrightarrow X$ a closed embedding, with $X$ nonsingular; and let
$f=\iota\circ \pi: Y \to X$ be the composition.
As explained above, $T^*_{f(Y)} X=T^*_S X$ may be viewed as an element of $\cL(S)$;
Proposition~\ref{prop:Sabpf} states that if $\pi$ is trim, then this is the image $\pi_*(T^*_YY)$
under $\pi_*: \cL(Y)\to \cL(S)$.
\qede\end{remark}

The rest of this section is devoted to the proof of Proposition~\ref{prop:Sabpf}.\smallskip

{\em A priori,\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles
$\sum_Z m_Z T^*_ZX$, with $Z\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.
The task is to show that if $Y\to f(Y)$ is trim and $Z\subsetneq f(Y)$ is a {\em proper\/}
subvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.
Following Sabbah, we write $f$ as the composition
\[
\xymatrix{
Y \ar[r]^-\gamma & Y\times X \ar[r]^-\rho & X
}
\]
where $\gamma$ is the graph of $f$ and $\rho$ is the (smooth) projection.
Let $p: \rho^*(T^* X) \to T^* X$ be the natural mophism. Then  (cf.~\cite[\S2]{MR804052})
the components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the
irreducible components of the image
\begin{equation}\label{eq:pint}
p \left(\rho^*(T^* X) \cap T^*_{\gamma(Y)}(Y\times X)\right)
\end{equation}

\label{intro}
Recall that a proper surjective stratified map $\pi: Y\to S$ of complex algebraic varieties
of the same dimension is {\em small\/} if for all proper strata $Z$ of $S$, $2d(Z)< \codim_Z S$,
where $d(Z)$ denotes the (common) dimension of the fibers of $\pi$ over points of $Z$.
We say that $\pi$ is a {\em small resolution\/} if $Y$ is nonsingular and $\pi$ is birational
and small.

We introduce a strengthening of this condition, which we name `trim'
(Definition~\ref{def:contained}).
Assume $\pi:Y\to S$ is a proper birational map, with $Y$ nonsingular.
For $d\ge 0$, denote by $Y_d\subseteq Y$ the locally closed subset along which
the rank of the differential of $\pi$ is~$d$. We say that $\pi$ is {\em trim\/} if
$\dim Y_d<d$ for all $d<\dim Y$.

Trim maps are small (Proposition~\ref{prop:consma}).
If $\pi:Y\to S$ is small, and for all strata~$Z$ of~$S$ the restriction $\pi^{-1}(Z)\to Z$ is a smooth
morphism, then $\pi$ is trim (\S\ref{Furrem}).
For instance, the standard Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve (which is small, cf.~\cite{MR1642745}) is trim.

Our main result is the following.

\label{Furrem}

1. In this note we have considered the notion of `trim' only for proper birational morphisms
because that is the case relevant to our application in Theorem~\ref{thm:mainthm}.
It co

\begin{defin}\label{def:contained}
Let $\pi:Y \to S$ be a proper birational morphism of varieties, with $Y$ nonsingular.
For $0\le d\le \dim Y$, let $Y_d$ denote the locus where the rank of the differential
$d\pi$ equals $d$. Then $\pi$ is a {\em trim resolution\/} if $\dim Y_d<d$
for all $0\le d<\dim Y$.
\qede\end{defin}

\begin{prop}\label{prop:consma}
Let $\pi: Y \to S$ be a trim proper birational map. Then $\pi$ is small.
\end{prop}

\begin{abstract}
We introduce {\em trim\/} resolutions of complex algebraic varieties, a strengthening
of the notion of small resolution. We prove that the characteristic cycle of the intersection
cohomology sheaf of a variety admitting a trim resolution is irreducible and that for
such varieties the stringy and Chern-Mather classes coincide.
\end{abstract}

Our main result is the following.

The definition of characteristic cycle will be recalled below (\S\ref{Lagrapf}).
`Stringy' Chern classes were defined in~\cite{MR2183846, MR2304329}
(see~\cite{MR2280127} for a lean account). If $\pi: Y\to S$ is a crepant resolution,
$\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle
of~$Y$; the equality stated in the theorem is
\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

Important invariants of $V$ may be expressed directly in terms of Lagrangian cycles
by means of intersection-theoretic operations, after taking the projective completion.
As above, realize $V$ as a closed subvariety of a
nonsingular variety $X$; the results will be independent of the choice of $X$.
Let $\pi: \Tbb^*_V X\to V$ be the natural projection; and let $\cO(1)$ be the
tautological line bundle on $\Tbb^* X=\Pbb(T^* X\oplus \one)$.
\begin{itemize}
\item[---]
The {\em local Euler obstruction\/} $\Eu_V: V\to \Zbb$ is
\[
\Eu_V(p)= (-1)^{\dim X-\dim V} \int c(\pi^* TX|_V) c(\cO(1))^{-1}\cap s(\pi^{-1}(p), \Tbb^*_V X)
\]
where $s(\pi^{-1}(p), \Tbb^*_V X)$ denotes the {\em Segre class\/} in the sense
of~\cite[Chapter~4]{85k:14004};
\item[---]
The {\em Chern-Mather class\/} $\cma(V)\in A_*V$ is
\[
\cma(V)=(-1)^{\dim X-\dim V} c(TX|_V)\cap \pi_* \left(c(\cO(1))^{-1}\cap [\Tbb^*_V X]\right)\saf.
\]
\end{itemize}
Equivalent results were established by C.~Sabbah (\cite[1.2.1, 1.2.2]{MR804052}); we also
address the reader to~\cite{MR1063344} and~\cite{MR2002g:14005} for clear treatments of these
formulas.

{\em A priori,\/} $f_*(T^*_Y Y)$ is an integer linear combination of conormal cycles
$\sum_Z m_Z T^*_ZX$, with $Z\subseteq f(Y)$. One of the summands is $T^*_{f(Y)}X$.
The task is to show that if $Y\to f(Y)$ is trim and $Z\subsetneq f(Y)$ is a {\em proper\/}
subvariety of $f(Y)$, then the coefficient of $T^*_Z X$ in $f_* (T^*_Y Y)$ is~$0$.
Following Sabbah, we write $f$ as the composition
\[
\xymatrix{
Y \ar[r]^-\gamma & Y\times X \ar[r]^-\rho & X
}
\]
where $\gamma$ is the graph of $f$ and $\rho$ is the (smooth) projection.
Let $p: \rho^*(T^* X) \to T^* X$ be the natural mophism. Then (cf.~\cite[\S2]{MR804052})
the components $T^*_ZX$ appearing in the decomposition of $f_*(T^*_YY)$ are the
irreducible components of the image
\begin{equation}\label{eq:pint}
p \left(\rho^*(T^* X) \cap T^*_{\gamma(Y)}(Y\times X)\right)
\end{equation}
where we view both $\rho^*(T^* X)$ and $T^*_{\gamma(Y)}(Y\times X)$
as subschemes of $T^* (Y\times X)$.
Let $T^*_ZX$ be one such component, and let $z$ be a general point of $Z$
and $\xi$ a general covector in the fiber $(T^*_ZX)_z$. For $(z,\xi)$ to be in
the image~\eqref{eq:pint}, there must be a point $(y,z)\in \gamma(Y)$ mapping to
$z$, such that
$
(0,\xi)\in T^*_y Y\oplus T^*_z X\cong T^*_{(y,z)} (Y\times X)
$
vanishes on $T_{(y,z)}\gamma(Y)$.

By Proposition~\ref{prop:consma}, $\pi$ is small. It follows (cf.~\cite[\S6.2]{MR696691})
that the intersection cohomology sheaf $\IC^\bullet_S$ is the direct image of the intersection
cohomology sheaf of $Y$. Since $Y$ is nonsingular, the latter is a shift of the constant
sheaf. Therefore,
\[
\IC^\bullet_S = R\pi_* \Qbb_Y[\dim Y]\saf.
\]
(Concerning the shift, we follow the modern convention as in e.g.,
\cite[Remark~4.2.4]{MR2525735}.)
The following formula for the stalk Euler characteristic of $\IC^\bullet_S$ at $z\in S$
is a consequence of standard properties of (derived) direct images:
\[
\chi_z(\IC^\bullet_S) =
\chi_z(R\pi_* \Qbb_Y[\dim Y]) =
(-1)^{\dim S} \sum_i (-1)^i \dim H^i(\pi^{-1}(z);\Qbb)
= (-1)^{\dim S}\chi(\pi^{-1}(z))\saf.
\]
By the first assertion in Theorem~\ref{thm:mainthm}, this implies
\[
\chi_z(\IC^\bullet_S) = (-1)^{\dim S}\Eu_S(z)\saf.
\]
Now embed $S$ in a nonsingular variety $X$.
The characteristic cycle of $\IC^\bullet_S$ is
\[
CC(\chi_z(\IC^\bullet_S)) = CC((-1)^{\dim S}\Eu_S(z)) = T^*_SX\saf,
\]
and this verifies it is irreducible, completing the proof of
Theorem~\ref{thm:mainthm}.

The definition of characteristic cycle will be recalled below (\S\ref{Lagrapf}).
`Stringy' Chern classes were defined in~\cite{MR2183846, MR2304329}
(see~\cite{MR2280127} for a lean account). If $\pi: Y\to S$ is a crepant resolution,
$\cstr(S)$ equals the push-forward of the total Chern class of the tangent bundle
of~$Y$; the equality stated in the theorem is
\begin{equation}\label{eq:strMa}
f_*(c(TY)\cap [Y]) = \cma(S)
\end{equation}
in the Chow group of $S$. In general, stringy Chern classes are defined for
normal varieties with a $\Qbb$-Cartier canonical divisor and at worst log-terminal
singularities, and take value in the Chow group with rational coefficients. These
additional stipulations are not needed for the notion considered in this note; we can
adopt the left-hand side of~\eqref{eq:strMa} as the definition of stringy Chern class
in the (integral) Chow group of $S$, compatibly with the more general definition.
The fact that the class is independent of the choice of trim resolution is
also a consequence of Theorem~\ref{thm:mainthm}.

We note that Theorem~\ref{thm:mainthm} implies that Batyrev's {\em stringy Euler
number\/} (\cite{MR2001j:14018}) of a variety admitting a trim resolution equals
its Euler characteristic weighted by the local Euler obstruction.

Theorem~\ref{thm:mainthm} is a consequence of considerations concerning Sabbah's
formalism of {\em conical Lagrangian cycles,\/} to which local Euler obstructions
and Chern-Mather classes relate directly (see~\S\ref{Lagrapf}). Concerning intersection
cohomology, recall that if $\pi:Y \to S$ is a small resolution, then the
intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant
sheaf on~$Y$ (\cite[\S6.2]{MR696691}). We evaluate the corresponding push-forward
at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.
More precisely, we prove (Proposition~\ref{prop:Sabpf}) that, for trim resolutions, the
Lagrangian push-forward of the zero-section~$T^*_YY$ of the cotangent bundle
$T^* Y$ equals the conormal cycle $T^*_S X$. The theorem follows from this
more basic observation, as we show in~\S\ref{thmpf}.

There is considerable interest in conditions implying that the characteristic cycle of
the intersection cohomology sheaf is irreducible. Lusztig (\cite[13.7, p.~414]{MR1088333})
expressed the `hope' that this may be the case for Schubert varieties in flag manifolds of
type A, D, E. A counterexample was constructed by Kashiwara and Saito for type A in
$F\ell(8)$ (\cite{MR1458969}, \cite{MR1896039}), while it holds in $F\ell(n)$ for $n\le 7$.
Irreducibility is also known for all Schubert
varieties in the standard Grassmannian (\cite{MR1084458}), and more generally
for all Schubert varieties in cominuscule Grassmannians of
types A, D, and E, see~\cite{MR1451256, MihalceaSingh}.

We hope that Theorem~\ref{thm:mainthm} may help in streamlining such verifications.
For instance, since the Abel-Jacobi resolution of the theta divisor of a non-hyperelliptic
curve is trim, the irreducibility of the $\IC$ characteristic cycle in this case (first established
in \cite{MR1642745}) follows directly from~Theorem~\ref{thm:mainthm}.

The connection between the irreducibility of the $\IC$ characteristic cycle and
the equality of Chern-Mather and stringy Chern classes was pointed out by B.~Jones
in~\cite[Remark 3.3.2]{MR2628830}, ultimately as an application of the microlocal
index formula of Dubson and Kashiwara.
We freely borrow ideas from~\cite{MR2628830}
in~\S\ref{thmpf}.

A condition on fibers of small resolutions is considered in~\cite{graham}, including a proof
that the condition implies the irreducibility of the IC characteristic cycles.\medskip

{\em Acknowledgments.} This work was supported in part by an award from the Simons
Foundation, SFI-MPS-TSM-00013681. The author also thanks David Massey and
Leonardo Mihalcea for useful conversations, and Caltech for hospitality as most of this
work was carried out.

Theorem~\ref{thm:mainthm} is said to be “a consequence of considerations concerning Sabbah's formalism of {\em conical Lagrangian cycles,\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\pi:Y\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves (Proposition~\ref{prop:Sabpf}) that for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown in~\S\ref{thmpf}.

To prove the main theorem, it is said to be “a consequence of considerations concerning Sabbah's formalism of {\em conical Lagrangian cycles,\/}” (to which “local Euler obstructions and Chern-Mather classes relate directly”). For intersection cohomology, using that for a small resolution $\pi:Y\to S$ “the intersection cohomology sheaf of~$S$ is the push-forward of a shift of the constant sheaf on~$Y$,” the paper “evaluate[s] the corresponding push-forward at the level of characteristic cycles after embedding $S$ in a nonsingular variety~$X$.” More precisely, it proves the following proposition.

\begin{prop}\label{prop:Sabpf}
Let $f: Y\to X$ be a proper morphism of nonsingular varieties, such that $Y \to f(Y)$ is
a trim resolution. Then $f_* (T^*_Y Y)=T^*_{f(Y)} X$.
\end{prop}

That is, for trim resolutions “the Lagrangian push-forward of the zero-section~$T^*_Y Y$ of the cotangent bundle $T^*Y$ equals the conormal cycle $T^*_S X$.” “The theorem follows from this more basic observation,” as shown later.
\label{thmpf}

Let $S$ be a variety admitting a trim resolution $\pi: Y\to S$; let $\iota: S\hookrightarrow X$
be an embedding in a nonsingular variety, and let $f=\iota\circ \pi:Y \to X$ be the compo

Question: Let $S$ be a complex algebraic variety admitting a trim resolution, meaning that there exists a proper birational morphism $\pi:Y\to S$ with $Y$ nonsingular such that, for each $0\le d\le \dim Y$, if $Y_d\subseteq Y$ denotes the locus where the differential $d\pi$ has rank $d$, then $\dim Y_d<d$ for every $d<\dim Y$. For such a variety, let $\Eu_S:S\to \mathbb Z$ be the local Euler obstruction, and in this setting let the stringy Chern class be given by $\cstr(S)=\pi_*(c(TY)\cap [Y])\in A_*S$ for a trim resolution $\pi$, while $\cma(S)$ denotes the Chern-Mather class of $S$. Which of the following conclusions about $S$ holds under these hypotheses?

For every $s\in S$, the local Euler obstruction satisfies $\Eu_S(s)=\chi(\pi^{-1}(s))$ for the fiber over $s$ in any trim resolution $\pi:Y\to S$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.

Choices:
B. For every $s\in S$, the local Euler obstruction satisfies $\Eu_S(s)=\chi(\pi^{-1}(s))$ for the fiber over $s$ in some trim resolution $\pi:Y\to S$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is supported on a single conormal component, though not necessarily irreducible.
C. For every $s\in S$, the local Euler obstruction satisfies $\Eu_S(s)=\chi(\pi^{-1}(s))$ for the fiber over $s$ in any trim resolution $\pi:Y\to S$, and moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$.
D. For every $s\in S$, the local Euler obstruction satisfies $\Eu_S(s)=\chi(\pi^{-1}(s))$ for the fiber over $s$ in any small resolution $\pi:Y\to S$; moreover whenever $S$ admits a small resolution one has $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the intersection cohomology sheaf of $S$ is irreducible.
E. For every $s\in S$, the local Euler obstruction satisfies $\Eu_S(s)=\chi(\pi^{-1}(s))$ for the fiber over $s$ in any trim resolution $\pi:Y\to S$; moreover $\cstr(S)=\cma(S)$ in the Chow group of $S$, and the characteristic cycle of the constant sheaf on $S$ is irreducible.


\begin{theorem}\label{main theorem}
Let $\Omega\subset\mathbb{C}^{n}$, with $n>1$,  be a (possibly unbounded) pseudoconvex domain.
If $\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point, \ then the Bergman metric of $\Omega$ is well-defined in a nonempty open subset of $\Omega$, denoted by $\Omega^* $,  and the Bergman metric  cannot be  Einstein on any open subset of  $\Omega^*$.
\end{theorem}

For any bounded domain $D\subset \mathbb{C}^n$, its Bergman metric is an invariant Kähler metric.
Cheng and Yau \cite{CY80} proved that every bounded pseudoconvex domain in $\mathbb{C}^n$ with a $C^2$-smooth boundary admits a unique complete K\"ahler–Einstein metric (up to a scaling factor) which is also biholomorphically invariant.
Later, Mok and Yau \cite{MY80} removed the boundary regularity assumption and proved the existence of such a metric for arbitrary bounded pseudoconvex domains.

A natural problem arising from these works is to determine under what circumstances these two important invariant metrics coincide. A classical conjecture of Yau \cite{Yau} states that the Bergman metric of a bounded pseudoconvex domain is Einstein if and only if it is biholomorphic to a bounded homogeneous domain. Earlier,
Cheng \cite{C79} had conjectured that the Bergman metric of a smoothly bounded strongly pseudoconvex domain is K\"ahler–Einstein if and only if the domain is biholomorphic to the unit ball.
Cheng's conjecture was confirmed in dimension two by Fu–Wong \cite{FW97} and Nemirovski--Shafikov \cite{NS06}, and was resolved in all dimensions by Huang–Xiao \cite{HX16} based on earlier work of many authors.
Subsequent generalizations were obtained for Stein manifolds and Stein spaces with compact strongly pseudoconvex boundaries; see Huang–Li \cite{HL23}, Ebenfelt–Xiao–Xu \cite{EXX22,EXX24}, and references therein.
Related variations of Cheng’s conjecture were also discussed by S. Li in his papers \cite{L1, L2, L3}.

In a more recent development, Savale and Xiao \cite{SX23} investigated Bergman –Einstein metrics on smoothly bounded pseudoconvex domains in $\mathbb{C}^2$.
They proved that a smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^2$, whose Bergman metric is Einstein, must be biholomorphic to the unit ball in $\mathbb{C}^2$.
A prior result by Fu–Wong \cite{FW97} established an analogous statement for smoothly bounded complete Reinhardt pseudoconvex domains of finite type in $\mathbb{C}^2$.

Despite these advances, not much is known about the Einstein property of Bergman metrics on unbounded pseudoconvex domains or in bounded pseudoconvex domains with rough boundary points. In this paper, we aim to conduct a study along these lines.
We will show that the Bergman metric of a pseudoconvex domain, possibly unbounded, which possesses a strongly pseudoconvex polyhedral boundary point (as defined in Definition \ref{def1-9-24}) is not Einstein.

\begin{definition}\label{def1-9-24}
Let $\Omega$ be a possibly unbounded domain in $\mathbb{C}^{n}$  with $n>1$ and let $p\in\partial\Omega$.
We say that $p$  is  a  strongly pseudoconvex polyhedral boundary  point if there exists a neighborhood $U$ of $p$ in $\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\rho_{1},\dots,\rho_{m}\colon U\to\mathbb{R}$ with $m>1$  such that
$\Omega\cap U=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\}$  and
$\{\partial\rho_{1}|_p,\dots,\partial\rho_{m}|_p\}$  are linearly independent over $\mathbb{C}$.
\end{definition}

One of the main tools used in the proof of Theorem \ref{main theorem} is the rescaling argument, which has been used to work on many related
problems. In particular, in connection with our present work, we
mention the papers by Wong \cite{W77},
Kim \cite{Kim}, Kim-Yu \cite{KY}, Krantz-Yu \cite{KYu}
and Boas-Straube-Yu \cite{BSY}, where the rescaling method has been used to study the boundary limit of various quantities associated with the Bergman metric. Indeed, our current work has benefited from their studies. A recent application of the rescaling method can also be found in Huang-Zhu \cite{HZ}, where it is employed in solving a CR transversality problem. Another recent application of the rescaling method was used in working on the pinched properties of a K\"ahler metric is a recent paper of Bracci-Gauthier-Zimmer \cite{BGZ}.

We next recall the definition of strongly pseudoconvex polyhedral boundary points for a domain $\Omega\subset{\mathbb C}^n$.
\begin{definition}\label{def1-9-24}
Let $\Omega$ be a possibly unbounded domain in $\mathbb{C}^{n}$ with $n>1$ and let $p\in\partial\Omega$.
We say that $p$ is a strongly pseudoconvex polyhedral boundary point if there exists a neighborhood $U$ of $p$ in $\mathbb C^n$ and $C^{2}$-smooth strongly plurisubharmonic functions $\rho_{1},\dots,\rho_{m}\colon U\to\mathbb{R}$ with $m>1$ such that
$\Omega\cap U=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\}$ and
$\{\partial\rho_{1}|_p,\dots,\partial\rho_{m}|_p\}$ are linearly independent over $\mathbb{C}$.
\end{definition}

\begin{proposition}\label{prop1-4-3}
Let $\Omega$ be a possibly unbounded pseudoconvex domain in $\mathbb C^n$ which is stronlgy pseudoconvex polyhedral at some boundary point $p\in\partial\Omega$. Let $U$ be a neighborhood of $p$ such that $U\cap\Omega$ is connected, and on which the Bergman metric $g_{\Omega}$ is well defined. Then its Bergman metric $g_\Omega$ is K\"ahler-Einstein on $U\cap\Omega$ if and only if its Bergman canonical invariant $J_{\Omega}\equiv (n+1)^n\frac{\pi^n}{n!}$ on $U\cap\Omega$.
\end{proposition}
\begin{proof}
By Corollary \ref{coro1-3-30}, for any smooth boundary point $q\in U\cap\partial\Omega$ near which $\partial\Omega$ is strongly pseudoconvex, one has
 \begin{equation}\label{e2-3-31}
 \lim_{z\rightarrow q}J_{\Omega}(z)=\frac{(n+1)^n\pi^n}{n!}.
 \end{equation}
 The Bergman metric $g_{\Omega}$ {is} K\"ahler-Einstein if its Ricci curvature $R_{\Omega}=c g_{\Omega}$ for some constant $c$. By Corollary \ref{coro1-3-30}, one has $c=-1$. Consequently, the K\"ahler-Einstein assumption implies that $\log J_{\Omega}$ is a pluriharmonic function on $U\cap\Omega$. Now, for any attached holomorphic disk $\phi:\Delta\rightarrow\Omega$ where $\phi$ is holomorphic in $\Delta:=\{t\in {\mathbb C}: |t|<1\}$, continuous up to $\overline {\Delta}$, and $\phi(\partial\Delta)$ is contained in the smooth part of $ U\cap \partial\Omega$, we have that $\log
 J_{\Omega}(\phi(t))$ is harmonic. Since it is constant on the strongly pseudoconvex part of the boundary by \eqref{e2-3-31}, it assumes the value
\[
\log\frac{(n+1)^n\pi^n}{n!}
\]
everywhere on $\Delta$. Now, since $\partial \Omega$ is strongly pseudoconvex near $q$, the union of such disks fills up an open subset of $\partial\Omega$ near $q$. Since $\log
 J_{\Omega}$ is well defined in $U\cap\Omega$ on which it is real analytic, we conclude that $\log J_{\Omega}\equiv
 \log\frac{(n+1)^n\pi^n}{n!}$ over $U\cap\Omega$ as $U\cap\Omega$ is connected by definition.
Conversely, if $J_\Omega(z)$ takes a constant value near $p$, then the Bergman metric is obviously K\"ahler-Einstien. Thus, we have the conclusion of the proposition.
\end{proof}
\begin{remark}\label{remark1-10-21}

Note that the zero set of the Bergman kernel function, denoted by $E$, is a complex analytic variety in $\Omega$.
Thus, $J_\Omega$ is a well-defined real-analytic function on $\Omega\setminus E$. Since $\Omega\setminus E$ is connected, $J_\Omega$ is constant if and only if it is constant on some nonempty open subset of $\Omega$. In particular, when $\Omega$ contains a $C^2$-smooth strongly pseudoconvex boundary point, the Bergman metric of the domain $\Omega$ is Kähler–Einstein wherever it is well-defined if and only if $J_\Omega=c$ is a constant on a certain open subset of $\Omega\setminus E$ .
 In this case, $c=\frac{(n+1)^n\pi^n}{n!}$, and the Bergman space $A^2(\Omega)$ separates holomorphic directions at any point in $\Omega\setminus E$ and thus the Bergman metric is well-defined in $\Omega\setminus E$.
\end{remark}
\section{Stability of the Bergman kernels}\label{sec2}

\begin{proof}[Proof of Theorem \ref{main theorem}]
Let $\Omega$ be a pseudoconvex domain which is strongly pseudoconvex polyhedral at a boundary point $p\in\partial\Omega$ as defined in Definition~\ref{def1-9-24}. After shrinking $U$ if necessary, we may assume that
\begin{enumerate}
\item[(i)] there are $C^{2}$-smooth strongly plurisubharmonic functions $\{\rho_{j}\}_{j=1}^{m}$ with $m>1$ on $U$ such that
\begin{equation}\label{9-24-a2}
 U\cap\Omega=\{z\in U:\rho_{1}(z)<0,\dots,\rho_{m}(z)<0\};
\end{equation}
\item[(ii)] the vectors $\partial\rho_{1}(q),\dots,\partial\rho_{m}(q)$ are linearly independent over $\mathbb C$ for $q\in U\cap \overline\Omega$.
\end{enumerate}
By Lemma \ref{25-9-23a1}, after a suitable change of coordinates on a small neighborhood $V\Subset U$ of $p$, we may assume that $p=0$ and
\begin{equation}\label{9-24-a1}
V\cap\Omega=\{(z_1,\cdots, z_n)\in V: {\rm Im~ }z_1>\Phi_1(z, \overline z), \cdots, {\rm Im}~ z_m>\Phi_m(z, \overline z)\}
\end{equation}
where $$\Phi_1(z, \overline z)=|z|^2+R_1(z), \Phi_j(z, \overline z)=\sum a^j_{\alpha\overline\beta} z_{\alpha}\overline z_{\beta}+R_j(z), \quad 2\leq j\leq m$$
with each remainder $R_j=\mathcal O(|z|^3)$. Write $$U_0=\{z\in\mathbb C^n: |z_j|<\varepsilon_0, 1\leq j\leq n\}$$ with $\varepsilon_0\ll 1$ such that $U_0\Subset V$ and on $U_0$ one has
$$|R_j(z)|\leq \frac{A_0}{2}|z|^2, \forall z\in U_0, 1\leq j\leq m,$$
where $A_0$ is defined in (\ref{10-27-a3}).
We first construct a bounded continuous plurisubharmonic function $\psi$ in $\Omega$ where $\psi$ is strictly plurisubharmonic near $p=0$ as follows:
\begin{lemma}\label{le3-28-1}
After shrinking $U_0$ if necessary, there exists a plurisubharmonic function $\psi:\Omega\rightarrow (-\infty, 0)$ such that $$\psi (z)>-c_0, ~\left(\frac{\partial^2\psi(z)}{\partial z_j\partial\overline z_k}\right)\geq c ~I_{n}, ~~z\in U_0\cap\Omega$$for some constants $c_0>0,~c>0$.
\end{lemma}
\begin{proof}
Let $(V, z)$ be the coordinates given as in (\ref{9-24-a1}) and set $\varphi=\frac{A_0}4|z|^2-y_1$. Then $\varphi$ is strictly plurisubharmonic on $V$ and satisfies $\varphi(0)=0, \varphi(z)<0$ when $z\in U_0\cap\overline\Omega\setminus \{0\}$. Take $r>0$ such that $\overline {\mathbb B^n(0, r)}\Subset U_0$. Set $M=\max\{\varphi(z): z\in \partial \mathbb B^n(0, r)\cap\overline \Omega\}$. Then $M<0$. Now we define $\psi$ as follows:
\begin{equation}
 \psi=\begin{cases}
 \max\{\varphi(z), M\}, &z\in \mathbb B^n(0, r)\cap\Omega\\
 M, &z\in\Omega\setminus \mathbb B^n(0, r).
 \end{cases}
\end{equation}
Then $\psi$ is a bounded and continuous plurisubharmonic function on $\Omega$ with $$\psi(0)=0, ~~\psi(z)<0, \forall z\in\overline\Omega\setminus\{0\}.$$
Furthermore, $\psi$ is equal to $\varphi$ near $0$ with $$\left(\frac{\partial^2\psi}{\partial z_i\partial\overline z_j}\right)= \frac{A_0}4 I_n $$
in some neighborhood $U$ of $0$ in $\mathbb C^n$. Moreover, $\psi>-c_0$ on $U$ for some positive constant $c_0$.
\end{proof}

\begin{corollary}
Let $\Omega\subset\mathbb{C}^{n}$, with $n>1$,  be a bounded pseudoconvex domain.  If $\Omega$  possesses a  strongly pseudoconvex polyhedral boundary point,then its  Bergman metric cannot be Einstein.
\end{corollary}

One of the main tools used in the proof of Theorem \ref{main theorem} is the rescaling argument, which has been  used  to work on many related
problems. In particular, in connection with our present work, we
mention   the papers by Wong \cite{W77},
Kim \cite{Kim},  Kim-Yu \cite{KY}, Krantz-Yu \cite{KYu}
and Boas-Straube-Yu \cite{BSY},  where the rescaling method has been used to study the boundary limit of  various quantities associated with  the Bergman metric. Indeed, our current work  has benefited   from  their studies.  A recent application of the rescaling method can also be found in Huang-Zhu \cite{HZ}, where it is employed in solving a CR transversality problem. Another recent  application of the rescaling method was used in working on the pinched  properties of a K\"ahler metric is a recent  paper of Bracci-Gauthier-Zimmer \cite{BGZ}.

The  ideas of our proof of the main theorem can be stated briefly as follows: First, we   show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball. Next we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point  tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.
To obtain such a sequence, we assign weight $2$ to one of the complex normal directions, weight $1.5$  to the other normal directions, and weight $1$  to the remaining CR directions. The main part of the paper is then devoted to showing that the Bergman invariant function of this product domain coincides with that of  $\Omega$.
A direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned  product domain, leading to a contradiction.  In this respect, our proof departs from earlier approaches to the Cheng conjecture and its generalizations, where the contradiction is derived via spherical CR geometry and the Qi-Keng Lu uniformization theorem \cite{HX20}.

The ideas of the proof of Theorem~\ref{main theorem} are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”

The ideas of the proof, in establishing the main theorem, are stated as follows. One main tool is a “rescaling argument.” First, “we show that if the Bergman metric of our domain is Einstein, then its Bergman invariant function is constant and equals that of the unit ball.” Next, “we carefully construct a special sequence of points approaching a strongly pseudoconvex polyhedral boundary point tangentially such that the limit domain is equivalent to the product of a ball and a bidisk of lower dimension.” To obtain such a sequence, “we assign weight $2$ to one of the complex normal directions, weight $1.5$ to the other normal directions, and weight $1$ to the remaining CR directions.” The main part then shows “that the Bergman invariant function of this product domain coincides with that of $\Omega$.” Finally, “a direct computation shows that the Bergman invariant function of the unit ball differs from that of the aforementioned product domain,” yielding “a contradiction.”

Question:
Let \(\Omega\subset \mathbb C^n\) with \(n>1\) be a pseudoconvex domain, possibly unbounded. Assume that \(\Omega\) has a strongly pseudoconvex polyhedral boundary point \(p\in \partial\Omega\), meaning that there exist a neighborhood \(U\) of \(p\) in \(\mathbb C^n\) and \(C^2\)-smooth strongly plurisubharmonic functions \(\rho_1,\dots,\rho_m:U\to\mathbb R\) with \(m>1\) such that
\[
\Omega\cap U=\{z\in U: \rho_1(z)<0,\dots,\rho_m(z)<0\},
\]
and the complex differentials \(\{\partial\rho_1|_p,\dots,\partial\rho_m|_p\}\) are linearly independent over \(\mathbb C\). Under these assumptions, which statement about the Bergman metric of \(\Omega\) holds?

There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric cannot be Einstein on any open subset of \(\Omega^*\).

Choices:
B. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and if the Bergman metric is Einstein on some open subset of \(\Omega^*\), then \(\Omega\) is biholomorphic to the unit ball.
C. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined.
D. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman metric is not Einstein on some nonempty open subset of \(\Omega^*\).
E. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well-defined, and the Bergman invariant function on \(\Omega^*\) is constant and equal to that of the unit ball.

conclusion-from-contradiction replaced by global classification

constant-ball invariant asserted unconditionally instead of under Einstein hypothesis


Question:
Let \(\Omega\subset \mathbb{C}^n\) with \(n>1\) be a pseudoconvex domain, possibly unbounded. Suppose \(\Omega\) has a strongly pseudoconvex polyhedral boundary point \(p\in \partial\Omega\), meaning that there exist a neighborhood \(U\) of \(p\) in \(\mathbb C^n\) and \(C^2\)-smooth strongly plurisubharmonic functions \(\rho_1,\dots,\rho_m:U\to\mathbb R\) with \(m>1\) such that
\[
\Omega\cap U=\{z\in U: \rho_1(z)<0,\dots,\rho_m(z)<0\},
\]
and the complex differentials \(\partial \rho_1|_p,\dots,\partial \rho_m|_p\) are linearly independent over \(\mathbb C\). The Bergman metric is the invariant K\"ahler metric associated to \(\Omega\) when it is defined, and a K\"ahler metric is called Einstein if its Ricci form is a constant multiple of the metric. Which statement holds for \(\Omega\)?

There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well defined, and that Bergman metric is not Einstein on any open subset of \(\Omega^*\).

Choices:
B. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well defined, and if that Bergman metric is Einstein on some open subset of \(\Omega^*\), then \(\Omega\) is biholomorphic to the unit ball.
C. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well defined.
D. If the Bergman metric of \(\Omega\) is well defined on a nonempty open subset \(\Omega^*\subset \Omega\), then there exists an open subset of \(\Omega^*\) on which the Bergman metric is Einstein.
E. There exists a nonempty open subset \(\Omega^*\subset \Omega\) on which the Bergman metric of \(\Omega\) is well defined, and its Bergman invariant function agrees on \(\Omega^*\) with that of the unit ball.


\label{thm-main}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	If $Q(A)$ is totally positive, then
	\begin{enumerate}[\rm(1)]
		\item $A$ is totally positive;
		\item $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive;
		\item the row generating function $\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\ge0$.
	\end{enumerate}

\begin{prob}\label{prob-TP}
	Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
	under which conditions are both $A$ and its reversal $\rev{A}:=[a_{n,n-k}]_{n,k\ge0}$ totally positive?
\end{prob}

\hspace*{\parindent}
Following Karlin \cite{Kar68},
a finite or infinite matrix $A=[a_{n,k}]_{n,k\ge0}$ is {\it totally positive}
if all its minors are nonnegative.
Such matrices have been widely studied in both pure and applied mathematics \cite{Bre95,Bre96,Kar68,Pin10}.
In recent five years there has been great interest concerning total positivity of combinatorial matrices \cite{CDD+21,CS24,LLY+22,LPW23,LWW23,MMW22,MW22,PSZ23,Sok23,Zhu21,Zhu24}.
However, there are still many combinatorial matrices that are experimentally totally positive but without proof.
For instance, a longstanding conjecture of Brenti \cite[Conjecture 6.10]{Bre96} asked the total positivity of the Eulerian triangle
$$
[A(n,k)]_{n,k\ge0}=
\left[
 \begin{array}{*{6}c}
 1 \\
 1 & 1 \\
 1 & 4 & 1 \\
 1 & 11 & 11 & 1 \\
 1 & 26 & 66 & 26 & 1 \\
 \vdots &&&&& \ddots
 \end{array}
\right],
$$
where the {\it Eulerian number} $A(n,k)$ counts the number of permutations of $[n+1]$ with $k$ descents,
and satisfies the recurrence
$$
A(n,k)=(n-k+1)A(n-1,k-1)+(k+1)A(n-1,k)
$$
with initial conditions $A(0,k)=\del_{0k}$.
Although the total positivity of the Eulerian triangle is hard to prove,
the first author {\it et al.} \cite{CDD+21},
by using a planar network approach,
showed the total positivity of the reversal of the Stirling triangle of the second kind
$$
\overleftarrow{S}:=[\overleftarrow{S}(n,k)]_{n,k\ge0}=
\left[
 \begin{array}{*{6}c}
 1 \\
 1 & 1 \\
 1 & 3 & 1 \\
 1 & 6 & 7 & 1 \\
 1 & 10 & 25 & 15 & 1 \\
 \vdots &&&&& \ddots
 \end{array}
\right],
$$
which enjoys a similar recurrence to the Eulerian triangle, i.e.,
$$
\overleftarrow{S}(n,k)=(n-k+1)\overleftarrow{S}(n-1,k-1)+\overleftarrow{S}(n-1,k)
$$
with initial conditions $\overleftarrow{S}(0,k)=\del_{0k}$.
On the other hand, it is known \cite{Bre95,Zhu14,CLW15rec} that the Stirling triangle of the second kind is totally positive.
So it is natural to ask the following question.

\begin{prob}\label{prob-TP}
 Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
 under which conditions are both $A$ and its reversal $\overleftarrow{A}:=[a_{n,n-k}]_{n,k\ge0}$ totally positive?
\end{prob}

One particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\dots$.
The reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\boldsymbol{\al}=(a_n)_{n\ge0}$:
$$
T_{\infty}(\boldsymbol{\al}):=[a_{i-j}]_{i,j\ge0}=
\left[
 \begin{array}{*{5}c}
 a_0 \\
 a_1 & a_0 \\
 a_2 & a_1 & a_0 \\
 a_3 & a_2 & a_1 & a_0 \\
 \vdots &&&& \ddots
 \end{array}
\right].
$$
An infinite nonnegative sequence $\boldsymbol{\al}$ is called a {\it P\'olya frequency sequence} (or shortly, {\it PF sequence}),
if its Toeplitz matrix $T_{\infty}(\boldsymbol{\al})$ is totally positive.
We identify a finite sequence $a_0,a_1,\dots,a_n$ with the infinite sequence $a_0,a_1,\dots,a_n,0,0,\dots$.
A fundamental characterization for PF sequences is due to Schoenberg and Edrei,
which states that the sequence $\boldsymbol{\al}=(a_n)_{n\ge0}$ is PF if and only if its generating function
\begin{equation}\label{eq-PF}
 \sum_{n\ge0} a_n x^n=C x^k e^{\gamma x} \frac{\prod_{j\ge0} (1+\al_j x)}{\prod_{j\ge0} (1-\be_j x)},
\end{equation}
where $C>0$, $k\in\mathbb{N}$, $\al_j,\be_j,\gamma\ge0$
and $\sum_{j\ge0}(\al_j+\be_j)<\infty$.
In particular, Aissen, Schoenberg and Whitney states that
a finite sequence of nonnegative numbers is PF if and only if its generating function is {\it real-rooted}, that is,
it is a constant polynomial or all of its zeros are real numbers (see \cite[p. 399, 412]{Kar68} for instance).
In this sense, we refer the generating functions of PF sequences as {\it PF formal power series}.
Hence, it gives rise to the another problem.

\begin{prob}\label{prob-RZ}
 Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
 under which conditions is the row generating function $\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\ge0$,
 or equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?
\end{prob}

The objective of this paper is to give a common answer to Problems \ref{prob-TP} and \ref{prob-RZ}.
An efficient method of proving total positivity of matrices is to prove the total positivity of their (right) production matrices \cite{DFR05},
which has been frequently used in literature \cite{CLW15rio,CLW15rec,PZ16,LLY+22,CS24,Zhu14,Zhu24}.
However, this method does not work when proving total positivity of matrix reversals.
So we turn to consider the total positivity of left production matrices,
and discover that it implies the total positivity of the matrix itself, its reversal, and the Toeplitz matrix of each row, simultaneously.

Let $A$ be a lower triangular matix with nonzero diagonals,
then $A$ is invertible.
Following Liang {\it et al.} \cite{LPW23}, call
\begin{equation}\label{eq-leftprod}
Q(A):=A\cdot
\left[
 \begin{array}{cc}
 1 & O \\
 O & A^{-1}
 \end{array}
\right]
\end{equation}
the {\it left production matrix} of $A$.
This implies that $Q(A)$ is also lower triangular with nonzero diagonals.
Moreover, we have
\begin{equation}\label{eq-A=Q}
A=Q(A)
\left[
 \begin{array}{cc}
 I_1 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_2 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_3 & \\
 & Q(A)
 \end{array}
\right]
\dots.
\end{equation}
Note that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \eqref{eq-A=Q}.
In such cases, we also call $Q(A)$ the left production matrix of $A$.

One particular example is the lower triangular matrix whose columns are constant numbers $a_0,a_1,a_2,\dots$.
The reversal of such a triangle is precisely the Toeplitz matrix of the sequence $\bal=(a_n)_{n\ge0}$:
$$
T_{\infty}(\bal):=[a_{i-j}]_{i,j\ge0}=
\left[
 \begin{array}{*{5}c}
 a_0 \\
 a_1 & a_0 \\
 a_2 & a_1 & a_0 \\
 a_3 & a_2 & a_1 & a_0 \\
 \vdots &&&& \ddots
 \end{array}
\right].
$$
An infinite nonnegative sequence $\bal$ is called a \defn{P\'olya frequency sequence} (or shortly, \defn{PF sequence}),
if its Toeplitz matrix $T_{\infty}(\bal)$ is totally positive.
We identify a finite sequence $a_0,a_1,\dots,a_n$ with the infinite sequence $a_0,a_1,\dots,a_n,0,0,\dots$.
A fundamental characterization for PF sequences is due to Schoenberg and Edrei,
which states that the sequence $\bal=(a_n)_{n\ge0}$ is PF if and only if its generating function
\begin{equation}\label{eq-PF}
 \sum_{n\ge0} a_n x^n=C x^k e^{\ga x} \frac{\prod_{j\ge0} (1+\al_j x)}{\prod_{j\ge0} (1-\be_j x)},
\end{equation}
where $C>0$, $k\in\mathbb{N}$, $\al_j,\be_j,\ga\ge0$
and $\sum_{j\ge0}(\al_j+\be_j)<\infty$.
In particular, Aissen, Schoenberg and Whitney states that
a finite sequence of nonnegative numbers is PF if and only if its generating function is \defn{real-rooted}, that is,
it is a constant polynomial or all of its zeros are real numbers (see \cite[p. 399, 412]{Kar68} for instance).
In this sense, we refer the generating functions of PF sequences as \defn{PF formal power series}.
Hence, it gives rise to the another problem.

\begin{prob}\label{prob-RZ}
 Given a lower triangular matrix $A=[a_{n,k}]_{n,k\ge0}$,
 under which conditions is the row generating function $\sum_{k=0}^n a_{n,k} x^k$ real-rooted for each $n\ge0$,
 or equivalently, the Toeplitz matrices of the rows of $A$ are all totally positive?
\end{prob}

Let $A$ be a lower triangular matix with nonzero diagonals,
then $A$ is invertible.
Following Liang {\it et al.} \cite{LPW23}, call
\begin{equation}\label{eq-leftprod}
Q(A):=A\cdot
\left[
 \begin{array}{cc}
 1 & O \\
 O & A^{-1}
 \end{array}
\right]
\end{equation}
the \defn{left production matrix} of $A$.
This implies that $Q(A)$ is also lower triangular with nonzero diagonals.
Moreover, we have
\begin{equation}\label{eq-A=Q}
A=Q(A)
\left[
 \begin{array}{cc}
 I_1 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_2 & \\
 & Q(A)
 \end{array}
\right]
\left[
 \begin{array}{cc}
 I_3 & \\
 & Q(A)
 \end{array}
\right]
\dots.
\end{equation}
Note that one can obtain the matrix $A$ (either invertible or not) from a given matrix $Q(A)$ by \eqref{eq-A=Q}.
In such cases, we also call $Q(A)$ the left production matrix of $A$.

Recently, Mao {\it et al.} \cite{MMW22} showed Theorem \ref{thm-main} (1) whenever $A$ is a proper Riordan array,
by using matrix decompositions and induction.
However, sometimes Theorem \ref{thm-main} (2) seems more useful.
In fact, it is possible that the total positivity of a given matrix $A$ is not easy to obtain
whenever neither its left nor right production matrix is totally positive (for instance, the matrix $\rev{S}$),
but we can turn to consider its reversal $\rev{A}$ which may have a totally positive left production matrix.
In this paper, we will prove the three results in Theorem \ref{thm-main} simultaneously,
by constructing a unified planar network that represent $A$, $\rev{A}$, and Toeplitz matrices of rows of $A$, respectively,
when selecting different sets of sources and sinks.
Furthermore, we will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$.

\begin{thm}\label{thm-T}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 Define
 \setlength{\arraycolsep}{3pt}
 $$
 M_{n,r}:=
 \left[
 \begin{array}{ccc}
 I_0 \\
 & Q_n(A) \\
 & & I_r
 \end{array}
 \right]
 \left[
 \begin{array}{ccc}
 I_1 \\
 & Q_n(A) \\
 & & I_{r-1}
 \end{array}
 \right]
 \dots
 \left[
 \begin{array}{ccc}
 I_r \\
 & Q_n(A) \\
 & & I_0
 \end{array}
 \right],
 $$
 where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
 Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
 Then for each $r\ge0$, we have
 $$
 T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
 $$
 where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
 and $^T$ stands for transpose.
\end{thm}

\begin{thm1}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 If $Q(A)$ is totally positive, then
 \begin{enumerate}[\rm(1)]
 \item $A$ is totally positive;
 \item $\rev{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive.
 \end{enumerate}
\end{thm1}

\begin{thm2}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 Define
 \setlength{\arraycolsep}{3pt}
 \begin{equation}\label{eq-M}
 M_{n,r}:=
 \left[
 \begin{array}{ccc}
 I_0 \\
 & Q_n(A) \\
 & & I_r
 \end{array}
 \right]
 \left[
 \begin{array}{ccc}
 I_1 \\
 & Q_n(A) \\
 & & I_{r-1}
 \end{array}
 \right]
 \dots
 \left[
 \begin{array}{ccc}
 I_r \\
 & Q_n(A) \\
 & & I_0
 \end{array}
 \right],
 \end{equation}
 where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
 Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
 Then for each $r\ge0$, we have
 \begin{equation}\label{eq-Toep}
 T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
 \end{equation}
 where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
 and $^T$ stands for transpose.
\end{thm2}

\begin{thm1}
 Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
 and $Q(A)$ be its left production matrix.
 If $Q(A)$ is totally positive, then
 \begin{enumerate}[\rm(1)]
 \item[\rm(3)] the row generating function $\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\ge0$.
 \end{enumerate}
\end{thm1}

\begin{equation}\label{eq-A=Q}
A=Q(A)
\left[
	\begin{array}{cc}
		I_1 &  \\
			& Q(A)
	\end{array}
\right]
\left[
	\begin{array}{cc}
		I_2 &  \\
			& Q(A)
	\end{array}
\right]
\left[
	\begin{array}{cc}
		I_3 &  \\
			& Q(A)
	\end{array}
\right]
\dots.
\end{equation}

\begin{thm}\label{thm-main}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	If $Q(A)$ is totally positive, then
	\begin{enumerate}[\rm(1)]
		\item $A$ is totally positive;
		\item $\rev{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive;
		\item the row generating function $\sum_{k=0}^n a_{n,k} x^k$ is real-rooted for each $n\ge0$.
	\end{enumerate}
\end{thm}

Recently, Mao {\it et al.} \cite{MMW22} showed Theorem \ref{thm-main} (1) whenever $A$ is a proper Riordan array,
by using matrix decompositions and induction.
However, sometimes Theorem \ref{thm-main} (2) seems more useful.
In fact, it is possible that the total positivity of a given matrix $A$ is not easy to obtain
whenever neither its left nor right production matrix is totally positive (for instance, the matrix $\overleftarrow{S}$),
but we can turn to consider its reversal $\overleftarrow{A}$ which may have a totally positive left production matrix.
In this paper, we will prove the three results in Theorem \ref{thm-main} simultaneously,
by constructing a unified planar network that represent $A$, $\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively,
when selecting different sets of sources and sinks.
Furthermore, we will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$.

For a given a matrix $A$, denote by
$$
A\submat{i_0, \ldots, i_r}{j_0, \ldots, j_r}
\quad \text{and} \quad
A\minor{i_0, \ldots, i_r}{j_0, \ldots, j_r}
$$
the submatrix and the minor of $A$ determined by
the rows indexed $i_0 < \cdots < i_r$ and
the columns indexed $j_0 < \cdots < j_r$, respectively.
Let
$$
A_r:=A	\left[
	\begin{array}{c}
		0,1,\dots,r \\
		0,1,\dots,r
	\end{array}
	\right]

$$
be the $r$th leading principal submatrix (of order $r+1$) of $A$.

\begin{thm}\label{thm-T}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	Define
	\setlength{\arraycolsep}{3pt}
	$$
	M_{n,r}:=
	\left[
		\begin{array}{ccc}
			I_0 \\
				& Q_n(A) \\
				&     & I_r
		\end{array}
	\right]
	\left[
		\begin{array}{ccc}
			I_1 \\
				& Q_n(A) \\
				&     & I_{r-1}
		\end{array}
	\right]
	\dots
	\left[
		\begin{array}{ccc}
			I_r \\
				& Q_n(A) \\
				&     & I_0
		\end{array}
	\right],
	$$
	where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
	Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
	Then for each $r\ge0$, we have
	$$
	T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
	$$
	where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
	and $^T$ stands for transpose.
\end{thm}

This paper is organized as follows.
In section 2, we introduce the Lindstr\"om-Gessel-Viennot lemma
and planar network characterizations of totally positive matrices.
In section 3, we give proofs of Theorem \ref{thm-main} and Theorem \ref{thm-T}
by constructing a unified planar network.
As applications, we present sufficient conditions for total positivity and real-rootedness properties in the exponential Riordan arrays and the iteration matrices (section 4),
and the $n$-recursive matrices (section 5), repectively.
Examples include matrices and polynomials formed by the Stirling numbers of both kinds (of type A and type B), the Lah numbers, the idempotent numbers, the Delannoy numbers, and the derangement numbers of type A and type B.

They propose to prove the three statements in Theorem~\ref{thm-main} "simultaneously, by constructing a unified planar network that represent $A$, $\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively, when selecting different sets of sources and sinks." They further state that they will "identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$." Concretely, Section 2 introduces the Lindstr\"om-Gessel-Viennot lemma and planar-network characterizations of totally positive matrices, and Section 3 gives the proofs of Theorem~\ref{thm-main} (and Theorem~\ref{thm-T}) "by constructing a unified planar network."

They propose to prove the three statements in the main theorem simultaneously, by constructing a unified planar network that represent $A$, $\overleftarrow{A}$, and Toeplitz matrices of rows of $A$, respectively, when selecting different sets of sources and sinks. They further state that they will identify the Toeplitz matrix of each row of $A$ with a submatrix of a certain matrix defined by means of the left production matrix $Q(A)$. Concretely, they first introduce the Lindstr\"om-Gessel-Viennot lemma and planar-network characterizations of totally positive matrices, and then they give the proofs of the main theorem and also establish the following theorem.

\begin{thm}\label{thm-T}
	Let $A=[a_{n,k}]_{n,k\ge0}$ be a lower triangular matix,
	and $Q(A)$ be its left production matrix.
	Define
	\setlength{\arraycolsep}{3pt}
	$$
	M_{n,r}:=
	\left[
		\begin{array}{ccc}
			I_0 \\
				& Q_n(A) \\
				&     & I_r
		\end{array}
	\right]
	\left[
		\begin{array}{ccc}
			I_1 \\
				& Q_n(A) \\
				&     & I_{r-1}
		\end{array}
	\right]
	\dots
	\left[
		\begin{array}{ccc}
			I_r \\
				& Q_n(A) \\
				&     & I_0
		\end{array}
	\right],
	$$
	where $I_k$ is the identity matrix of order $k$ with the convention that $I_0$ is an empty block.
	Let $\bal_n:=(a_{n,0},a_{n,1},\dots,a_{n,n},0,0,\dots)$ be the $n$th row of $A$.
	Then for each $r\ge0$, we have
	$$
	T_r(\bal_n)^T=M_{n,r}\submat{n,n+1,n+2,\dots,n+r}{0,1,2,\dots,r},
	$$
	where $T_r(\bal_n)$ is the $r$th leading principal submatrix of $T_{\infty}(\bal_n)$,
	and $^T$ stands for transpose.
\end{thm}

They do this by constructing a unified planar network.

Question:
Let $A=[a_{n,k}]_{n,k\ge 0}$ be a lower triangular matrix, and define its left production matrix by
$$
Q(A):=A\begin{bmatrix}1&O\\ O&A^{-1}\end{bmatrix},
$$
where $O$ denotes zero blocks. A matrix is called totally positive if all of its minors are nonnegative, and the reversal of $A$ is the lower triangular matrix $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge 0}$. Which statement holds for every such matrix $A$ for which $Q(A)$ is totally positive?

$A$ is totally positive, $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive, and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ is real-rooted.

Choices:
B. $A$ is totally positive and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ is real-rooted, but the reversal $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ need not be totally positive.
C. $A$ is totally positive and $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive.
D. $A$ is totally positive, $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive, and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ has only nonnegative coefficients.
E. If $A$ is totally positive, $\overleftarrow{A}=[a_{n,n-k}]_{n,k\ge0}$ is totally positive, and for every $n\ge0$ the row generating polynomial $\sum_{k=0}^n a_{n,k}x^k$ is real-rooted, then $Q(A)$ is totally positive.

unified planar-network transfer to the reversal

dropped the real-rootedness conclusion for row generating polynomials

real-rootedness of row generating polynomials replaced by coefficient nonnegativity

reversed the implication from consequences back to total positivity of the left production matrix