\begin{theorem}\label{main}
	Let $G$ be a controllable graph and $p$ be an odd prime such that  $\rank_p W(G)=n-1$. Then $v_p(L(G))\le \frac{1}{2} v_p(\det W(G))$.
\end{theorem}
	The main innovation of this paper lies in the shift of the underlying algebraic structure from the vector space over the field $\mathbb{Z}/p\mathbb{Z}$ (as used in \cite{wang2013EJC,qiu2023}) to the module over the local ring $\mathbb{Z}/p^k\mathbb{Z}$. While working over a ring introduces the complexity of zero divisors, many basic results from linear algebra over fields still hold for the ring $\mathbb{Z}/p^k\mathbb{Z}$. In particular, it possesses the ``Basis Extension Property'' (Steinitz Property), which allows us to generalize arguments from linear algebra.

	The rest of this paper is organized as follows.  In Sec.~\ref{pre}, we recall some  preliminaries on Smith normal forms and modules over $\mathbb{Z}/p^k \mathbb{Z}$.  The proof of Theorem \ref{main} is given in Sec.~\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \cite{wang2006}, combined with some improvements and simplifications developed in \cite{wang2020}.   Some direct applications of Theorem \ref{main} are presented in Sec.~\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \ref{main}.
\section{Preliminaries}
\label{pre}
Let $R$ be a commutative principal ideal ring with identity.  It is well known \cite{stanley2016JCTA} that any matrix $M$ over $R$ has a Smith normal form (SNF); that is, there exist invertible matrices $U$ and $V$ over $R$ such that
\begin{equation*}
	UMV=\begin{bmatrix}
		\diag(d_1,\ldots,d_r)&0\\
		0&0
	\end{bmatrix},
\end{equation*}
where the elements $d_i$ ($i=1,\ldots,r$) are nonzero and satisfy $d_1\mid d_2\mid \cdots\mid d_r$. The nonzero elements $d_1,\ldots,d_r$ are called the \emph{invariant factors} of $M$.

Throughout this paper, we fix an odd prime $p$ and consider three kinds of rings: $\mathbb{Z}$, $\mathbb{Z}/p\mathbb{Z}$ and $\mathbb{Z}/p^k\mathbb{Z}$, where $k\ge 2$. Let $\mathbb{Z} \to \mathbb{Z}/p^k \mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}$ be the natural projections. For a matrix $M$ over $\mathbb{Z}$, the SNFs of the projections of $M$ are determined naturally by the SNF of $M$ over $\mathbb{Z}$ via $p$-adic valuations. For example, suppose $M$ has the Smith normal form $\diag(2,10,30,270)$ over $\mathbb{Z}$. Then, over the field $\mathbb{Z}/3\mathbb{Z}$, the SNF of $M$ is $\diag(1,1,0,0)$, whereas over the local ring $\mathbb{Z}/3^2\mathbb{Z}$, its SNF is $\diag(1,1,3,0)$.

In this paper, we are mainly concerned with the SNFs of matrices over the local ring $\mathbb{Z}/p^k \mathbb{Z}$. Note that the invariant factors are unique only up to multiplication by units. To ensure uniqueness, we conventionally assume that each nonzero invariant factor takes the form $p^{c}$ for some integer $0 \le c < k$. We formalize this in the following lemma.

\begin{lemma}
	For any matrix $M$ over $\mathbb{Z}/p^k\mathbb{Z}$, there exist invertible matrices $U$ and $V$ over $\mathbb{Z}/p^k\mathbb{Z}$ such that
	\begin{equation*}
		UMV = \begin{bmatrix}
			\diag(p^{c_1}, \ldots, p^{c_r})&0\\
			0&0
		\end{bmatrix},
	\end{equation*}
	where $0 \le c_1 \le \cdots \le c_r < k$.
\end{lemma}
A classic result in linear algebra states that the linear system $Mx=b$ over a field has a solution if and only if $\rank M = \rank (M,b)$. The corresponding question over an arbitrary ring is generally more complicated, but for our purposes, the existence of the SNF gives a straightforward extension.

\begin{proposition}\label{ssnf}
	The linear system $Mx=b$ over $\mathbb{Z}/p^k\mathbb{Z}$ has a solution if and only if $M$ and $(M,b)$ have the same invariant factors.
\end{proposition}

\begin{proof}
	The ``only if'' part is clear. Let $\mathcal{C}(M)$ and $\mathcal{C}(M,b)$ be the modules generated by the columns of $M$ and $(M,b)$, respectively. Since $M$ and $(M,b)$ have the same invariant factors, the two modules $\mathcal{C}(M)$ and $\mathcal{C}(M,b)$ are isomorphic. As $\mathcal{C}(M,b)$ is finite, it cannot be isomorphic to any proper submodule. Since $\mathcal{C}(M)$ is a submodule of $\mathcal{C}(M,b)$, the isomorphism $\mathcal{C}(M)\cong \mathcal{C}(M,b)$ implies that they must be equal. Thus, $b\in\mathcal{C}(M)$, i.e., $Mx=b$ has a solution.
\end{proof}
The SNF is particularly useful for determining the structure of the solution space (kernel) of a linear system. It is important to note that the kernel is not always a free module. However, under specific conditions on the invariant factors, it is.

\begin{proposition}\label{kf}
	Let $M$ be an $m \times n$ matrix over the ring $\mathbb{Z}/p^k\mathbb{Z}$. Let the invariant factors of $M$ be $p^{c_1}, \ldots, p^{c_r}$ satisfying $0 \le c_1 \le \dots \le c_r < k$. Then the kernel of $M$ is given, up to isomorphism, by:
	\begin{equation*}
		\ker(M) \cong \bigoplus_{i=1}^r \left( \mathbb{Z}/p^{c_i}\mathbb{Z} \right) \oplus \left( \mathbb{Z}/p^k \mathbb{Z} \right)^{n-r}.
	\end{equation*}
	In particular, if $c_1=\cdots=c_r=0$, then $\ker(M)\cong (\mathbb{Z}/p^k\mathbb{Z})^{n-r}$, which is a free module of rank $n-r$.
\end{proposition}

The following proposition is a direct generalization of \cite[Lemma 7]{wang2013EJC}, where the case $m=n,k=2$ was considered. The original proof in \cite{wang2013EJC} remains valid for the general setting and is therefore omitted here.

\begin{proposition}[\cite{wang2013EJC}]\label{dnk}
	Let $M$ be an $m\times n~(m\ge n)$ integral matrix whose SNF is
	\begin{equation*}
		\begin{bmatrix}
		\diag(d_1,\ldots,d_n)\\
		0_{(m-n)\times n}
		\end{bmatrix}.
	\end{equation*} Then the equation $Mz\equiv 0\pmod{p^{k}}$ has a solution $z\not\equiv 0\pmod{p}$ if and only if $p^{k}\mid d_n.$
\end{proposition}

A set of vectors $v_1, \ldots, v_m \in (\mathbb{Z}/p^k\mathbb{Z})^n$ is called \emph{linearly independent} if $\sum_{i=1}^m c_i v_i = 0$ implies $c_i = 0$ for all $i$. It is known that $v_1,\ldots, v_m$ are linearly independent if and only if their projections in the vector space $(\mathbb{Z}/p\mathbb{Z})^n$ are linearly independent (over the field $\mathbb{Z}/p\mathbb{Z}$). A key property of the ring $\mathbb{Z}/p^k\mathbb{Z}$ is the so-called Basis Extension Property (or Steinitz Property \cite{mcdonald}), which states that any linearly independent vectors in a free $\mathbb{Z}/p^k\mathbb{Z}$-module $M$ can always be extended to a basis of $M$. Since we only require the finite-rank case, we formalize it in the following proposition.

\begin{proposition}\label{bas}
	Let $M$ be a free submodule of rank $m$ in $(\mathbb{Z}/p^k\mathbb{Z})^n$. Let $v_1, \ldots, v_k \in M$ ($k\le m$) be linearly independent. Then:

	\textup{(i)} If $k=m$, the set $\{v_1, \ldots, v_k\}$ constitutes a basis of $M$.

	\textup{(ii)}  If $k< m$, there exist $m-k$ vectors $v_{k+1}, \ldots, v_{m} \in M$ such that $\{v_1,\ldots,v_m\}$ constitutes a basis of $M$.
\end{proposition}
\section{Proof of Theorem \ref{main}}\label{pf}
We begin with the fundamental matrix characterization of generalized cospectrality for graphs.

\begin{lemma}[\cite{johnson,wang2006}]
	Let $G$ and $H$ be two graphs of the same order. Then $G$ and $H$ are generalized cospectral if and only if there exists a regular orthogonal matrix $Q$ such that $Q^\mathsf{T} A(G) Q=A(H)$. Moreover, if $G$ is controllable, then $Q^\mathsf{T}=W(H)(W(G))^{-1}$, and hence $Q$ is unique and rational.
\end{lemma}

In the following, let $G$ be an $n$-vertex controllable graph and $p$ be an odd prime factor of $\det W(G)$ such that $\rank_p W(G)=n-1$. Denote $\tau=v_p(L(G))$, i.e., $\tau=\max\{v_p (\ell(Q))\colon\, Q\in \mathcal{Q}(G)\}$. Note that if $\tau=0$, then Theorem \ref{main} clearly holds. Thus, we may assume that $\tau\ge 1$. For simplicity, the adjacency matrix $A(G)$ and the walk matrix $W(G)$ will be denoted by $A$ and $W$, respectively, when there is no confusion.

\begin{lemma}\label{fourcong}
	There exist an integral vector $z_0$ with $z_0\not\equiv 0\pmod{p}$ and an integer $\lambda_0$ such that $z_0^\mathsf{T} z_0\equiv 0\pmod{p^{2\tau}}$, $z_0^\mathsf{T} A z_0\equiv 0\pmod{p^{2\tau}}$, $W^\mathsf{T} z_0\equiv 0\pmod{p^\tau}$, and $Az_0\equiv \lambda_0z_0\pmod{p^\tau}$.
\end{lemma}

\begin{proof}
	Let $Q\in \mathcal{Q}(G)$ be such that $v_p(\ell(Q))=\tau$, and let $H$ be the corresponding graph. Let $\ell=\ell(Q)$ and $\hat{Q}=\ell Q$. Then $\hat{Q}$ is an integral matrix and $\hat{Q}\not\equiv 0\pmod{p}$. Let $z_0$ be a column of $\hat{Q}$ such that $z_0\not\equiv 0\pmod{p}$. Since $Q$ is an orthogonal matrix and $p^\tau\mid \ell$, we have $\hat{Q}^\mathsf{T} \hat{Q}=\ell^2 I$, and hence $z_0^\mathsf{T} z_0\equiv 0\pmod{p^{2\tau}}$. Similarly, as $Q^\mathsf{T} A Q$ is a $(0,1)$-matrix, we have $\hat{Q}^\mathsf{T} A\hat{Q}\equiv 0\pmod{p^{2\tau}}$, and hence $z_0^\mathsf{T} A z_0\equiv 0\pmod{p^{2\tau}}$. Moreover, noting that $W^\mathsf{T} Q$ equals $(W(H))^\mathsf{T}$, which is an integral matrix, we find that $W^\mathsf{T} \hat{Q}\equiv 0\pmod{\ell}$, and hence $W^\mathsf{T} z_0\equiv 0\pmod{p^\tau}$.

	Let $M=\{z\in \mathbb{Z}^n\colon\, W^\mathsf{T} z\equiv 0\pmod {p^\tau}\}$. Since $\rank_p W=n-1$ and $p^\tau \mid \det W$, we see that, over $\mathbb{Z}/p^\tau \mathbb{Z}$, the SNF of $W^\mathsf{T}$ is $\diag(1,1,\ldots,1,0)$. It follows from Proposition \ref{kf} that $M$ is a free $\mathbb{Z}/p^\tau \mathbb{Z}$-module of rank one. Since $z_0\in M$ and $z_0\not\equiv 0\pmod{p}$, we see that $\{z_0\}$ forms a basis for $M$ over $\mathbb{Z}/p^\tau \mathbb{Z}$ by Proposition \ref{bas} (i). On the other hand, by the Cayley-Hamilton Theorem, it is easy to see that $M$ is an $A$-invariant $\mathbb{Z}/p^\tau \mathbb{Z}$-submodule. This implies that there exists an integer $\lambda_0$ such that $Az_0\equiv \lambda_0 z_0\pmod{p^\tau}$. This completes the proof.
\end{proof}
\begin{lemma}\label{wy}
	If $(A-\lambda_0 I)y\equiv s p^j z_0\pmod{p^{j+\tau}}$ for some integer $s$ and integer $j\ge 0$, then
	\begin{equation*}
		W^\mathsf{T} y\equiv (e^\mathsf{T} y)(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T} \pmod{p^{j+\tau}}.
	\end{equation*}
\end{lemma}

\begin{proof}
	We proceed by induction on $k$ to show that $e^\mathsf{T} A^k y\equiv \lambda_0^k e^\mathsf{T} y\pmod{p^{j+\tau}}$ for $k=0,1,\ldots,n-1$. The base case $k=0$ is trivial. Assume that the congruence holds for some integer $k$ with $0 \le k < n-1$; we proceed to verify it for $k+1$.

	By Lemma \ref{fourcong}, we have $W^\mathsf{T} z_0\equiv 0\pmod{p^\tau}$, which implies that $e^\mathsf{T} A^k z_0\equiv 0\pmod{p^\tau}$. Consequently, we obtain $e^\mathsf{T} A^k (sp^j z_0)\equiv 0\pmod{p^{j+\tau}}$. By the hypothesis of this lemma, $Ay \equiv  sp^j z_0+\lambda_0 y \pmod{p^{j+\tau}}$, which implies that
	\[
	e^\mathsf{T} A^{k+1}y \equiv e^\mathsf{T} A^k (sp^j z_0+\lambda_0 y) \equiv \lambda_0 e^\mathsf{T} A^k y \equiv \lambda_0^{k+1}e^\mathsf{T} y \pmod{p^{j+\tau}}.
	\]
	This completes the proof.
\end{proof}
\begin{lemma}\label{rA}
	Let $S=\diag(f_1,\ldots,f_n)$ be the SNF of $A-\lambda_0 I$. Then $f_{n-2}\not \equiv 0\pmod{p}$ and $f_n\equiv 0\pmod{p^\tau}$.
\end{lemma}

\begin{proof}
	By Lemma \ref{fourcong}, we have $(A-\lambda_0 I)z_0\equiv 0\pmod{p^\tau}$ and $z_0\not\equiv 0\pmod{p}$. It follows from Proposition \ref{dnk} that $f_n\equiv 0\pmod{p^\tau}$. 	It remains to show that $f_{n-2}\not \equiv 0\pmod{p}$. Suppose to the contrary that $f_{n-2}\equiv 0\pmod{p}$. Then $\rank_p (A-\lambda_0 I)\le n-3$.
	Let \begin{equation*}
	B=\begin{bmatrix} A-\lambda_0 I\\e^\mathsf{T}\end{bmatrix}.\end{equation*}
	Then, we have $\rank_p B\le n-2$. Recall from Lemma \ref{fourcong} that $W^\mathsf{T} z_0 \equiv 0 \pmod{p}$, which implies $e^\mathsf{T} z_0 \equiv 0 \pmod{p}$. Together with $(A-\lambda_0 I)z_0 \equiv 0 \pmod{p}$, this yields $Bz_0 \equiv 0 \pmod{p}$. Since the nullity of $B$ over $\mathbb{Z}/p\mathbb{Z}$ is at least $n - (n-2) = 2$, the equation $Bz\equiv 0\pmod {p}$ must have a solution $z_1$ such that $z_0$ and $z_1$ are linearly independent over $\mathbb{Z}/p\mathbb{Z}$.

	Noting that $Az_1\equiv \lambda_0 z_1 \pmod{p}$ and $e^\mathsf{T} z_1\equiv 0\pmod{p}$, we clearly have
	\[
	e^\mathsf{T} A^{k}z_1\equiv \lambda_0^k e^\mathsf{T} z_1\equiv 0\pmod{p}
	\]
	for $k=0,1,\ldots, n-1$. This means that $W^\mathsf{T} z_1\equiv 0\pmod{p}$. By Lemma \ref{fourcong}, we also have $W^\mathsf{T} z_0\equiv 0\pmod{p}$. Therefore, $\rank_p W^\mathsf{T} \le n-2$. This contradicts the assumption that $\rank_p W = n-1$, which completes the proof.
\end{proof}
\begin{lemma} \label{lc}
		Let $M = [A-\lambda_0 I, z_0]$. Then $z_0^\mathsf{T} M \equiv 0 \pmod{p^\tau}$ and the SNF of $M$ is $[\diag(I_{n-1},0),0]$. Moreover, any integral  vector $z$ satisfying $z_0^\mathsf{T} z \equiv 0 \pmod{p^\tau}$ can be expressed as a linear combination of the columns of $M$ over $\mathbb{Z}/p^\tau \mathbb{Z}$.
	\end{lemma}
	\begin{proof} By Lemma \ref{fourcong}, we have $(A-\lambda_0 I)z_0\equiv 0\pmod{p^\tau}$ and $z_0^\mathsf{T} z_0\equiv 0\pmod{p^\tau}$. As  $A-\lambda_0 I$ is symmetric, we find that
		\begin{equation}\label{mz}
			M^\mathsf{T} z_0=\begin{bmatrix}
				A-\lambda_0 I\\
				z_0^\mathsf{T}
			\end{bmatrix}z_0\equiv 0\pmod{p^\tau}.
		\end{equation} Let the SNF of $M$ be $S= [\diag(p^{c_1},p^{c_2}, \dots, p^{c_n}), 0]$. Clearly, the SNF of $M^\mathsf{T}$ is $S^\mathsf{T}$. Note that $z_0\not\equiv 0\pmod{p}$. It follows from Eq.~\eqref{mz} and Proposition \ref{dnk}  that $p^{c_n}$ is zero over $R$. Thus, the SNF of $M$ can be simplified as
		\begin{equation}\label{ss}
			S=[\diag(p^{c_1},\ldots,p^{c_{n-1}},0),0].
		\end{equation}

\noindent\textbf{Claim}: $p^{c_{n-1}}$ is a unit in $R$, i.e., $c_{n-1}= 0$.

		Suppose to the contrary that $c_{n-1}\ge 1$. Then, we have $\rank_p S\le n-2$, or equivalently, $\rank_p M\le n-2$. On the other hand, by Lemma \ref{rA}, we see that $\rank_p (A-\lambda_0 I)\ge n-2$ and hence  $\rank_p M \ge n-2$. Thus, we must have  $\rank_p M=\rank_p (A-\lambda_0 I)=n-2$. It follows that there exists an integral vector $z_1$ such that $(A-\lambda_0 I)z_1\equiv z_0\pmod{p}$.

		As $\rank_p  (A-\lambda_0 I)=n-2$, $(A-\lambda_0I)z\equiv 0\pmod{p}$ has two  solutions $z_2$ and $z_3$ that are linearly  independent over $\mathbb{Z}/p\mathbb{Z}$. Since $(A-\lambda_0 I)z_1\equiv z_0\not \equiv 0\pmod{p}$, $z_1$ cannot be written as a linear combination of $z_2$ and $z_3$. This implies that $z_1,z_2,z_3$ are linearly independent. Consider the equation $e^\mathsf{T}(k_1z_1+k_2z_2+k_3z_3)\equiv 0 \pmod{p}$ with three unknowns $k_1,k_2,k_3$. Clearly, it has at least two independent solutions over $\mathbb{Z}/p\mathbb{Z}$. Let $(a_1,a_2,a_3)^\mathsf{T}$ and  $(b_1,b_2,b_3)^\mathsf{T}$  be  two such solutions and write $\alpha=a_1z_1+a_2z_2+a_3z_3$ and $\beta=b_1z_1+b_2z_2+b_3z_3$. It is easy to see that $\alpha$ and $\beta$ are linearly independent over $\mathbb{Z}/p\mathbb{Z}$. Note that $(A-\lambda_0I)\alpha\equiv a_1z_0$ and $e^\mathsf{T}\alpha\equiv 0\pmod{p}$. Using a similar argument as in the proof of  Lemma~\ref{wy}, we find  that $W^\mathsf{T}\alpha \equiv 0\pmod{p}$. Also,  $W^\mathsf{T} \beta\equiv 0\pmod{p}$. Thus, we have found two linearly independent solutions of $W^\mathsf{T} z\equiv 0\pmod{p}$. This contradicts the fact that $\rank_p W^\mathsf{T}=n-1$ and hence completes the proof of the Claim.

		By the Claim, we see that Eq.~\eqref{ss} can be further reduced to
		\begin{equation}\label{s3}
			S=[\diag(I_{n-1},0),0].
		\end{equation}
		Let $\mathcal{C}(M)$ be the $R$-module generated by the columns of $M$ and $N$ be the module $\{z\in R^n\colon\, z_0^\mathsf{T} z=0\}$. By Eq.~\eqref{s3}, we know that $\mathcal{C}(M)$ is isomorphic to the free module $R^{n-1}$. Since $z_0\not\equiv 0\pmod{p}$, we find that $N$ is also isomorphic to $R^{n-1}$. Thus, the two $R$-modules $\mathcal{C}(M)$ and $N$ are isomorphic. Since $z_0^\mathsf{T} M=0$ over $R$, we know that $\mathcal{C}(M)$ is a submodule of $N$. As $R$ is a finite ring, we must have $\mathcal{C}(M)=N$, which completes the proof of Lemma \ref{lc}.
	\end{proof}

\begin{lemma}\label{ez}
	There exists an  integral vector $z_1$ such that $e^\mathsf{T} z_1\not\equiv 0\pmod {p}$ and $(A-\lambda_0 I)z_1\equiv p^cz_0\pmod{p^\tau}$ for some $c\in \{0,1,\ldots,\tau\}$.
\end{lemma}
\begin{proof}
	Let $R=\mathbb{Z}/p^\tau \mathbb{Z}$.  By Lemma \ref{rA}, we know that the SNF of $A-\lambda_0 I$ is
	\begin{equation}\label{smA}
		S=\diag(1,1,\ldots,1,p^c,0) \text{~for some~} c\in\{0,1,\ldots,\tau\}.
	\end{equation}
Let $M_i=[A-\lambda_0 I, p^i z_0]$ for $i\in\{0,1,\ldots,c\}$.

\noindent\textbf{Claim 1}: The SNF of $M_i$ is $[\diag(1,1,\ldots,1,p^i,0),0]$ for $i\in\{0,1,\ldots,c\}$.

Let $S_i$ be the SNF of $M_i$. Noting that $\rank_p M_i\ge \rank_p (A-\lambda_0 I)\ge n-2$, we conclude that $S_i$ must have the form
\begin{equation}\label{si}
	S_i=[\diag(1,\ldots,1,p^{t_{i}},p^{t'_{i}}),0].
\end{equation}
Since $ M_i^\mathsf{T} z_0\equiv 0\pmod{p^\tau}$ and $z_0\not\equiv 0\pmod{p}$, Proposition \ref{dnk} indicates that $p^{t'_i}$ is the zero element in $R$. Thus, Eq.~\eqref{si} can be simplified as
\begin{equation}\label{st}
	S_i=[\diag(1,\ldots,1,p^{t_{i}},0),0].
\end{equation}
We need to show that $t_i=i$ for each $i\in\{0,1,\ldots,c\}$. Clearly, $t_0=0$ by Lemma \ref{lc}. Thus, it suffices to show that the sequence $\{t_i\}_{0\le i\le c}$ satisfies
\begin{equation}\label{tc}
	t_c=c
\end{equation}
and
\begin{equation}\label{icr}
	t_i\le t_{i+1}\le t_{i}+1\text{~for~} 0\le i<c.
\end{equation}
Now regard $A-\lambda_0 I$ and $M_i$ as matrices over $\mathbb{Z}$. Let $\Delta$ and $\Delta^{(i)}$ be the $(n-1)$-th determinantal factors of $A-\lambda_{0}I$ and $M_i$, respectively. Note that Eq.~\eqref{tc} holds for the case $c=\tau$. Indeed, if $c=\tau$ then $M_c=[A-\lambda_0 I,p^\tau z_0]=[A-\lambda_0 I,0]$ over $R$, which implies that the SNF of $M_c$ is $S_c=[S,0]$, i.e., $t_c=c$ as desired. Now assume $c<\tau$. From Eq.~\eqref{smA}, we know that $v_p(\Delta)=c$. Since $M_c$ contains $A-\lambda_0 I$ as a submatrix, we must have $v_p(\Delta^{(c)})\le v_p(\Delta)$ and hence  $v_p(\Delta^{(c)})\le c$. Let $D$ be any nonzero $(n-1)$-th minor of $M_c$. If $D$ does not include the last column of $M_c$, then $D$ is also a minor of $A-\lambda_0 I$ and hence $v_p(D)\ge v_p(\Delta)\ge c$. Otherwise,  each entry of the last column of $D$ is a multiple of $p^c$ and hence $v_p(D)\ge c$. Thus, we always have $v_p(D)\ge c$, which implies  $v_p(\Delta^{(c)})\ge c$. This means $v_p(\Delta^{(c)})= c$. But by Eq.~\eqref{st} for the case $i=c$, we know that $v_p(\Delta^{(c)})=t_c$ and hence $t_c=c$. This proves Eq.~\eqref{tc}.

We proceed to verify Eq.~\eqref{icr}. We may assume $c>0$ since otherwise we have nothing to show. Let $D^{(i)}$ and $D^{(i+1)}$ be any two corresponding $(n-1)$-th minors of $M_i$ and $M_{i+1}$, respectively. By the constructions of $M_i$ and $M_{i+1}$, we have either $D^{(i+1)}=D^{(i)}$ or $D^{(i+1)}=pD^{(i)}$. It follows that $v_p(\Delta^{(i)})\le v_p(\Delta^{(i+1)}) \le v_p(\Delta^{(i)}) +1$, i.e., $t_i\le t_{i+1}\le t_i+1$. Thus,  Claim 1 follows.

To complete the proof of Lemma \ref{ez}, we consider the following three cases:

\noindent\emph{Case 1}:  $c=0$. Since $A-\lambda_0 I$ and $M_0=[A-\lambda_0 I,z_0]$ have the same invariant factors over $R$, Proposition \ref{ssnf} implies that there exists an integral vector $z_1$ such that  \begin{equation}\label{as}
(A-\lambda_0 I)z_1\equiv z_0\pmod{p^\tau}.
\end{equation} By Lemma \ref{wy}, we have
	\begin{equation*}
	W^\mathsf{T} z_1\equiv e^\mathsf{T} z_1(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T} \pmod{p^{\tau}}.
\end{equation*}
Suppose to the contrary that $e^\mathsf{T} z_1\equiv 0\pmod {p}$. Then we have $W^\mathsf{T} z_1\equiv 0\pmod{p}$.  Since $\rank_p W=n-1$ and $z_0$ is a nonzero (over $\mathbb{Z}/p\mathbb{Z}$) solution of $W^\mathsf{T} z\equiv 0\pmod{p}$, we conclude that $z_1\equiv kz_0\pmod{p}$ for some integer $k$. But this would imply
\begin{equation*}
(A-\lambda_0 I) z_1\equiv k(A-\lambda_0 I) z_0\equiv 0\pmod{p},
\end{equation*}
which, combining with Eq.~\eqref{as}, leads to $z_0\equiv 0\pmod{p}$. This is a contradiction and hence we must have  $e^\mathsf{T} z_1\not\equiv 0\pmod {p}$.

\noindent\emph{Case 2}: $c=\tau$. As the SNF of $A-\lambda_0 I$ is $S=\diag(I_{n-1},0,0)$, we know that the kernel of $(A-\lambda_0 I)$ is a free $R$-module of rank $2$ by Proposition \ref{kf}. Let $K=\ker (A-\lambda_0 I)$ be the kernel. Since $z_0\in K$ and $z_0\not\equiv 0\pmod{p}$, Proposition \ref{bas} (ii) implies that there exists an integer vector $z_1$ such that $\{z_0,z_1\}$ (over $R$) constitutes a basis of the free module $K$. In particular, $(A-\lambda_0 I)z_1\equiv 0\pmod{p^\tau}$. Using Lemma \ref{wy}, we conclude that
	\begin{equation*}
	W^\mathsf{T} z_1\equiv e^\mathsf{T} z_1(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T} \pmod{p^{\tau}}.
\end{equation*}
If $e^\mathsf{T} z_1\equiv 0\pmod{p}$ then we would have 	$W^\mathsf{T} z_1\equiv 0\pmod{p}$. But since $z_0$ and $z_1$ are independent over the field $\mathbb{Z}/p\mathbb{Z}$, we must have $\rank_p W^\mathsf{T}\le n-2$. This is a contradiction and hence $e^\mathsf{T} z_1\not\equiv 0\pmod{p}$.

\noindent{\emph{Case 3}}:  $0<c<\tau$. By Claim 1, we see that $A-\lambda_0 I$ and $M_c=[A-\lambda_0 I, p^c z_0]$ have the same invariant factors. By Proposition \ref{ssnf}, there exists an integral vector $z_1$ such that
\begin{equation}\label{az1}
	(A-\lambda_0 I)z_1\equiv p^c z_0\pmod{p^\tau}.
\end{equation}

\noindent\textbf{Claim 2}: $z_1$ and $z_0$ are linearly independent over $\mathbb{Z}/p\mathbb{Z}$.

Suppose to the contrary that $z_1$ and $z_0$ are linearly dependent over $\mathbb{Z}/p\mathbb{Z}$. As $z_0\not\equiv 0\pmod{p}$, there exist an integer $k$ and an integral vector $\tilde{z}_1$ such that $z_1=kz_0+p\tilde{z}_1$. Noting that $(A-\lambda_0 I)z_0\equiv 0\pmod{p^\tau}$, we obtain from Eq.~\eqref{az1} that $(A-\lambda_0 I)p\tilde{z}_1\equiv p^c z_0\pmod{p^\tau}$, i.e.,
\begin{equation}\label{az1s}
	(A-\lambda_0 I)\tilde{z}_1\equiv p^{c-1} z_0\pmod{p^{\tau-1}}.
	\end{equation}
Let $R'=\mathbb{Z}/p^{\tau-1}\mathbb{Z}$. By Eq.~\eqref{az1s}, we see that, over $R'$, the two matrices $A-\lambda_0 I$ and $M_{c-1}=[A-\lambda_0 I, p^{c-1}z_0]$ have the same invariant factors. Since $c\le \tau-1$, we see that $p^c$ and $p^{c-1}$ are not associates over $R'$. Thus, Claim 1 means that $A-\lambda_0 I$ and $M_{c-1}=[A-\lambda_0 I, p^{c-1}z_0]$ cannot have the same invariant factors. This contradiction completes the proof of Claim 2.

By Eq.~\eqref{az1} and Lemma \ref{wy}, we have
\begin{equation*}
	W^\mathsf{T} z_1\equiv e^\mathsf{T} z_1(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T}\pmod{p^\tau}.
\end{equation*}
By Claim 2 and the same argument as in Case 2, we must have $e^\mathsf{T} z_1\not\equiv 0\pmod{p}$.

Combining the three cases,  Lemma \ref{ez} follows.
\end{proof}
Now we are in a position to present a proof of Theorem \ref{main}.

\begin{proof}[Proof of Theorem \ref{main}]  Let $\tau=v_p(L(G))$. Let $z_0$ and $\lambda_0$ be the integral vector and the integer as described in  Lemma \ref{fourcong}. Then we have
\begin{equation*}
	z_0^\mathsf{T}(A-\lambda_0 I)z_0\equiv 0\pmod{p^{2\tau}}
\end{equation*}
and hence
\begin{equation*}
	z_0^\mathsf{T}\frac{(A-\lambda_0 I)z_0}{p^\tau}\equiv 0\pmod{p^{\tau}}.
\end{equation*}
It follows from Lemma \ref{lc} that
\begin{equation}\label{aps}
\frac{(A-\lambda_0 I)z_0}{p^\tau}\equiv (A-\lambda_0 I)y+s z_0\pmod{p^{\tau}}
\end{equation}
for some $y\in \mathbb{Z}^{n}$ and $s\in \mathbb{Z}$.
Multiplying both sides of Eq.~\eqref{aps} by $p^\tau$ and rearranging the terms, we obtain
\begin{equation*}
	(A-\lambda_0 I)(z_0-p^\tau y)\equiv sp^\tau z_0\pmod{p^{2\tau}}.
\end{equation*}
Consequently, by Lemma \ref{wy}, we have
\begin{equation}\label{wp}
	W^\mathsf{T} (z_0-p^\tau y)\equiv e^\mathsf{T} (z_0-p^\tau y)(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T}\pmod{p^{2\tau}}.
\end{equation}
By Lemma \ref{ez}, there exist an integral vector $z_1$ and an integer $c\in\{0,1,\ldots,\tau\}$ such that $e^\mathsf{T} z_1\not\equiv 0\pmod{p}$ and
 \begin{equation}\label{apta}
(A-\lambda_0 I)z_1\equiv p^c z_0\pmod{p^\tau}.
\end{equation} As $e^\mathsf{T} (z_0-p^\tau y)\equiv e^\mathsf{T} z_0\equiv 0\pmod{p^\tau}$, we see that ${p^{-\tau}}e^\mathsf{T} (z_0-p^\tau y) $ is an integer. Note that $e^\mathsf{T} z_1$ is a unit in $\mathbb{Z}/p^{\tau}\mathbb{Z}$. Thus, there exists some integer $g$ such that
\begin{equation}\label{ep}
	\frac{e^\mathsf{T} (z_0-p^\tau y)}{p^\tau}\equiv g e^\mathsf{T} z_1\pmod{p^{\tau}}.
\end{equation}
We know from Eq.~\eqref{apta} and Lemma \ref{wy} that
\begin{equation*}
	W^\mathsf{T} z_1\equiv e^\mathsf{T} z_1(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T} \pmod{p^\tau}.
\end{equation*}
This, together with Eq.~\eqref{ep}, leads to
\begin{equation*}
	W^\mathsf{T} (g z_1)\equiv g e^\mathsf{T} z_1(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T}\equiv 	\frac{e^\mathsf{T} (z_0-p^\tau y)}{p^\tau}(1,\lambda_0,\ldots,\lambda_0^{n-1})^\mathsf{T} \pmod{p^\tau}.
\end{equation*}
Multiplying both sides by $p^\tau$ and using Eq.~\eqref{wp}, we find that
\begin{equation*}
W^\mathsf{T} (gp^\tau z_1) \equiv W^\mathsf{T} (z_0-p^\tau y)\pmod{p^{2\tau}},
\end{equation*}
i.e.,
\begin{equation*}
W^\mathsf{T} (z_0-p^\tau y-gp^\tau z_1) \equiv 0\pmod{p^{2\tau}}.
\end{equation*}
As $z_0-p^\tau y-g p^\tau z_1\equiv z_0\not\equiv 0\pmod{p}$, Proposition \ref{dnk} implies that $d_n\equiv 0\pmod{p^{2\tau}}$, where $d_n$ is the $n$-th invariant factor of the integral matrix $W^\mathsf{T}$. As $\rank_p W=n-1$, we have $v_p(\det W) =v_p(d_n)$ and hence $v_p(\det W)\ge 2\tau$. This completes the proof of Theorem \ref{main}.
\end{proof}
\section{Applications and future work}\label{dis}
As a natural extension of the arithmetic criterion of DGS-graphs established in \cite{wang2017JCTB}, Wang et al. \cite{wang2023ejc} defined, for each odd prime $p$,  a family of graphs
 \begin{equation*}
\mathcal{F}_{n,p}=\{\text{$n$-vertex graphs~} G\colon\, 2^{-\lfloor n/2 \rfloor}\det W =p^2b\text{~and~} \rank_p W=n-1\},
 \end{equation*}
where $b$ is odd, square-free and $p\nmid b$. It was shown that each graph in $\mathcal{F}_{n,p}$ has at most one generalized cospectral mate \cite{wang2023ejc}. Using Theorem \ref{main}, we can show the same conclusion for a larger family of graphs:
 \begin{equation*}
 \widetilde{\mathcal{F}}_{n,p}=\{\text{$n$-vertex graphs~} G\colon\, 2^{-\lfloor n/2 \rfloor}\det W \in\{p^2b, p^3b\} \text{~and~} \rank_p W=n-1\},
 \end{equation*}
 where $p$ and $b$ satisfy the same restrictions. Indeed, let $G\in \widetilde{\mathcal{F}}_{n,p}$ and  $Q$ be any matrix in $\mathcal{Q}(G)$. Then, by Propositions \ref{oddcase} and \ref{evencase}, the only possible prime factor of $\ell(Q)$ is $p$. It follows from Theorem \ref{main} that
 \begin{equation*}
v_p(\ell(Q))\le \frac{1}{2}v_p(\det W)\le 3/2,
 \end{equation*}
 i.e., $v_p(\ell(Q))\in\{0,1\}$. In other words, $Q$ is either a permutation matrix, or a matrix with level exactly $p$. Now, the original argument in \cite{wang2023ejc} shows that $G$ has at most one generalized cospectral mate; refer to \cite{wang2023ejc} for details.

 The following small example illustrates the optimality of the family $\widetilde{\mathcal{F}}_{n,p}$ in the sense that $G$ may have two or more generalized cospectral mates if the $v_p(\det W)\ge 4$ for some odd prime $p$.

 \noindent\textbf{Example 1}. Let $n=10$ and $G$ be the $n$-vertex graph with adjacency matrix
\begin{equation*}
	A={\footnotesize\begin{bmatrix}
		0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\
	1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\
	0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\
	1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
	1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
	0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
	1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
	0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\
	0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
	1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
	\end{bmatrix}}.
\end{equation*}
It can be verified  that $2^{-\lfloor\frac{n}{2}\rfloor} \det W=3^4\times 19$ and $\rank_3 W=n-1$. By Theorem \ref{main}, we know that $v_3(L(G))\le \frac{1}{2} v_3(\det W)=2$. Indeed, using the program in \cite{wang2025}, we find that, besides the permutation matrices,  there exist essentially two rational regular orthogonal matrices
\begin{equation*}
	Q_1=\frac{1}{3}{\footnotesize
	\begin{bmatrix}
		3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
		0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
		0 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 1 \\
		0 & 0 & 0 & 0 & 1 & -2 & 1 & 1 & 1 & 1 \\
		0 & 0 & 0 & 0 & 1 & 1 & -2 & 1 & 1 & 1 \\
		0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
		0 & 0 & 0 & 0 & 1 & 1 & 1 & -2 & 1 & 1 \\
		0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 1 \\
		0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & -2 \\
		0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\
	\end{bmatrix}}
\end{equation*}
and
\begin{equation*}
Q_2=\frac{1}{9}{\footnotesize\begin{bmatrix}
		-3 & -3 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 6 \\
		3 & 3 & 6 & -3 & -3 & 0 & 0 & 0 & 0 & 3 \\
		1 & 1 & 2 & 2 & 2 & -6 & 3 & 3 & 3 & -2 \\
		1 & 1 & 2 & 2 & 2 & 3 & -6 & 3 & 3 & -2 \\
		-5 & 4 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\
		3 & 3 & -3 & 6 & -3 & 0 & 0 & 0 & 0 & 3 \\
		1 & 1 & 2 & 2 & 2 & 3 & 3 & -6 & 3 & -2 \\
		1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & -6 & -2 \\
		4 & -5 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\
		3 & 3 & -3 & -3 & 6 & 0 & 0 & 0 & 0 & 3 \\
\end{bmatrix}}
\end{equation*}
such that both $Q_1^\mathsf{T} A Q_1$ and $Q_2^\mathsf{T} A Q_2$ are adjacency matrices. Thus, $G$ has two generalized cospectral mates.

Recently, as a significant improvement of the criterion for a graph to have at most one cospectral mate established in \cite{wang2023ejc}, Raza et al. \cite{raza2026} obtained the following upper bound on the number of generalized cospectral mates of a graph.
\begin{theorem} [\cite{raza2026}]\label{ngm}
	Let $G$ be a controllable graph and $d_1,\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\lceil\frac{n}{2}\rceil}=1$ and $d_{n-1}=2$. Let $d_n=\prod_{i=1}^k p_i^{m_i}$ be the prime factorization of $d_n$. Then $G$ has at most $\left(\prod_{i=1}^{k}m_i\right)-1$ generalized cospectral mates.
	\end{theorem}

For a graph $G$, the \emph{spectrum} of $G$ is the multiset of all eigenvalues of its adjacency matrix. Two graphs are \emph{cospectral} if they share the same spectrum. We say two graphs $G$ and $H$ are \emph{generalized cospectral} if (i) $G$ is cospectral with $H$ and (ii) $\overline{G}$ is cospectral with $\overline{H}$, where $\overline{G}$ denotes the complement of $G$. A graph $G$ is \emph{determined by its generalized spectrum} (DGS) if any graph generalized cospectral with $G$ is isomorphic to $G$. An orthogonal matrix $Q$ is \emph{regular} if the sum of each row is 1. A classic result of Johnson and Newman \cite{johnson} states that two graphs $G$ and $H$ are generalized cospectral if and only if their adjacency matrices are similar via a regular orthogonal matrix.

For an $n$-vertex graph $G$ with adjacency matrix $A$, the \emph{walk matrix} of $G$ is \begin{equation*}
W(G):=[e,Ae,\ldots,A^{n-1}e],
\end{equation*} where $e$ is the all-ones vector. A graph is \emph{controllable} \cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.

We are mainly concerned with controllable graphs in this paper. We use $\RO_n(\mathbb{Q})$ to denote the group of rational regular orthogonal matrices of order $n$. For a controllable graph $G$ with adjacency matrix $A$, we write
\begin{equation*}
 \mathcal{Q}(G)=\{Q\in \RO_n(\mathbb{Q})\colon\, Q^\mathsf{T} AQ \text{~is a $(0,1)$-matrix} \}.
 \end{equation*}
It is easy to show that any $(0,1)$-matrix of the form $Q^\mathsf{T} A Q$ must be the adjacency matrix of some graph (see \cite[Remark 1]{qiu2023}). Let $\mathcal{C}(G)$ be the set of all graphs that are generalized cospectral with $G$. As there exists a one-to-one correspondence between $\mathcal{C}(G)$ and $\mathcal{Q}(G)$, the problem of determining the set $\mathcal{C}(G)$ can naturally be reduced to that of determining the set $\mathcal{Q}(G)$. For example, $\mathcal{Q}(G)$ always contains all permutation matrices of order $n$ which correspond to graphs that are isomorphic to $G$. Moreover, if $\mathcal{Q}(G)$ contains a non-permutation matrix, then $\mathcal{C}(G)$ contains a graph that is not isomorphic to $G$, i.e., $G$ is not DGS. The following notion introduced in \cite{wang2006} is crucial to investigate the set $\mathcal{Q}(G)$.
\begin{definition}[\cite{wang2006}] \normalfont
 Let $Q$ be a rational matrix. The \emph{level} of $Q$, denoted by $\ell(Q)$, is the
smallest positive integer $k$ such that $kQ$ is an integral matrix.
\end{definition}
Let
\begin{equation*}
L(G)=\lcm\{\ell(Q)\colon\,Q\in \mathcal{Q}(G)\},
\end{equation*}the least common multiple of all levels $\ell(Q)$ for $Q\in \mathcal{Q}(G)$. Clearly, if $L(G)=1$ then $\mathcal{Q}(G)$ contains only permutation matrices which implies that $G$ is DGS. A theorem of Wang \cite{wang2013EJC,wang2017JCTB} states that if the integer $2^{-\lfloor\frac{n}{2}\rfloor}\det W(G)$ is odd and square-free, then $L(G)=1$ and hence $G$ is DGS. Wang proves this theorem by excluding the possibilities of $p\mid L(G)$ for odd primes $p$ and for $p=2$ separately.
\begin{proposition}[\cite{wang2013EJC}]\label{oddcase}
 If $p$ is an odd prime with $p^2\nmid \det W(G)$ then $p\nmid L(G)$.
\end{proposition}
\begin{proposition}[\cite{wang2017JCTB}]\label{evencase}
 If $2^{-\lfloor\frac{n}{2}\rfloor}$ is odd then $L(G)$ is odd.
\end{proposition}
Recently, Qiu et al.~\cite{qiu2023} improved both Proposition \ref{oddcase} and Proposition \ref{evencase} using a new and much simpler argument. The current paper is a further improvement of the result in \cite{qiu2023}, but only for the odd prime case. The result of Qiu et al.~\cite{qiu2023} for the odd prime case is the following.
\begin{proposition}[\cite{qiu2023}]\label{qiuodd} If $p$ is an odd prime with $\rank_p W(G)=n-1$ then $L(G)\mid \frac{\det W(G)}{p}$.
\end{proposition}
For a prime $p$ and a nonzero integer $m$, we use $v_p(m)$ to denote the \emph{$p$-adic valuation} \cite{gouvea} of $m$, that is, the maximum nonnegative integer $k$ such that $p^k$ divides $m$. Note that the conclusion of Proposition \ref{qiuodd} can be rewritten as the inequality $v_p(L(G))\le v_p(\det W(G))-1$. The main aim of this paper is to give a better upper bound of $v_p(L(G))$ under the same assumption of Proposition \ref{qiuodd}.

\begin{proposition}[\cite{wang2017JCTB}]\label{evencase}
	If $2^{-\lfloor\frac{n}{2}\rfloor}$ is odd then $L(G)$ is odd.
\end{proposition}

\begin{proposition}[\cite{wang2013EJC}]\label{oddcase}
	If $p$ is an odd prime with $p^2\nmid \det W(G)$ then $p\nmid L(G)$.
\end{proposition}

For an $n$-vertex graph $G$ with adjacency matrix $A$, the \emph{walk matrix} of $G$ is \begin{equation*}
W(G):=[e,Ae,\ldots,A^{n-1}e],
\end{equation*} where $e$ is the all-ones vector. A graph is \emph{controllable} \cite{godsil} if $W(G)$ is nonsingular. We remark that the set of controllable graphs is \emph{closed} under generalized cospectrality, that is, any graph generalized cospectral with a controllable graph must be controllable. A key observation of Wang \cite{wang2006} is that, when restricted to controllable graphs, the regular orthogonal matrix connecting the adjacency matrices of two generalized cospectral graphs $G$ and $H$ is unique and rational.

The rest of this paper is organized as follows. In Sec.~\ref{pre}, we recall some preliminaries on Smith normal forms and modules over $\mathbb{Z}/p^k \mathbb{Z}$. The proof of Theorem \ref{main} is given in Sec.~\ref{pf}, which can be seen as a module-theoretical version of the original argument of Wang \cite{wang2006}, combined with some improvements and simplifications developed in \cite{wang2020}. Some direct applications of Theorem \ref{main} are presented in Sec.~\ref{dis}. The paper concludes with a conjecture proposing a further possible improvement to Theorem \ref{main}.
\section{Preliminaries}
\label{pre}
Let $R$ be a commutative principal ideal ring with identity. It is well known \cite{stanley2016JCTA} that any matrix $M$ over $R$ has a Smith normal form (SNF); that is, there exist invertible matrices $U$ and $V$ over $R$ such that
\begin{equation*}
 UMV=\begin{bmatrix}
 \diag(d_1,\ldots,d_r)&0\\
 0&0
 \end{bmatrix},
\end{equation*}
where the elements $d_i$ ($i=1,\ldots,r$) are nonzero and satisfy $d_1\mid d_2\mid \cdots\mid d_r$. The nonzero elements $d_1,\ldots,d_r$ are called the \emph{invariant factors} of $M$.

In the following, let $G$ be an $n$-vertex controllable graph and $p$ be an odd prime factor of $\det W(G)$ such that $\rank_p W(G)=n-1$. Denote $\tau=v_p(L(G))$, i.e., $\tau=\max\{v_p (\ell(Q))\colon\, Q\in \mathcal{Q}(G)\}$. Note that if $\tau=0$, then Theorem \ref{main} clearly holds. Thus, we may assume that $\tau\ge 1$. For simplicity, the adjacency matrix $A(G)$ and the walk matrix $W(G)$ will be denoted by $A$ and $W$, respectively, when there is no confusion.

\noindent\textbf{Example 1}. Let $n=10$ and $G$ be the $n$-vertex graph with adjacency matrix
\begin{equation*}
 A={\footnotesize\begin{bmatrix}
 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 1 \\
 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\
 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\
 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \\
 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
 1 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
 \end{bmatrix}}.
\end{equation*}
It can be verified that $2^{-\lfloor\frac{n}{2}\rfloor} \det W=3^4\times 19$ and $\rank_3 W=n-1$. By Theorem \ref{main}, we know that $v_3(L(G))\le \frac{1}{2} v_3(\det W)=2$. Indeed, using the program in \cite{wang2025}, we find that, besides the permutation matrices, there exist essentially two rational regular orthogonal matrices
\begin{equation*}
 Q_1=\frac{1}{3}{\footnotesize
 \begin{bmatrix}
 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & -2 & 1 & 1 & 1 & 1 & 1 \\
 0 & 0 & 0 & 0 & 1 & -2 & 1 & 1 & 1 & 1 \\
 0 & 0 & 0 & 0 & 1 & 1 & -2 & 1 & 1 & 1 \\
 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 1 & 1 & 1 & -2 & 1 & 1 \\
 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & -2 & 1 \\
 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & -2 \\
 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\
 \end{bmatrix}}
\end{equation*}
and
\begin{equation*}
Q_2=\frac{1}{9}{\footnotesize\begin{bmatrix}
 -3 & -3 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 6 \\
 3 & 3 & 6 & -3 & -3 & 0 & 0 & 0 & 0 & 3 \\
 1 & 1 & 2 & 2 & 2 & -6 & 3 & 3 & 3 & -2 \\
 1 & 1 & 2 & 2 & 2 & 3 & -6 & 3 & 3 & -2 \\
 -5 & 4 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\
 3 & 3 & -3 & 6 & -3 & 0 & 0 & 0 & 0 & 3 \\
 1 & 1 & 2 & 2 & 2 & 3 & 3 & -6 & 3 & -2 \\
 1 & 1 & 2 & 2 & 2 & 3 & 3 & 3 & -6 & -2 \\
 4 & -5 & -1 & -1 & -1 & 3 & 3 & 3 & 3 & 1 \\
 3 & 3 & -3 & -3 & 6 & 0 & 0 & 0 & 0 & 3 \\
\end{bmatrix}}
\end{equation*}
such that both $Q_1^\mathsf{T} A Q_1$ and $Q_2^\mathsf{T} A Q_2$ are adjacency matrices. Thus, $G$ has two generalized cospectral mates.

Recently, as a significant improvement of the criterion for a graph to have at most one cospectral mate established in \cite{wang2023ejc}, Raza et al. \cite{raza2026} obtained the following upper bound on the number of generalized cospectral mates of a graph.
\begin{theorem} [\cite{raza2026}]\label{ngm}
 Let $G$ be a controllable graph and $d_1,\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\lceil\frac{n}{2}\rceil}=1$ and $d_{n-1}=2$. Let $d_n=\prod_{i=1}^k p_i^{m_i}$ be the prime factorization of $d_n$. Then $G$ has at most $\left(\prod_{i=1}^{k}m_i\right)-1$ generalized cospectral mates.
 \end{theorem}
Using the result of Qiu et al. \cite{qiu2023}, it is known that, for graphs $G$ as described in Theorem \ref{ngm}, the number $L(G)$ must be a factor of $\prod_{i=1}^k p_i^{m_i-1}$. Thus, $\ell(Q)$ can take at most $\prod_{i=1}^k m_i$ different values (including 1) when $Q$ runs through $\mathcal{Q}(G)$. A key contribution of Raza et al.~\cite{raza2026} is to show that, under the assumptions of Theorem \ref{ngm}, there exists at most one matrix $Q$ in $\mathcal{Q}(G)$ for each possible level $\ell$ (in the sense of column permutations). Noting that the matrix $Q\in \mathcal{Q}(G)$ with level $\ell=1$ is the permutation matrix, the number of generalized cospectral mates of $G$ is upper bounded by the number of possible levels, excluding 1. Thus, Theorem \ref{ngm} holds.

Using Theorem \ref{main} for odd primes, we can obtain a better bound on $L(G)$. Indeed, let $d_n=2^{m_1}\prod_{i=2}^kp_i^{m_i}$ be the factorization of $d_n$, where $p_2,\ldots,p_k$ are distinct odd primes. Then for each odd prime $p_i$, we have $v_{p_i}(L(G)) \le \lfloor\frac{1}{2}{m_i}\rfloor$ by Theorem \ref{main}. For $p=2$, we only have $v_2(L(G))\le m_1-1$ by the result of Qiu et al.~\cite{qiu2023}. Thus, the number of possible levels for matrices in $\mathcal{Q}(G)$, other than 1, is at most $m_1\left(\prod_{i=2}^k\left( \lfloor\frac{1}{2}m_i\rfloor+1\right)\right)-1$. This gives an improvement of Theorem \ref{ngm} under the same assumptions. We state it as the following theorem.
\begin{theorem}
 Let $G$ be a controllable graph and $d_1,\ldots, d_n$ be the invariant factors of the walk matrix $W(G)$. Suppose $d_{\lceil\frac{n}{2}\rceil}=1$ and $d_{n-1}=2$. Let $d_n=2^{m_1}\prod_{i=2}^k p_i^{m_i}$ be the prime factorization of $d_n$, where $p_2,\ldots,p_k$ are distinct odd primes. Then $G$ has at most $m_1\left(\prod_{i=2}^k\left( \lfloor\frac{1}{2}m_i\rfloor+1\right)\right)-1$ generalized cospectral mates.
\end{theorem}
We end this paper by proposing a conjecture that the technical assumption on $\rank_p W(G)$ in Theorem \ref{main} is unnecessary for the conclusion. Moreover, the upper bound in terms of the determinant may be refined using the last two invariant factors.
 \begin{conjecture}
 Let $G$ be a controllable graph and $p$ be an odd prime factor of $\det W(G)$. Let $d_{n-1}$ and $d_n$ be the last two invariant factors of $W(G)$. Then $v_p(L(G))\le \frac{1}{2}( v_p(d_{n})+v_p(d_{n-1}))$, and in particular, $v_p(L(G))\le \frac{1}{2} v_p(\det W(G))$.
 \end{conjecture}

The proof of Theorem~\ref{main} is said to be "a module-theoretical version of the original argument of Wang \cite{wang2006}, combined with some improvements and simplifications developed in \cite{wang2020}." The paper’s approach shifts "from the vector space over the field $\mathbb{Z}/p\mathbb{Z}$" to "the module over the local ring $\mathbb{Z}/p^k\mathbb{Z}$" and uses that this ring has the "Basis Extension Property'' (Steinitz Property), which "allows us to generalize arguments from linear algebra."

Question:
Let $G$ be an $n$-vertex controllable graph with adjacency matrix $A$ and walk matrix
\[
W(G)=[e,Ae,\ldots,A^{n-1}e],
\]
where $e$ is the all-ones vector and controllable means that $W(G)$ is nonsingular. Define
\[
\mathcal{Q}(G)=\{Q\in \mathrm{RO}_n(\mathbb{Q}) : Q^\mathsf{T}AQ \text{ is a }(0,1)\text{-matrix}\},
\]
where $\mathrm{RO}_n(\mathbb{Q})$ is the set of rational regular orthogonal matrices (orthogonal matrices whose row sums are all $1$). For a rational matrix $Q$, its level $\ell(Q)$ is the smallest positive integer $k$ such that $kQ$ is integral, and
\[
L(G)=\operatorname{lcm}\{\ell(Q):Q\in \mathcal{Q}(G)\}.
\]
For a prime $p$, let $v_p(m)$ denote the $p$-adic valuation of a nonzero integer $m$, and let $\operatorname{rank}_p W(G)$ denote the rank of $W(G)$ over $\mathbb{F}_p$. If $p$ is an odd prime such that $\operatorname{rank}_p W(G)=n-1$, which quantitative estimate holds?

\[v_p(L(G))\le \frac{1}{2}\,v_p(\det W(G)).\]

Choices:
B. \[v_p(L(G))\le v_p(\det W(G))-1.\]
C. \[v_p(L(G))\le v_p(\det W(G)).\]
D. \[v_p(L(G))\le \frac{1}{2}\,v_p(\det W(G))-1.\]
E. If \(p\) is an odd prime such that \(\operatorname{rank}_p W(G)=n-1\), then \(p\nmid L(G)\).

half-valuation bound replaced by earlier linear-minus-one bound

inserted an extra subtractive constant into the valuation estimate

collapsed rank condition to the stronger conclusion \(p\nmid L(G)\)


Question:
Let $G$ be a controllable $n$-vertex graph with adjacency matrix $A$, meaning that its walk matrix
\[
W(G)=[e,Ae,\ldots,A^{n-1}e]
\]
is nonsingular, where $e$ is the all-ones vector. For such a graph, define
\[
\mathcal{Q}(G)=\{Q\in \mathrm{RO}_n(\mathbb{Q}) : Q^\mathsf{T}AQ \text{ is a }(0,1)\text{-matrix}\},
\]
where $\mathrm{RO}_n(\mathbb{Q})$ is the set of rational orthogonal matrices of order $n$ whose row sums are all $1$. If $Q$ is rational, its level $\ell(Q)$ is the smallest positive integer $k$ such that $kQ$ is integral, and
\[
L(G)=\operatorname{lcm}\{\ell(Q):Q\in \mathcal{Q}(G)\}.
\]
Let $p$ be an odd prime such that $\operatorname{rank}_p W(G)=n-1$, where $\operatorname{rank}_p$ denotes rank over $\mathbb{F}_p$. For a nonzero integer $m$, let $v_p(m)$ be the $p$-adic valuation of $m$. Under these assumptions, which statement about $L(G)$ holds?

Choices:
B. \[v_p(L(G))\le v_p(\det W(G))-1.\]
C. \[v_p(L(G))\le v_p(\det W(G)).\]
D. \[v_p(L(G))\le \frac{1}{2}\,v_p(\det W(G))\quad\text{for every prime }p\text{ with }\operatorname{rank}_p W(G)=n-1.\]
E. \[v_p(L(G))< \frac{1}{2}\,v_p(\det W(G)).\]

half-valuation bound replaced by older minus-one bound

dropped the factor \(\tfrac12\) from the upper bound

odd-prime hypothesis broadened to all primes

non-strict inequality strengthened to strict inequality


\begin{theorem}[Embedded NC-preserving resolution]
\label{thm:embedded-NC}
Let $(X,E)$ be a smooth irreducible variety, or more generally a smooth
irreducible Deligne-Mumford stack, over a field of characteristic~$0$,
equipped with an SNC divisor $E$, and let
$Y \subset X$ be a closed subscheme (respectively, a closed substack).

There exists a canonical sequence of weighted blow-ups
defined by the invariant $\inv_{(X,E)}(Y)$.
\[
   \Pi: X_N \longrightarrow \cdots \longrightarrow X_1
   \longrightarrow X_0 = X
\]
with maximal admissible centers disjoint from the normal crossings locus
$X_*^{\mathrm{nc}}$, such that:

\begin{enumerate}
\item Each center coincides with the locus where the invariant
$\inv_{(X,E)}(Y)$ attains its maximal value outside $X_*^{\mathrm{nc}}$.

\item The invariant strictly decreases after each blow-up away from
the NC locus. In particular, the process terminates.

\item The morphism $\Pi$ restricts to an isomorphism over
$X_*^{\mathrm{nc}}$.

\item The strict transform $\Pi^s(Y)\subset X_N$ has normal crossings
adapted to the SNC divisor $E' \cup D$, where
$D$ is the exceptional divisor and $E'=\Pi^s(E)$.
\end{enumerate}
\end{theorem}

Hironaka's theorem (1964) shows that every algebraic variety over a field
of characteristic zero admits a resolution of singularities, and that the
exceptional divisor can be arranged to have simple normal crossings (snc).
Subsequent functorial algorithms refined this statement by producing
simplified canonical resolutions compatible with smooth morphisms.

However, classical desingularization procedures do not distinguish between
genuinely singular points and those that already possess normal crossings
structure. From both geometric and birational perspectives this distinction
is essential: normal crossings singularities behave, in many respects,
as terminal objects of desingularization theory.
They arise naturally as boundary strata in moduli problems
(e.g.\ stable curves, KSBA compactifications),
in Hodge theory (semistable reduction and degenerations),
and in birational geometry (log pairs $(X,D)$ in the Minimal Model Program).
Blowing them up further often obscures intrinsic structure rather than
simplifying it.

The guiding principle of this work is that the normal crossings locus
should be fixed by a canonical desingularization procedure.
We therefore seek a resolution algorithm that removes all singularities
outside the normal crossings locus while leaving that locus untouched.
This leads to a notion of \emph{resolution up to normal crossings}.
Resolution except for SNC singularities was established in
\cite{BM12,BLM12,BDMP14,BBL22},
where the desingularization invariant is refined so as to avoid blowing up
snc points and to isolate minimal residual singularities.
In that setting the splitting of components is already visible
Zariski-locally.

The problem treated here is fundamentally different.
For nc singularities the splitting of branches may occur only after
an \'etale or Galois extension.
Thus the nc condition is intrinsically non-local.
This forces the introduction of stack-theoretic methods and more flexible
birational transformations, such as weighted blow-ups.
More generally, resolution except for \emph{locally toric}
singularities was obtained in~\cite{W22t} using logarithmic and combinatorial methods,
but the \'etale-locally toric case requires new structural tools.

The purpose of this article is to construct a canonical
resolution algorithm that removes all singularities outside the nc locus
while preserving every pre-existing nc structure.
The method is based on weighted blow-ups adapted to the combinatorial
type of the singularity.
Applied canonically, these transformations simplify several components
simultaneously and guide the resolution process in a controlled manner.

{\it While completing this proof, I became aware of a parallel paper in preparation by
A.~Belotto~da~Silva and E.~Bierstone addressing, in particular, the same result as Theorem \ref{thm:embedded-NC} in the case of  reduced normal crossings and in a non-SNC divisorial setting for exceptional divisors,
using different techniques~\cite{BB26}.}

 Their approach combines formal splitting
techniques of Newton--Puiseux type with a combinatorial analysis of circulant
singularities near the normal crossings locus, together with weighted blow-ups.

Our construction provides a functorial resolution that avoids the reduced or non-reduced nc locus and is governed
by the following principle.

\subsection{Center Theorem}

Let $X^{\mathrm{nc}}(\mathcal{I})$ denote the locus of points where an ideal
$\mathcal{I}$ (reduced or not) has at most normal crossings singularities
on a smooth variety $X$.

The NC-preserving algorithm rests on two ingredients.

First, we use the weighted blow-up resolution framework of
McQuillan and Abramovich-Temkin-W\l odarczyk
\cite{ATW-weighted,Marzo-McQuillan},
together with its logarithmic refinement developed in
\cite{W22,W23}.
The centers are chosen as canonical maximal admissible centers
supported on the maximal locus of the logarithmic invariant
$\mathrm{inv}_{(X,E)}$, introduced in  \cite{W22,W23}, which refines the invariant of~\cite{ATW-weighted}.

Second and this is the key structural property we show that such
canonical weighted centers are rigid with respect to the nc locus.
Unlike classical smooth centers, they cannot intersect
$X^{\mathrm{nc}}(\mathcal{I})$ partially.

Hironaka's theorem (1964) shows that every algebraic variety over a field
of characteristic zero admits a resolution of singularities, and that the
exceptional divisor can be arranged to have simple normal crossings (snc).
Subsequent functorial algorithms refined this statement by producing
simplified canonical resolutions compatible with smooth morphisms.

The guiding principle of this work is that the normal crossings locus
should be fixed by a canonical desingularization procedure.
We therefore seek a resolution algorithm that removes all singularities
outside the normal crossings locus while leaving that locus untouched.
This leads to a notion of \emph{resolution up to normal crossings}.
Resolution except for SNC singularities was established in
\cite{BM12,BLM12,BDMP14,BBL22},
where the desingularization invariant is refined so as to avoid blowing up
snc points and to isolate minimal residual singularities.
In that setting the splitting of components is already visible
Zariski-locally.

The purpose of this article is to construct a canonical
resolution algorithm that removes all singularities outside the nc locus
while preserving every pre-existing nc structure.
The method is based on weighted blow-ups adapted to the combinatorial
type of the singularity.
Applied canonically, these transformations simplify several components
simultaneously and guide the resolution process in a controlled manner.

Let $X^{\mathrm{nc}}(\mathcal{I})$ denote the locus of points where an ideal
$\mathcal{I}$ (reduced or not) has at most normal crossings singularities
on a smooth variety $X$.

First, we use the weighted blow-up resolution framework of
McQuillan and Abramovich-Temkin-W\l odarczyk
\cite{ATW-weighted,Marzo-McQuillan},
together with its logarithmic refinement developed in
\cite{W22,W23}.
The centers are chosen as canonical maximal admissible centers
supported on the maximal locus of the logarithmic invariant
$\mathrm{inv}_{(X,E)}$, introduced in \cite{W22,W23}, which refines the invariant of~\cite{ATW-weighted}.

Second and this is the key structural property we show that such
canonical weighted centers are rigid with respect to the nc locus.
Unlike classical smooth centers, they cannot intersect
$X^{\mathrm{nc}}(\mathcal{I})$ partially.

First, we use the weighted blow-up resolution framework of
McQuillan and Abramovich-Temkin-W\l odarczyk
\cite{ATW-weighted,Marzo-McQuillan},
together with its logarithmic refinement developed in
\cite{W22,W23}.
The centers are chosen as canonical maximal admissible centers
supported on the maximal locus of the logarithmic invariant
$\mathrm{inv}_{(X,E)}$, introduced in \cite{W22,W23}, which refines the invariant of~\cite{ATW-weighted}.

Second and this is the key structural property we show that such
canonical weighted centers are rigid with respect to the nc locus.
Unlike classical smooth centers, they cannot intersect
$X^{\mathrm{nc}}(\mathcal{I})$ partially.

This rigidity forces the algorithm to preserve normal crossings:
centers chosen outside the nc locus remain disjoint from it.
The proof of the theorem forms the main technical part of the paper.
\subsection{Algorithmic Rule}

The algorithm proceeds as follows:
\begin{itemize}
 \item The nc locus $X^{\mathrm{nc}}(\mathcal{I})$ is open.
 \item At each step, one blows up a canonical admissible center associated with the
 maximum of the invariant $\mathrm{inv}$ on the closed complement
 $X \setminus X^{\mathrm{nc}}(\mathcal{I})$. By the Center Theorem, the associated
 center is disjoint from the nc locus.
 \item The process stops when every point lies in the nc locus, so
 that $X = X^{\mathrm{nc}}(\mathcal{I})$.
\end{itemize}

There exists a canonical semi-continuous invariant
\[
\inv_{(X,E)}(\mathcal{I}) : X \to \Sigma,
\]
with values in a well-ordered set (see Section~\ref{sec:invariant},
Definition~\ref{cntr}),
and a canonical sequence of weighted blow-ups
\[
 \Pi\colon X_N \longrightarrow \cdots \longrightarrow X_1
 \longrightarrow X_0 = X
\]
with smooth centers disjoint from the normal crossings locus
$X^{\mathrm{nc}}_*$ such that:

There exists a canonical sequence of weighted blow-ups
defined by the invariant $\inv_{(X,E)}(Y)$.
\[
 \Pi: X_N \longrightarrow \cdots \longrightarrow X_1
 \longrightarrow X_0 = X
\]
with maximal admissible centers disjoint from the normal crossings locus
$X_*^{\mathrm{nc}}$, such that:

\begin{remark}
\label{rem:inv-compat-QIDL}
Let
\[
\mathcal{A}_J
=
\mathcal{O}_{X,p}
\bigl[x_1 t^{1/a_1},\ldots,x_k t^{1/a_k}\bigr]^{\mathrm{int}}
=
\bigoplus_{q\in\QQ_{\ge0}} (\mathcal{A}_J)_q t^q
\]
be a weighted center and let
\(
J=(x_1^{a_1},\ldots,x_k^{a_k})
\)
be the corresponding $\QQ$-ideal.
Then the degree $1$ component satisfies
$
(\mathcal{A}_J)_1 = J_{X,p}.
$
Hence
\[
\mathcal{I}_p\, t \subset \mathcal{A}_{J,p}
\quad\Longleftrightarrow\quad
\mathcal{I}_p \subset J_{X,p}.
\]
Therefore the definition of $\inv_p(\mathcal{I})$ via weighted centers
coincides with the previous $\QQ$-ideal definition.
\end{remark}

In general, the process yields principalization of $\mathcal{I}$:
its total transform becomes a locally monomial ideal supported on an SNC divisor.
\end{theorem}\subsection{Invariant associated with NC singularities}
\begin{lemma} \label{invar} Let $\cI$ be a Normal Crossing of codimension $r+1$ at $p\in X$ so that in an \'etale neighborhood $\cI$ admits a presentation
\[
\cI = (u_1, \ldots, u_r, m),
\qquad
m = u_{r+1}^{a_{r+1}} \cdots u_k^{a_k},
\]
where $u_1, \ldots, u_k$ form a partial system of local parameters and
$a_{r+1} + \cdots + a_k = d$, where $u_1,\ldots,u_{r+t}$ are free and $u_{r+t+1},\ldots,u_{r}$, are divisorial.
Then the maximal admissile center at $p\in X$ is given by $$\cJ=(u_1,\ldots,u_{r},u^d_{r+1},\ldots,u^d_{r+t},u^d_{r+t+1},\ldots,u^d_{k}),$$ and the invariant is given by $$\inv_p(\cI)=\inv(\cJ)=(\underbrace{1, \ldots, 1}_{r \text{ times}}, \underbrace{d, \ldots, d}_{t \text{ times}},\underbrace{d_+, \ldots, d_+}_{k-r-t \text{ times}})$$
\end{lemma}
\begin{proof} We follow the algorithm from \cite[Section 3.3]{ W22}. We compute the invariant in the given \'etale neighborhood. Consider the algebra $R_1=\cO_X[\cI t]$.
Its order is equal $$\ord_p(R_1)=\ord_p(\cI)=1,$$ with the tangent ideal $$T(\cI)=\cD^{1-1}(\cI)=\cD^0(\cI)=\cI,$$ and a graded multiple maximal contact $\overline{u}_1t$ where $$\overline{u}_1:=(u_1,\ldots,u_{r}).$$
Then $$R_2:=C_{\overline{u}_1t}(R_1)=(mt)$$ is generated by the only coefficient $m$ with respect to presenataion in $\overline{u}_1$. Thus $$\ord_p(R_2)=\ord_p(mt)=\ord_p(m)=d,$$ In this case multiple maximal contact is given by
$\overline{u}_2t^{1/d}$, where $$\overline{u}_2:=(u_{r+1},\ldots,u_k)\subset T(m)=\cD^{d-1}(m).$$ Note that
$R_3=C_{\overline{u}_2t^{1/d}}(R_2)=0$. This determines the Rees algebra of the center $$\cA_{\cJ}=\cO_X[\overline{u}_1t,\overline{u}_2t^{1/d}],$$ associated with $\QQ$-ideal center $$\cJ=(\overline{u}_1,\overline{u}_2^d)=(u_1,\ldots,u_r,u^d_{r+1},\ldots,u^d_k)$$ and the invariant $$\inv_p(\cI)=\inv(\cJ)=(\underbrace{1, \ldots, 1}_{r \text{ times}},, \underbrace{d, \ldots, d}_{t \text{ times}},\underbrace{d_+, \ldots, d_+}_{k-r-t \text{ times}}).$$

This rigidity forces the algorithm to preserve normal crossings:
centers chosen outside the nc locus remain disjoint from it.
The proof of the theorem forms the main technical part of the paper.
\subsection{Algorithmic Rule}

The algorithm proceeds as follows:
\begin{itemize}
  \item The nc locus $X^{\mathrm{nc}}(\mathcal{I})$ is open.
  \item At each step, one blows up a canonical admissible center associated with the
  maximum of the invariant $\mathrm{inv}$ on the closed complement
  $X \smallsetminus X^{\mathrm{nc}}(\mathcal{I})$. By the Center Theorem, the associated
  center is disjoint from the nc locus.
  \item The process stops when every point lies in the nc locus, so
  that $X = X^{\mathrm{nc}}(\mathcal{I})$.
\end{itemize}

\subsection{Galois groups, weights, and stacks}

The obstruction to eliminating nc singularities without modifying the nc locus
is already visible in classical examples, most notably the pinch point
(Whitney umbrella), as explained by Koll\'ar
\cite[Paragraph 8]{Kollar}; see also
\cite[Corollary 3.6.10]{Fujino} and \cite[Example 1.7]{BM12}.
At a nonzero point of the axis, the pinch point has two analytic branches
which are interchanged when one loops around the origin.
Any birational morphism that is an isomorphism over the generic point
of the axis preserves this monodromy.
Hence the singularity cannot be eliminated without blowing up the axis itself.

This example reflects a general feature of nc singularities.
Although they are locally defined by monomial equations,
their branches may split only after passing to a finite Galois extension.
In particular, a normal crossings (nc) singularity becomes
simple normal crossings (snc) on a finite Galois cover,
and the original nc structure is recovered as the quotient by a finite
group acting by permutation of branches.

\medskip

\noindent
\textbf{Structural principle.}
Normal crossings singularities are \'etale-locally simple normal crossings.
Equivalently, nc singularities are quotients of snc singularities
by finite Galois groups permuting their components.

Thus nc geometry is inherently equivariant geometry.
Smooth blow-ups performed equivariantly on the splitting cover
descend to weighted blow-ups on the base,
with weights encoding the stabilizer and ramification data
of the finite group action.

Since these stabilizers are intrinsic to the nc structure,
the natural ambient category is that of Deligne-Mumford stacks:
the stack quotient records the finite isotropy groups,
while the coarse space corresponds to the weighted transform.
Stacks therefore arise canonically in nc-preserving resolution.

\medskip

To make this structure canonical, we develop a Galois theory
of multivariate splitting forms.

Let
\[
F(x_1,\ldots,x_k)=\prod_i \ell_i^{a_i}\in K[x_1,\ldots,x_k]
\]
be a homogeneous form whose linear factors $\ell_i$
split only after a finite extension.
We associate to $F$ a canonical splitting field $K_F$
with Galois group
\[
G_F=\mathrm{Gal}(K_F/K),
\]
where $K$ is the fraction field of a ring $R$,
and denote by $R_F$ the integral closure of $R$ in $K_F$.
The morphism
\[
\operatorname{Spec} R_F[x_1,\ldots,x_k]\longrightarrow \operatorname{Spec} R[x_1,\ldots,x_k]
\]
encodes the splitting behavior of $F$.
Here $R$ is the coordinate ring of the canonical center,
and $x_1,\ldots,x_k$ generate its defining ideal.

We prove that the locus where $F$ has distinct linear factors
coincides with the \'etale locus of this morphism.
Thus the nc locus is precisely the locus where the associated
Galois cover is unramified.
On the splitting cover the initial form splits and the singularity becomes formally simple normal crossings along the canonical center; equivariant smooth centers resolving the ramification locus descend to canonical weighted centers downstairs.

This Galois-weight-stack correspondence is the structural input
in the proof of the Center Theorem and explains
why canonical weighted centers cannot partially intersect
the nc locus.

This rigidity forces the algorithm to preserve normal crossings: centers chosen outside the nc locus remain disjoint from it (the proof “forms the main technical part of the paper”). The “structural input in the proof of the Center Theorem” is that nc singularities are “\'etale-locally simple normal crossings,” i.e. “quotients of snc singularities by finite Galois groups permuting their components,” so nc geometry is “inherently equivariant geometry.” One develops a Galois theory of “multivariate splitting forms” by associating to a homogeneous form $F=\prod_i \ell_i^{a_i}$ a canonical splitting field $K_F$ with Galois group $G_F$ and integral closure $R_F$, and considers the morphism $\Spec R_F[x]\to \Spec R[x]$ encoding splitting. The key identification is: “the locus where $F$ has distinct linear factors coincides with the \'etale locus of this morphism,” hence “the nc locus is precisely the locus where the associated Galois cover is unramified.” On the splitting cover “the initial form splits and the singularity becomes formally simple normal crossings along the canonical center;” then “equivariant smooth centers resolving the ramification locus descend to canonical weighted centers downstairs.” This Galois–weight–stack correspondence “explains why canonical weighted centers cannot partially intersect the nc locus,” yielding the Center Theorem’s conclusion that a canonical maximal admissible weighted center meets $X^{\mathrm{nc}}(\mathcal I)$ only if it is contained in it.

This rigidity forces the algorithm to preserve normal crossings: centers chosen outside the nc locus remain disjoint from it (the proof “forms the main technical part of the paper”). The “structural input in the proof of the Center Theorem” is that nc singularities are “\'etale-locally simple normal crossings,” i.e. “quotients of snc singularities by finite Galois groups permuting their components,” so nc geometry is “inherently equivariant geometry.” One develops a Galois theory of “multivariate splitting forms” by associating to a homogeneous form $F=\prod_i \ell_i^{a_i}$ a canonical splitting field $K_F$ with Galois group $G_F$ and integral closure $R_F$, and considers the morphism $\Spec R_F[x]\to \Spec R[x]$ encoding splitting. The key identification is: “the locus where $F$ has distinct linear factors coincides with the \'etale locus of this morphism,” hence “the nc locus is precisely the locus where the associated Galois cover is unramified.” On the splitting cover “the initial form splits and the singularity becomes formally simple normal crossings along the canonical center;” then “equivariant smooth centers resolving the ramification locus descend to canonical weighted centers downstairs.” This Galois–weight–stack correspondence “explains why canonical weighted centers cannot partially intersect the nc locus,” yielding the conclusion needed in establishing the main theorem: a canonical maximal admissible weighted center meets $X^{\mathrm{nc}}(\mathcal I)$ only if it is contained in it.

[Center Theorem]\label{center}
A canonical maximal admissible weighted center
intersects the nc locus $X^{\mathrm{nc}}(\mathcal{I})$
only if it is entirely contained in it.,

Question: Let $X$ be a smooth variety, let $\mathcal I$ be an ideal sheaf on $X$, and let $X^{\mathrm{nc}}(\mathcal I)$ denote the locus of points of $X$ where $\mathcal I$ has at most normal crossings singularities. Consider a canonical maximal admissible weighted center, meaning a weighted blow-up center chosen canonically by the resolution procedure and maximal among admissible centers. Which statement about its relation to the normal crossings locus is valid?

A canonical maximal admissible weighted center can intersect $X^{\mathrm{nc}}(\mathcal I)$ only if the entire center is contained in $X^{\mathrm{nc}}(\mathcal I)$; equivalently, it cannot meet the normal crossings locus only partially.

Choices:
B. A canonical maximal admissible weighted center may intersect $X^{\mathrm{nc}}(\mathcal I)$ in a proper closed subset, but whenever this happens the intersection is itself a union of strata of the normal crossings locus.
C. If a canonical maximal admissible weighted center is disjoint from $X^{\mathrm{nc}}(\mathcal I)$, then it does not meet the normal crossings locus.
D. A canonical maximal admissible weighted center intersects $X^{\mathrm{nc}}(\mathcal I)$ only if, after passing to an étale neighborhood of each intersection point, the induced center is entirely contained in the pullback of $X^{\mathrm{nc}}(\mathcal I)$.
E. A canonical maximal admissible weighted center intersects $X^{\mathrm{nc}}(\mathcal I)$ only if its reduced support is entirely contained in $X^{\mathrm{nc}}(\mathcal I)$.

global rigidity against partial intersection

drops the nontrivial implication from meeting to containment and keeps only the disjointness case

global descent from étale-local containment

scheme-theoretic center versus reduced support


\begin{rem}\label{rem:thm B}
    The hypothesis on the Sylow subgroups of $G/N$ being cyclic is only used in the second-to-last paragraph of the proof of Theorem \ref{thm:ppalblock gam}, to apply \cite[Thm. 3.4]{Nav04}. In general, we could assume only that, for every block $B$ of maximal defect of $G/N$ with Brauer correspondent $b$, there is a bijection
    $\Omega_B:\irrtau(B)\rightarrow\irrtau(b)$. Moreover, if one assumes $\Omega_B(\psi)(1)\leq \psi(1)$ for all $\psi\in\irrtau(B)$ and all blocks $B$ of maximal defect, then the proof of Theorem \ref{thm:ppalblock gam} gives a bijection $\Omega:\Irr_{0,\tau}(B_0(G))\rightarrow\Irr_{0,\tau}(B_0(\norm G P))$ with $\Omega(\chi)(1)\leq\chi(1)$. The existence of such bijections has been proposed recently in \cite{Gia25}.
\end{rem}

\begin{theoA}\label{thm:B}
Let $G$ be a finite group of order divisible by $p$, let $N\normal G$ be an abelian $p$-group and assume $G/N$ has cyclic Sylow $p$-subgroups and let $\tau\in\mathcal{H}$ be a $p$-power order element. Then $\tau$ fixes the same number of height-zero characters in $\Irr(B_0(G))$ as in $\Irr(B_0(\norm G P))$, where $P\in{\rm Syl}_p(G)$.
\end{theoA}

In Problem 12 of his famous list of problems from 1963, Brauer asked what information could be gleaned about a Sylow $p$-subgroup $P$ of a finite group $G$ from its character table. In particular, the question of whether the number of generators of $P$ could be obtained in this way has been studied in recent years, see e.g. \cite{Riz-Sch-Val20, Nav-Riz-Sch-Val20, Mor-Sam23, Va23}. Many questions in this realm seem linked to the action of the Galois automorphism $\sigma\in{\rm Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ that fixes $p'$-roots of unity and sends $p$-power roots of unity to their $1+p$ power. For any cyclotomic extension $\mathbb{Q}_n:=\mathbb{Q}(e^{2\pi i/n})$ of $\mathbb{Q}$, we keep the notation $\sigma$ for the restriction of this automorphism to $\mathbb{Q}_n$.
Here, we contribute to this line of problems by proving the following direction of the main conjecture from \cite{Nav-Riz-Sch-Val20}.

\section{Introduction}

As noted in \cite{Nav-Riz-Sch-Val20}, Theorem \ref{thm:A} (and its converse) would follow from the Alperin--McKay--Navarro conjecture \cite[Conj.~B]{Nav04}, or even the more restrictive version \cite[Conj.~D]{Isa-Nav02}. Hence, this result gives further evidence for these elusive conjectures. We remark that although the block-free version \cite[Conj.~C]{Isa-Nav02} was recently established in \cite{RSF25}, the blockwise version (even for principal blocks) is much further from completion and has not yet been reduced to a problem on simple groups. For this reason, finding additional evidence of this blockwise version remains pertinent.

We begin by collecting some basic facts about principal blocks. Throughout, given a prime $p$, $B_0(G)$ denotes the principal $p$-block of the finite group $G$. We denote by $\Irr(B_0(G))$ the set of irreducible characters in $B_0(G)$ and by $\Irr_0(B_0(G))$ the subset of those with height zero (that is, $p'$-degree). The set of $\sigma$-invariant characters in $\Irr_0(B_0(G))$ will be denoted $\irrsigma(B_0(G))$, and we set $\ksigma(B_0(G)):=|\irrsigma(B_0(G))|$.

\begin{lem}\label{lem:kernel of sigmas}
Let $G$ be a finite group with $\oh{p'}G=1$ and let $P\in\Syl_p(G)$. Then
$$K=\bigcap_{\chi\in\irrsigma(B_0(G))}\ker\chi\leq \Phi(P).$$
\end{lem}
\begin{proof}
 Let $\lambda\in\Irr(P/\Phi(P))$. Write $P\cent G P=P\times X$ and let $\psi=\lambda\times 1_X\in\irrsigma(B_0(P\cent G P))$. By Lemma \ref{lem:3.7MMSV} we have that $\lambda^G$ contains a constituent $\chi\in\irrsigma(B_0(G))$. By Frobenius reciprocity, $\chi_{P\cent G P}$ contains $\lambda\times 1_X$ so $\chi_P$ contains $\lambda_P$. Moreover $\chi_{P\cap K}=\chi(1)1_{P\cap K}$ contains $\lambda_{P\cap K}$ which shows that $\ker{\lambda}$ contains $P\cap K$. Therefore $$P\cap K\sbs\bigcap_{\lambda\in\Irr(P/\Phi(P))}\ker\lambda=\Phi(P).$$
By Tate's theorem \cite[Cor.~6.14]{Nav18}, this implies that $K$ is $p$-nilpotent. Then if $K$ is not a $p$-group, $\oh{p'}K>1$, which contradicts $\oh{p'}G=1$. Therefore $K$ is a normal $p$-subgroup so $K=P\cap K\sbs\Phi(P)$, as desired.
\end{proof}

\begin{pro}\label{pro:noelia relative sigma} Let $G$ be a finite group, $p\in\{2,3\}$, and $P\in\Syl_p(G)$. Let $N\normal G$ and assume $p$ divides $|G:N|$. Suppose that $\theta\in\Irr(B_0(N))$ is $P\times\langle \sigma\rangle$-invariant. Then $p$ divides $ \ksigma(B_0(G)|\theta)$. In particular, if $\theta$ extends to $\hat\theta\in\irrsigma(B_0(PN))$ then $\ksigma(B_0(G)|\theta)\geq p$.
\end{pro}
\begin{proof}
 Arguing as in \cite[Thm.~2.7]{Gia-Riz-Sch-Val24} we get that
 $$\sum_{\chi\in\Irr_0(B_0(G)|\theta)}\chi(1)^2\equiv 0\mod p.$$

\begin{pro}\label{thm:ppalblock gam2}
 Let $N\normal G$ be perfect and let $P\in\Syl_p(G)$. Assume every $P$-invariant character $\theta\in\irrsigma(B_0(N))$ that extends to a $\sigma$-invariant character of $PN$ also extends to a $\sigma$-invariant character of $B_0(G_\theta)$ and assume $G/N$ has cyclic Sylow $p$-subgroups. Then $|\irrsigma(B_0(G))|=|\irrsigma(B_0(N\norm G P))|$.
\end{pro}
\begin{proof}
We may argue as in the proof of Theorem \ref{thm:ppalblock gam} that every $\chi\in\irrsigma(B_0(G))$ lies over a $P$-invariant $\theta\in\Irr(B_0(N))$. Since $\theta$ has a number of $G$-conjugates not divisible by $p$ and $\langle \sigma \rangle$ acts on the set of $G$-conjugates of $\theta$, it follows that $\theta$ is also $\sigma$-invariant. Arguing again as in Theorem \ref{thm:ppalblock gam} we conclude that $\chi\in\Irr_{0}(B_0(G))$ is $\sigma$-invariant if and only if its Clifford correspondent $\psi\in\Irr_{0}(B_0(G_\theta))$ is $\sigma$-invariant.

\begin{thm}
 Let $G$ be a finite group, let $B_0(G)$ be the principal 3-block of $G$, and let $P\in\syl 3 G$. Suppose that $k_{0,\sigma}(B_0(G))\in\{6,9\}$. Then $|P:\Phi(P)|=9.$
\end{thm}
\begin{proof}
 We argue by induction on $|G|$. Since $\irrsigma(B_0(G))=\irrsigma(B_0(G/\oh{p'}G))$ by \cite[Thm.~9.9]{Nav98}, we may assume $\oh{p'}G=1$. We can also assume that $P$ is not cyclic by \cite[Thm.~A]{Riz-Sch-Val20}.

\begin{thm}\label{thm:simple group conditions}
Let $S$ be a nonabelian simple group of order divisible by $3$ and let $X\in\Syl_3(\Aut(S))$.
\begin{enumerate}
 \item One of the following holds.
 \begin{enumerate}
 \item There exist $1_S\neq\theta_1,\theta_2\in\irrsigma(B_0(S))$ nonconjugate in $\Aut(S)$ and invariant under $X$, or
 \item There is an $X$-invariant $1_S\neq \theta\in\irrsigma(B_0(S))$ that extends to some $\sigma$-invariant character in $B_0(T)$ for all $S\leq T\leq \Aut(S)_\theta$.
 \end{enumerate}
 \item The set of degrees of characters in $\irrsigma(B_0(S))$ has size at least $3$.
 \item If $S$ has noncyclic Sylow $3$-subgroups, then there exist $1_S\neq \theta_1,\theta_2,\theta_3\in\irrsigma(B_0(S))$ nonconjugate in $\Aut(S)$.
\end{enumerate}
\end{thm}

\begin{theoA}\label{thm:A}
    Let $G$ be a finite group and $P\in\Syl_3(G)$. Then $|P:\Phi(P)|=9$ if the principal $3$-block $B_0(G)$ contains exactly $6$ or $9$ $\sigma$-invariant characters of degree coprime to $3$.
\end{theoA}

As noted in \cite{Nav-Riz-Sch-Val20}, Theorem \ref{thm:A} (and its converse) would follow from the Alperin--McKay--Navarro conjecture \cite[Conj.~B]{Nav04}, or even the more restrictive version  \cite[Conj.~D]{Isa-Nav02}. Hence, this result gives further evidence for these elusive conjectures. We remark that although the block-free version \cite[Conj.~C]{Isa-Nav02} was recently established in \cite{RSF25}, the blockwise version (even for principal blocks) is much further from completion and has not yet been reduced to a problem on simple groups. For this reason, finding additional evidence of this blockwise version remains pertinent.

Theorem \ref{thm:A} can be thought of simultaneously as an extension of the main result of \cite{Nav-Riz-Sch-Val20}, which addressed the analogous question for $p=2$, and of the main result of \cite{Gia-Riz-Sch-Val24}, which showed that $[P:P']=9$ if and only if $B_0(G)$ contains exactly 6 or 9 irreducible characters of degree coprime to $3$.

Our proof of Theorem \ref{thm:A} uses the recent work of Ketchum \cite{eden}, which shows that the statement of Theorem \ref{thm:A} (and its converse) hold for almost simple groups. It also uses a reduction theorem and additional results on almost simple groups analogous to those in \cite{Gia-Riz-Sch-Val24}. Moreover, the reduction relies on beautiful work of R. Brauer \cite{Bra76} on the structure of groups having cyclic Sylow $p$-subgroups.

As mentioned above, \cite[Conj.~C]{Isa-Nav02} has been completed in \cite{RSF25}. Our next main theorem considers the principal block version, which is part of \cite[Conj.~D]{Isa-Nav02}, and is a key step towards the proof of Theorem \ref{thm:A}. We believe it may be of independent interest. Let $\mathcal{H}\leq {\rm Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ be the Galois group considered in \cite{Nav04}.

\begin{theoA}\label{thm:B}
Let $G$ be a finite group of order divisible by $p$, let $N\triangleleft\, G$ be an abelian $p$-group and assume $G/N$ has cyclic Sylow $p$-subgroups and let $\tau\in\mathcal{H}$ be a $p$-power order element. Then $\tau$ fixes the same number of height-zero characters in $\operatorname{Irr}(B_0(G))$ as in $\operatorname{Irr}(B_0(\norm G P))$, where $P\in{\rm Syl}_p(G)$.
\end{theoA}

In particular, \cite[Conj.~D]{Isa-Nav02} holds for principal blocks of groups satisfying the hypotheses of Theorem \ref{thm:B}. In fact, as explained in Remark \ref{rem:thm B}, the same result holds by assuming only that every block of maximal defect of $G/N$ satisfies \cite[Conj.~D]{Isa-Nav02}, or \cite[Conj.~B]{Nav04}. (See also \cite[Prop.~5.4]{Mar-Mar-Sch-Val24} for another related, but restrictive, case.)

The paper is structured as follows: Section \ref{sec:aux} contains preliminary lemmas and results, as well as a Galois version of \cite[Thm.~B]{Gia-Riz-Sch-Val24}. Theorem \ref{thm:B} is proved in Section \ref{sec:thm B}. Theorem \ref{thm:A} is reduced to simple groups in Section \ref{sec:reduction} and finally completed in Section \ref{sec:simples}, where we also provide some evidence for the version of Theorem \ref{thm:A} for arbitrary blocks.

Our proof of Theorem~\ref{thm:A} uses the recent work of Ketchum \cite{eden}, which shows that the statement of Theorem~\ref{thm:A} (and its converse) hold for almost simple groups. It also uses a reduction theorem and additional results on almost simple groups analogous to those in \cite{Gia-Riz-Sch-Val24}. Moreover, the reduction relies on beautiful work of R.\ Brauer \cite{Bra76} on the structure of groups having cyclic Sylow $p$-subgroups.

A key step toward Theorem~\ref{thm:A} is the “next main theorem” Theorem~\ref{thm:B}, a principal block version (part of \cite[Conj.~D]{Isa-Nav02}). The overall structure is: prove Theorem~\ref{thm:B} (Section~\ref{sec:thm B}); then reduce Theorem~\ref{thm:A} to simple groups (Section~\ref{sec:reduction}); and finally complete the proof in the simple/almost simple case (Section~\ref{sec:simples}).

Question: Let $G$ be a finite group, let $P\in \Syl_3(G)$ be a Sylow $3$-subgroup, and let $B_0(G)$ denote the principal $3$-block of $G$. Let $\Phi(P)$ be the Frattini subgroup of $P$. Let $\sigma$ be the Galois automorphism that fixes all roots of unity of order prime to $3$ and sends every $3$-power root of unity to its fourth power; a character is called $\sigma$-invariant if it is fixed by this automorphism on its values. Assume that $B_0(G)$ contains exactly $6$ or exactly $9$ irreducible characters $\chi$ such that $3\nmid \chi(1)$ and $\chi$ is $\sigma$-invariant. Which conclusion about $P$ holds?

The Frattini quotient of $P$ has order $9$; equivalently, $|P:\Phi(P)|=9$.

Choices:
B. The Frattini quotient of $P$ has order at most $9$; equivalently, $|P:\Phi(P)|\in\{3,9\}$.
C. The Sylow $3$-subgroup $P$ is noncyclic.
D. The Frattini quotient of $P$ has order $9$ or $27$; equivalently, $|P:\Phi(P)|\in\{9,27\}$.
E. The subgroup $P$ itself has order $9$; equivalently, $P\cong C_9$ or $P\cong C_3\times C_3$.

drop the exact order $|P:\Phi(P)|=9$ and keep only noncyclicity

replace the exact Frattini quotient size by a larger nearby range

confuse the order of $P$ with the order of its Frattini quotient


[Local density of $\mathcal{S}$ in short intervals]\label{thm:main-elements}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
\#\left(\mathcal{S}\cap(x,x+h]\right)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.

\begin{theorem}[Large multiplicities in short intervals]\label{thm:erdos-short-interval}
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item There exist constants $c_1,c_2>0$ such that for $X$ sufficiently large there exists $n\in\left(X,X+X\exp(-c_1\sqrt{\log X})\right]$ with
\[
  f_{\mathrm{Rom}}(n)\ \geqslant\ c_2\,\log_2 n.
\]
\item Let $0<\theta<1$ and set $h:=X^{\theta}$. There exist constants $c_1(\theta),c_2(\theta)>0$ such that, for $X$ sufficiently large, for at least $c_1(\theta)\,X$ integers $x\in(X,2X]$ there exists $n\in(x,x+h]$ with
\[
  f_{\mathrm{Rom}}(n)\ \geqslant\ c_2(\theta)\,\log_2 n.
\]
\end{enumerate}
\end{theorem}

Let $\mathbb{P}$ denote the set of prime numbers.
Romanoff proved that the sumset $\mathbb{P}+\{2^k:k\geqslant 0\}$ has positive lower density~\cite{Romanoff1934}, following earlier questions of de~Polignac~\cite{dePolignac1849a,dePolignac1849b}.
Erd\H{o}s initiated a systematic study of multiplicities and related variants~\cite{Erdos1950} (see also van~der~Corput~\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\cite{ChenDing2023}.
See also Elsholtz and Schlage-Puchta~\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.

In this paper we study a short-interval refinement for the structured additive set $\cA_{\lambda}$ introduced by Ding and Zhai~\cite{DingZhai}.
For $h=X^{\theta}$ in the range of \Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\in[X,2X]$ the interval $(x,x+h]$ contains $\asymp h$ elements of the sumset $\mathbb{P}+\cA_{\lambda}(X)$, and the number of representations in such windows is also $\asymp h$.
In the model case of the Romanoff representation function
$$ f_{\mathrm{Rom}}(n):=\#\left\{(p,k)\in\mathbb{P}\times\mathbb N_0:\ p+2^k=n\right\}, $$
we additionally record a short-interval Erd\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.

Let $X\geqslant 3$ and set $h=X^{\theta}$ with $\theta\in(0,1)$.
Fix an integer $s\geqslant 2$ and positive real numbers $r_1,\dots,r_s>1$ such that
\begin{equation}\label{eq:reciprocal-sum}
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}<1\leqslant \frac{1}{r_1}+\cdots+\frac{1}{r_s}.
\end{equation}
Let $\lambda>0$ be defined by
\begin{equation}\label{eq:lambda-balance}
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}+\frac{1}{\lambda r_s}\ =\ 1,
\end{equation}
so that in particular $\lambda\geqslant 1$ by \eqref{eq:reciprocal-sum}.
Define the (Ding--Zhai~\cite{DingZhai}) additive set
\begin{equation}\label{eq:def-A}
\cA_{\lambda}\ :=\ \left\{2^{\lfloor k_1^{r_1}\rfloor}+\cdots+2^{\lfloor k_{s-1}^{r_{s-1}}\rfloor}+2^{\lfloor \lfloor k_s^{\lambda}\rfloor^{r_s}\rfloor}\ :\ k_1,\dots,k_s\in\mathbb{N}\right\}\subset\mathbb{N}.
\end{equation}
For $X\geqslant 3$ we truncate
\[
\cA_{\lambda}(X)\ :=\ \cA_{\lambda}\cap\left[1,2X\right],
\qquad
\mathcal{S}\ :=\ \mathbb{P}+\cA_{\lambda}(X).
\]
For real $x\geqslant 1$ we denote by
\[
A_{\lambda}(x)\ :=\ \#\left(\cA_{\lambda}\cap\left[1,x\right]\right)
\]
the associated counting function. The bound $A_{\lambda}(x)\asymp\log x$ holds for all $x\geqslant 3$ (see Ding--Zhai~\cite{DingZhai}, ultimately due to Chen--Xu~\cite{ChenXu2024}).
In particular,
\begin{equation}\label{eq:sizeA}
\#\cA_{\lambda}(X)\ \asymp\ \log X.
\end{equation}

In what follows, the parameters $s,r_1,\dots,r_s$ (and hence $\lambda$ and the set $\cA_{\lambda}$) are fixed, and all implied constants are allowed to depend on this data.

Define the representation function
\begin{equation}\label{eq:def-f}
f(n)\ :=\ \sum_{a\in\cA_{\lambda}(X)} \1_{\mathbb{P}}(n-a),
\end{equation}
so that $f(n)$ counts representations $n=p+a$ with $p\in\mathbb{P}$ and $a\in\cA_{\lambda}(X)$.
For real $x$ define the window sums
\begin{equation}\label{eq:def-R}
R(x)\ :=\ \sum_{x<n\leqslant x+h} f(n),
\qquad
Q(x)\ :=\ \sum_{x<n\leqslant x+h} f(n)^2,
\end{equation}
and the corresponding count of \emph{elements} of the sumset $\mathcal{S}$,
\begin{equation}\label{eq:def-Sx}
S(x)\ :=\ \#\left(\mathcal{S}\cap(x,x+h]\right)\ =\ \sum_{x<n\leqslant x+h}\1_{f(n)\geqslant 1}.
\end{equation}

\medskip

\begin{abstract}
Let $X$ be large and let $\PP$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda>0$ determined by a balancing relation, and let $\cA_\lambda(X)\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\cS_\lambda:=\PP+\cA_\lambda(X)$.
Fix $\varepsilon>0$, choose $\theta$ with $2/15+\varepsilon<\theta<1-\delta_0$, where $\delta_0>0$ is an absolute constant, and set $h=X^{\theta}$.
We prove that for all but $O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ values of $x\in[X,2X]$, the short interval $(x,x+h]$ contains $\asymp_{\varepsilon} h$ integers of the form $p+a$ with $p\in\PP$ and $a\in\cA_\lambda(X)$.
\end{abstract}

\medskip

Theorem~\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the
sumset $\cS$ meets almost all intervals of length $h$ with the expected order of magnitude.
To access it through Cauchy--Schwarz, it is convenient to also record the corresponding
first-moment statement for the representation function $f$.

\begin{theorem}[Typical number of representations]\label{thm:main-rep}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
R(x)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.
\end{theorem}

\begin{lemma}[Guth--Maynard, almost-all short intervals]\label{lem:GM}
Fix $\varepsilon>0$. There exists an absolute constant $\delta_0>0$ such that the following holds. Let $X\geqslant 3$ and let $y$ satisfy
\[
X^{2/15+\varepsilon}\ \leqslant\ y\ \leqslant\ X^{1-\delta_0}.
\]
Then there exists a constant $c=c_{\varepsilon}>0$ such that for all but
$O_{\varepsilon}\left(X\exp\left(-c(\log X)^{1/4}\right)\right)$ integers $t\in[X,2X]$ one has
\[
\pi(t+y)-\pi(t)\ =\ \frac{y}{\log t}\ +\ O_{\varepsilon}\left(y\exp\left(-c(\log X)^{1/4}\right)\right).
\]
This follows from \cite[Cor.~1.4]{GuthMaynard2024}; the corollary is stated for real $x$, and restricting to integer $t$ gives the same exceptional-set bound.
\end{lemma}

\begin{proposition}[First moment lower bound]\label{prop:first-moment}
Fix $\varepsilon>0$ and let $\delta_0>0$ be as in \cref{lem:GM}. Let $\theta$ satisfy $2/15+\varepsilon\leqslant\theta<1-\delta_0$ and set $h=X^{\theta}$. Then there exists a set
$\mathcal{E}\subset[X,2X]\cap\Z$ with
\[
\#\mathcal{E}\ \ll_{\varepsilon,\lambda}\ X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)
\]
such that
\[
R(x)\ \gg\ h
\]
for all $x\in([X,2X]\cap\Z)\setminus\mathcal{E}$.
\end{proposition}

We apply \cref{lem:GM} twice: first with the scale parameter $X' := X/2$ to control $\pi(t+h)-\pi(t)$ for $t\in[X/2,X]$, and then with the scale parameter $X' := X$ to control it for $t\in[X,2X]$. Since $\theta\geqslant 2/15+\varepsilon$, for $X$ sufficiently large we have $(X/2)^{2/15+\varepsilon/2}\leqslant h$. Moreover, since $\theta<1-\delta_0$ with $\delta_0>0$ fixed, for $X$ sufficiently large we also have $h=X^{\theta}\leqslant (X/2)^{1-\delta_0}$. Consequently the hypotheses of \cref{lem:GM} apply (with $\varepsilon$ replaced by $\varepsilon/2$) on both dyadic blocks. We therefore obtain an exceptional set $\mathcal{E}_a\subset[X,2X]\cap\Z$ with
\[
\#\mathcal{E}_a\ \ll_{\varepsilon}\ X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)
\]
such that for all $x\in([X,2X]\cap\Z)\setminus\mathcal{E}_a$ we have
\begin{equation}\label{eq:GM-shifted}
\pi(x+h-a)-\pi(x-a)\ =\ \frac{h}{\log(x-a)}\ +\ O_{\varepsilon}\left(h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right).
\end{equation}

For $x\in([X,2X]\cap\Z)\setminus\mathcal{E}$, summing \eqref{eq:GM-shifted} over $a\in\cA_{\lambda}(X;h)$ yields
\[
\begin{aligned}
R(x)
&\geqslant\ \sum_{a\in\cA_{\lambda}(X;h)}\left(\pi(x+h-a)-\pi(x-a)\right)
\\
&=\ h\sum_{a\in\cA_{\lambda}(X;h)}\frac{1}{\log(x-a)}\ +\ O_{\varepsilon}\left(A_h h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right).
\end{aligned}
\]
Since $x-a\asymp X$ uniformly for $x\in[X,2X]$ and $a\leqslant h/2\leqslant X/2$, the main term satisfies
\[
\sum_{a\in\cA_{\lambda}(X;h)}\frac{1}{\log(x-a)}\ \asymp\ \frac{A_h}{\log X}\ \asymp\ \frac{\log h}{\log X},
\]
and the error term is negligible. Since $h=X^{\theta}$, we have $\log h=\theta \log X$, hence $R(x)\gg h$ for all $x\notin\mathcal{E}$.
\end{proof}

\begin{theorem}[Local density of $\cS$ in short intervals]\label{thm:main-elements}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
\#\left(\cS\cap(x,x+h]\right)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.
\end{theorem}

Theorem~\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the
sumset $\mathcal{S}$ meets almost all intervals of length $h$ with the expected order of magnitude.
To access it through Cauchy--Schwarz, it is convenient to also record the corresponding
first-moment statement for the representation function $f$.

\begin{theorem}[Typical number of representations]\label{thm:main-rep}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
R(x)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.
\end{theorem}

\medskip

\noindent\emph{Strategy.} The pointwise upper bound $R(x)\ll h$ follows from Brun--Titchmarsh together with $\#\cA_{\lambda}(X)\asymp\log X$.
The almost-all lower bound $R(x)\gg h$ is obtained by restricting to $a\leqslant h/2$ and applying the Guth--Maynard~\cite[Cor.~1.4]{GuthMaynard2024} asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).
To pass from representations to elements we combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\ll h$.
The latter is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\in\cA_{\lambda}(X)$.

\medskip

\noindent\emph{Organization of the paper.} Section~2 establishes the first-moment estimate for the local representation count $R(x)$ in almost all short intervals. Section~3 proves a uniform local mean-square bound, leading to the required second-moment control. In Section~4 we combine these inputs with Cauchy--Schwarz to deduce Theorems~\ref{thm:main-elements} and~\ref{thm:main-rep}. Finally, Section~5 proves Theorem~\ref{thm:erdos-short-interval} on large short-interval multiplicities in the Romanoff model.

“To access [Theorem~\ref{thm:main-elements}] through Cauchy--Schwarz,” the paper first proves a first-moment statement for the representation function $f$, namely Theorem~\ref{thm:main-rep} that $R(x)\asymp h$ for almost all $x\in[X,2X]$. For this, the “pointwise upper bound $R(x)\ll h$ follows from Brun--Titchmarsh together with $\#\cA_{\lambda}(X)\asymp\log X$,” while the “almost-all lower bound $R(x)\gg h$ is obtained by restricting to $a\leqslant h/2$ and applying the Guth--Maynard ... asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).”

“To pass from representations to elements” (i.e. to deduce Theorem~\ref{thm:main-elements}), they “combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\ll h$.” This uniform second-moment bound “is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\in\cA_{\lambda}(X)$.”

Question:
Let \(\mathbb P\) denote the prime numbers. Fix an integer \(s\ge 2\) and real numbers \(r_1,\dots,r_s>1\) such that
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}<1\le \frac{1}{r_1}+\cdots+\frac{1}{r_s},
\]
and define \(\lambda>0\) by
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}+\frac{1}{\lambda r_s}=1.
\]
Set
\[
\mathcal A_\lambda:=\left\{2^{\lfloor k_1^{r_1}\rfloor}+\cdots+2^{\lfloor k_{s-1}^{r_{s-1}}\rfloor}+2^{\lfloor \lfloor k_s^{\lambda}\rfloor^{r_s}\rfloor}: k_1,\dots,k_s\in\mathbb N\right\},
\]
\[
\mathcal A_\lambda(X):=\mathcal A_\lambda\cap[1,2X],\qquad \mathcal S:=\mathbb P+\mathcal A_\lambda(X).
\]
Fix \(\varepsilon>0\), choose \(\theta\) with \(2/15+\varepsilon<\theta<1-\delta_0\), where \(\delta_0>0\) is an absolute constant, and set \(h=X^{\theta}\). Which statement holds for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\)?

\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
that is, for all but that exceptional set of integers \(x\), the interval \((x,x+h]\) contains on the order of \(h\) integers of the form \(p+a\) with \(p\in\mathbb P\) and \(a\in\mathcal A_\lambda(X)\); the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).

Choices:
B.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
for every integer \(x\in[X,2X]\cap\mathbb Z\); equivalently, there is no exceptional set, and the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
C.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\gg h,\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\); the implied constant depends at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
D.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) real numbers \(x\in[X,2X]\); the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
E.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)=h+O_{\varepsilon}\!\left(h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right),\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\), with implied constants depending at most on \(\lambda\), \(\varepsilon\), and \(\theta\).


Question:
Let \(\mathbb{P}\) be the set of prime numbers. Fix an integer \(s\ge 2\) and real numbers \(r_1,\dots,r_s>1\) such that
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}<1\le \frac{1}{r_1}+\cdots+\frac{1}{r_s},
\]
and let \(\lambda>0\) be defined by
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}+\frac{1}{\lambda r_s}=1.
\]
Define
\[
\mathcal A_\lambda:=\left\{2^{\lfloor k_1^{r_1}\rfloor}+\cdots+2^{\lfloor k_{s-1}^{r_{s-1}}\rfloor}+2^{\lfloor \lfloor k_s^{\lambda}\rfloor^{r_s}\rfloor}: k_1,\dots,k_s\in\mathbb N\right\},
\]
its truncation \(\mathcal A_\lambda(X):=\mathcal A_\lambda\cap[1,2X]\), and the sumset
\[
\mathcal S:=\mathbb P+\mathcal A_\lambda(X)=\{p+a: p\in\mathbb P,\ a\in\mathcal A_\lambda(X)\}.
\]
Fix \(\varepsilon>0\), let \(\theta\) satisfy \(2/15+\varepsilon<\theta<1-\delta_0\) for some absolute constant \(\delta_0>0\), and set \(h=X^{\theta}\). As \(X\to\infty\), which asymptotic statement holds for the number of elements of \(\mathcal S\) in the short interval \((x,x+h]\) for integers \(x\in[X,2X]\)?

For all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\), one has
\[
\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,
\]
where the implied constants in \(\asymp\) depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).

Choices:
B.
For all integers \(x\in[X,2X]\), one has
\[
\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,
\]
with implied constants depending at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
C.
For all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\), one has
\[
\#\bigl(\mathcal S\cap(x,x+h]\bigr)\gg h,
\]
where the implied constant depends at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
D.
For all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\), one has
\[
\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp \frac{h}{\log X},
\]
where the implied constants in \(\asymp\) depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
E.
There exists a constant \(c_{\varepsilon}>0\) such that for every integer \(x\in[X,2X]\), one has
\[
\#\bigl(\mathcal S\cap(x,x+h]\bigr)= h+O_{\lambda,\varepsilon,\theta}\!\left(h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right).
\]

pointwise asymptotic replacing almost-all comparability