\begin{theorem}\label{thm_main}
Let $M$ be a closed topological manifold of dimension $2n$ and let $b = \dim H^2(M,\mathbb{C})$.

Then there exist non-zero elements $e_1,\ldots,e_q \in H^2(M,\mathbb{C})$ with $q \leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\colon X \to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \in \{1, \dots, q\}$ such that $[E] = e_i$.
\end{theorem}

This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \ref{thm:W_F}, where we prove the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.

\medskip

The structure of the paper is as follows. In Section \ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \ref{s_low} and \ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many  classes $e\in H^2(M,\mathbb C)$ such that
$e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.
(cf. Theorem \ref{thm:W_F}). Section \ref{s_blowups} establishes our main theorem by examining the rank  of the associated  form in the presence of a divisorial contraction. In Section \ref{s_volume},  we present some related open problems.

\medskip

\textbf{Acknowledgements:}  The first and third authors are members of the GNSAGA - Istituto Nazionale di Alta Matematica.
The first author was also partially supported by the PRIN ``Vartietà reali e complesse: geometria, topologia e analisi armonica".
The second author is partially supported by a Simons collaboration grant  and would like to thank the National Center for Theoretical Sciences in Taiwan and Professor Jungkai Chen for
their  hospitality, where some of the work for this paper was completed.
The third author is partially supported by the PRIN2020 research grant ``2020KKWT53”.
The authors would also like to thank the referee for careful reading the paper and for several useful comments.

\section{Preliminaries}\label{s_preliminaries}
We work over the field of complex numbers $\mathbb C$. A \emph{form} over $\mathbb C$ is a homogeneous polynomial $F\in \mathbb C[x_0,\ldots,x_b]$.

\subsection{Tensors} Let $a_1,\dots,a_n$ be positive integers. A \emph{tensor} of type $a_1\times \ldots \times a_n$ is a multilinear map
$$T\colon \mathbb C^{a_1}\times \ldots \times \mathbb C^{a_n}\to \mathbb C.$$
For any positive integer $a$, denote $[a]:=\{ 1,2,\ldots, a \}$. Then a tensor $T$ of type  $a_1\times \ldots \times a_n$ is determined uniquely by a function
$$\tilde T\colon [a_1]\times \cdots \times [a_n] \to \mathbb{C}.$$
We will refer to $\tilde T$ as the \emph{hypermatrix associated to $T$}.
Given positive integers
$a_j^{'} \le a_j$ for $1 \le j \le n,$ and strictly increasing functions $f_j\colon [a_j^{'}] \to [a_j]$,
we define a \emph{sub-tensor} $T'$ of type $a'_1\times\ldots\times a'_n$ so that if $\tilde T'$ is the hypermatrix associated to $T'$ then
$$\tilde T'=\tilde T(f_1,\dots,f_n)\colon [a'_1]\times \cdots \times [a'_n] \to \mathbb{C}.$$
Note that if $v_i\in \mathbb C^{a_i}$ for $i=1,\dots,n$, then there exists a natural tensor of type $a_1\times \ldots \times a_n$ defined by
$v_1\otimes \ldots \otimes v_n$.

\begin{definition}
A nonzero tensor $T$ of type  $a_1\times \ldots \times a_n$ has {\it rank one} if there are nonzero vectors $v_i \in \mathbb{C}^{a_i}$ such that $T=v_1 \otimes \cdots \otimes v_n $. We define the
{\it rank} of a nonzero tensor $T$ ($=\rk(T)$) to be the minimum positive integer $r$
such that there exist $r$ tensors $T_1, \cdots , T_r$ of rank one with

$$T = T_1 + \cdots + T_r .$$
\end{definition}

Note that the rank of any sub-tensor of $T$ cannot be larger than the rank of $T$.
\begin{definition}\label{d_cubic}
A \emph{cubic tensor} $T$ is a tensor of type $d^{\times n} =d \times \cdots \times d$. Let $A$ be the hypermatrix associated to $T$.
We denote by $\det A$ the \emph{hyperdeterminant} of  $A$ (see \cite[Chapter 14]{GKZ} for the definition and some of its main properties). In particular, $\det A$  is a polynomial  in the entries of $A$.

\end{definition}

The following theorem is useful to compute the rank of a tensor:
\begin{theorem}{\cite[Thm. 3.1.1.1]{Landsberg-book}} \label{thm:slices}
Let $A_1,\dots,A_n$ be $l\times m$ matrices and consider the  tensor $A=[A_1,\ldots,A_n]$ of type $l\times m\times n$.

Then $\rk(A)$ is equal to  the minimum number of rank one matrices needed to span a vector space containing $\langle A_1,\ldots, A_n \rangle$.
\end{theorem}

\begin{proof}
Let $r=\rk(A)$ and let $B_1,\ldots, B_r$ be rank one matrices such that
$$\langle A_0,\ldots, A_q\rangle\subset \langle B_1,\ldots, B_r\rangle,$$
as in Theorem \ref{thm:slices}.
Thus, for any $j=0,\ldots,q$, we may write
$$
A_j= \sum_{i=1}^r \lambda_{ij}B_i
$$
for $\lambda_{ij} \in \C$ for $i=1,\ldots,r$ and $j=0,\dots,q$. Since $A_1,\ldots, A_q$ are linearly independent, after possibly reordering the $B_i$'s, we may assume that the $q\times q$-matrix
$$L=(\lambda_{ij})_{i,j=1,\ldots,q}$$
 is of maximal rank $q$. Thus, the linear system
$$L \cdot X =(-\lambda_{10},-\lambda_{20},\ldots,-\lambda_{q0})^t$$
admits a solution $(\mu_1,\ldots,\mu_q)\in \mathbb C^q$.
It follows that
 $$A_0 + \sum_{i=1}^q \mu_i A_i=\sum_{j=q+1}^{r} s_j B_j$$ for some  $s_{q+1},\ldots,s_r \in \C$.
 Since the rank of $\sum_{j=q+1}^{r} s_j B_j$ is at most $r-q$, our assumption implies that $r-q \ge t$  and the claim follows.
\end{proof}

\section{Points of low rank}\label{s_low}

\begin{definition}
	Given a form $F \in \C[x_0,\ldots,x_b]$ of degree $n \ge 3$ we consider the tensor
of type $(b+1)^{\times (n-1)}$
		defined by the hypermatrix $\mathcal H_F$ of the $(n-1)$-th order derivatives, whose entries are linear forms in $x_0, \ldots, x_b$.
	We say that $F$ is \emph{honest} if for any non-zero element $v \in \mathbb C^{b+1}$, we have  $\mathcal H_F(v) \ne 0$.
    We say that $F$ is  \emph{non-degenerate} if $ \det \mathcal H_F(v) \ne 0$ for some
 non-zero $v \in \mathbb C^{b+1}$
 (note that this definition is consistent with \cite[Page 7929]{CT18} but not with \cite[14.1.A]{GKZ}).
\end{definition}

Note that the locus $p \in \mathbb P^b$ such that  $\mathcal H_F(p) = 0$ is well defined. By \cite[Remark 6.3.5]{Bocci-Chiantini}, the rank $\rk \mathcal H_F(p)$ is also well-defined.

\begin{example}\label{ex:degenerate}
	Consider the  form
	$$
	F(x_0,\ldots,x_4)=\frac{x_0x_1^2}{2} + x_1x_3x_4 + \frac{x_2x_3^2}{2}.
	$$
	Then

	$$
	\mathcal H_F=\begin{pmatrix}
	0   & x_1 & 0   & 0   & 0   \\
	x_1 & x_0 & 0   & x_4 & x_3 \\
	0   & 0   & 0   & x_3 & 0   \\
	0   & x_4 & x_3 & x_2 & x_1 \\
	0   & x_3 & 0   & x_1 & 0
	\end{pmatrix}
	$$
Thus, the form $F$ is honest, but it is degenerate since $\det \mathcal H_F$ is identically zero.
\end{example}

Vaguely speaking, an honest form can be characterised by the fact that it  depends on all the variables. More precisely, we have:
\begin{lemma}\label{lem:H_F(p)=0}
	Let $F \in \C[x_0,\ldots,x_b]$ be a form of degree $n$ and let
    $p:=[1,0,\ldots,0] \in \mathbb P^b$.

    Then $\mathcal H_F(p) = 0$ if and only if $x_0$ does not appear in the expression of $F$.  In particular, a non-degenerate form is honest.
\end{lemma}

\begin{proof}
	Assume first that $\mathcal H_F(p) = 0$. By  the Euler formula for homogeneous polynomials we have that
	$$
	(n-k)\partial_{i_1 \ldots i_k}F(x_0,\ldots,x_b)= \sum_{\ell=0}^b x_\ell \partial_{\ell, i_1 \ldots i_k}F(x_0, \ldots, x_b)
	$$
	for any $k=0,\ldots, n-1$ and $i_j \in \{0,\ldots, b\}$.
    Proceeding by induction and using the fact that $\mathcal H_F(p) = 0$, we get that $\partial_{i_1 \ldots i_k}F(p)=0$ for any $k=0,\ldots, n-1$ and $i_j \in \{0,\ldots, b\}$. Thus, $x_0$ does not appear in $F(x_0, \ldots, x_n)$.
	The converse is a simple computation.

\medskip

Assume now that $F(x_0, \ldots, x_n)$ is a non-degenerate form. If by contradiction there exists $p\in \mathbb P^b$ such that $\mathcal H_F(p) = 0$ then, up to a  change of coordinates, we may assume $p=[1,0,\ldots,0]$. This means that $F$ does not depends on $x_0$ and so the face of $\mathcal H_F(p)=0$  corresponding to $\partial_0$ is trivial. By \cite[Corollary XIV.1.5(d)]{GKZ}, it follows that $\det \mathcal H_F$ is identically zero, which is a contradiction.
\end{proof}

\begin{pmatrix}
0 & x_1   \\
x_1 & x_0   \\
\end{pmatrix}.
$$
By \cite[Proposition XIV.1.7]{GKZ}, it follows that $F_X$ is degenerate.
\end{example}

\section{Blow-ups}\label{s_blowups}

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form  of degree $i$, for $i=1,\dots,n-k$.

\end{lemma}
\begin{proof}
This follows easily from the projection formula.
\end{proof}

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1}  If $n\ge 3$ and  $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\end{enumerate}
\end{proposition}
\begin{proof}
We use the same notation as in Lemma \ref{lem:blowup}.

We first prove \eqref{k=0}. We have
$$
F_Y(x_0,\ldots,x_n)=ax_0^n  + F_X(x_1, \ldots, x_b)
$$
where $a=E^n \ne 0$. Hence $\rk \mathcal H_{F_Y}(p)=1$.
Thus, \eqref{k=0} follows.
\medskip

We now prove \eqref{k=1}. We have
$$
F_Y(x_0,\ldots,x_n)=ax_0^n+ x_0^{n-1}\cdot \left(\sum_{i=1}^{b}\lambda_ix_i \right )  + F_X(x_1, \ldots, x_b )
$$
where $a=E^n$ and $\lambda_i=Z \cdot \gamma_i$, for $i=1,\dots,b$. After taking a base change that fixes $(1,0,\ldots,0)$, we may assume $\lambda_i=0$ for any $i=2, \ldots ,b$, i.e.

$$
F_Y(x_0,\ldots,x_n)=ax_0^n+ \lambda_1 x_0^{n-1}x_1  + F_X(x_1, \ldots, x_b ),
$$
where $\lambda_1 \ne 0$ because we are blowing up a curve $Z$ in a K\"ahler manifold and so there exists at least one class in $H^2(X,\mathbb Z)$ with non-zero intersection with $Z$. Thus, it is easy to check that the subtensor given by
$$(\partial_0^{n-3}\partial_i\partial_j)_{i,j=0,\dots,b}$$
has rank two,  which implies that $\rk \mathcal H_{F_Y}(p) \ge 2$.

\medskip

We finally prove \eqref{k=2}. Let $j\colon Z \hookrightarrow X$ be the inclusion.
Let  $V \subset H^2(Z, \C)$ be the subspace generated by $j^*\gamma_1, \ldots, j^*\gamma_b$ and let $q$ be the rank of the quadratic form $Q$ obtained restrincting to $V$ the quadratic form of $H^2(Z,\mathbb C)$.
 Note that $q \ge 1$ because a K\"ahler class of $X$ restricts to a K\"ahler class on $Z$.
After a base change that diagonalises $Q$, we can write
$$
F=F_Y(x_0,\ldots,x_b)=\frac{ax_0^n}{n!} + \frac{x_0^{n-1}}{(n-1)!}(c_1x_1+\ldots +c_bx_b) + \frac{x_0^{n-2}}{2(n-2)!}(x_1^2+\ldots+ x_q^2) + F_X(x_1, \ldots, x_b)
$$
for some    $c_1,\dots,c_b \in \C$.

Consider the $(b+1)\times (b+1) \times (b+1)$ subtensor $A$ of $\mathcal H_{F}(p)$ given by the slices $A_0, \ldots, A_b$ where
 $$
A_h= ( \partial^{n-4}_0 \partial_h \partial_i \partial_j F(p))_{i,j=0,\dots,b}
 $$
 for $h=0,\dots,b$.

It is enough to prove that $\rk A \ge 2q$, as it immediately implies that $\rk \mathcal H_{F}(p) \ge 2q$.
We have
$$
A_0 =
\begin{pmatrix}
1 & c_1 & \ldots & c_q & c_{q+1} & \ldots & c_b \\
c_1 & 1  & 0 & \ldots & \ldots & \dots & 0 \\
\vdots & 0 & \ddots  & 0 & \cdots & \cdots & \vdots \\
c_q & 0 & \ldots & 1 & 0 & \ldots & 0  \\
c_{q+1} &\vdots & \vdots  &\vdots & 0 & \ldots & 0 \\
\vdots&\vdots & \vdots  &\vdots & \vdots & \ddots & \vdots \\
c_b & \cdots & \cdots & \cdots & 0 & \cdots & 0 \\
\end{pmatrix}

\label{s_volume}

The goal of this section is to present some open problems on the relationship between birational invariants and topological invariants of a smooth complex projective variety.

\sub

Let $X$ be a compact K\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.

For $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.

Despite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \ref{s:intersection} for the definition of $\mathcal H_{F_X}$ and some of its basic properties).

\medskip

Our main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:

\label{s:intersection}

Let $M$ be a closed (i.e.\ compact and without boundary) oriented topological manifold of dimension $d=2n$.
Consider the symmetric
$n$-multilinear map defined by the cup prod

Our main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:

This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \ref{thm:W_F}, where we prove the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.

The structure of the paper is as follows. In Section \ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \ref{s_low} and \ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that
$e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.
(cf. Theorem \ref{thm:W_F}). Section \ref{s_blowups} establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction. In Section \ref{s_volume}, we present some related open problems.

\subsection{Tensors} Let $a_1,\dots,a_n$ be positive integers. A \emph{tensor} of type $a_1\times \ldots \times a_n$ is a multilinear map
$$T\colon \mathbb C^{a_1}\times \ldots \times \mathbb C^{a_n}\to \mathbb C.$$
For any positive integer $a$, denote $[a]:=\{ 1,2,\ldots, a \}$. Then a tensor $T$ of type $a_1\times \ldots \times a_n$ is determined uniquely by a function
$$\tilde T\colon [a_1]\times \cdots \times [a_n] \to \mathbb{C}.$$
We will refer to $\tilde T$ as the \emph{hypermatrix associated to $T$}.
Given positive integers
$a_j^{'} \le a_j$ for $1 \le j \le n,$ and strictly increasing functions $f_j\colon [a_j^{'}] \to [a_j]$,
we define a \emph{sub-tensor} $T'$ of type $a'_1\times\ldots\times a'_n$ so that if $\tilde T'$ is the hypermatrix associated to $T'$ then
$$\tilde T'=\tilde T(f_1,\dots,f_n)\colon [a'_1]\times \cdots \times [a'_n] \to \mathbb{C}.$$
Note that if $v_i\in \mathbb C^{a_i}$ for $i=1,\dots,n$, then there exists a natural tensor of type $a_1\times \ldots \times a_n$ defined by
$v_1\otimes \ldots \otimes v_n$.

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form of degree $i$, for $i=1,\dots,n-k$.

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1} If $n\ge 3$ and $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\begin{example}
Let $X$ be a complex projective manifold of dimension $n$ and $b_2(X)=1$. Let $f\colon Y \to X$ be the blow-up along a smooth curve $Z$ of $X$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor. Then we can write
$$
F_Y(x_0,\ldots,x_n)=\frac{a}{n!}x_0^n+ \frac{x_0^{n-1}x_1}{(n-1)!} + x_1^n,
$$
for some $a \in \mathbb Q$. If $n=3$, then
$$
\mathcal H_{F_Y}(p)=
\begin{pmatrix}
a & 1 \\
1 & 0 \\
\end{pmatrix}
$$
has rank 2. If $n=4$, then
$$
\mathcal H_{F_Y}(p)=
\left[ \begin{pmatrix}
a & 1 \\
1 & 0 \\
\end{pmatrix},
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
\right]
$$
has rank at least 3 by Lemma \ref{lem:trick}. On the other hand, 3 is the maximal rank for a $2\times 2 \times 2$ tensor (see, for instance, \cite[Section 3]{Kruskal}). Thus, $\mathcal H_{F_Y}(p)$ has rank $3$.
\end{example}

\begin{question}\label{q:volume}
Let $M$ be a closed topological manifold of dimension $2n$. Is there a constant $C$, depending only on $M$, such that for any smooth complex projective variety $X$ whose underlying topological space is $M$, we have
\[ \vol(X) \le C? \]
\end{question}

The proof of Theorem~\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that $e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one” (cf. Theorem~\ref{thm:W_F}), which itself comes from “the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Then Section~\ref{s_blowups} “establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\colon X\to Y$ to one of those finitely many rank-one Hessian classes, yielding the finite list $e_1,\ldots,e_q$.

The proof of Theorem~\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\in H^2(M,\mathbb C)$ such that $e^n\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.” We first record the precise finiteness statement:

\begin{theorem}\label{thm:W_F}
Let $F \in \C[x_0, x_1, \ldots,x_b]$ be a form of degree $n \ge 3$.
\begin{enumerate}
\item If $F$ is honest, then $W_F \cap \{F \ne 0\}$ is a finite set, and more precisely,
$$
|W_F  \cap \{F \ne 0\} | \le b+1.
$$

\item If $F$ is non-degenerate, then $W_F$ is a finite set.
\end{enumerate}
\end{theorem}

This itself comes from “the finiteness of rank one points of $\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Next we study blow-ups and use them to connect exceptional divisor classes to these finitely many rank-one Hessian classes. In particular, we use the following structural description of the intersection form under blow-up:

\begin{lemma}\label{lem:blowup}
Let $X$ be a smooth projective variety of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$.
Let $[E], \beta_1:=f^*\gamma_1, \ldots, \beta_b:=f^*\gamma_b$ be a basis of $H^2(Y,\Q)$ where $E$ is the exceptional divisor and $\gamma_1,\ldots,\gamma_b$ is a basis of $H^2(X,\Q)$.
Then, with respect to this basis, we may write
$$
F_Y(x_0,\ldots,x_n)=ax_0^n +\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \ldots, x_b)
$$
where $a=E^n$ and $R_i \in \mathbb Q[x_1,\ldots,x_b]$ is a form  of degree $i$, for $i=1,\dots,n-k$.

\end{lemma}

We then control the rank of the Hessian at the exceptional divisor class $p=[E]$ in low-dimensional centers of blow-up via the following result:

\begin{proposition}\label{prop:blowuprank}
Let $X$ be a smooth K\"ahler manifold of dimension $n$ and let $f\colon Y \to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \le 2$. Let $p=[E] \in H^2(Y,\C)$ be the class of the exceptional divisor.
\begin{enumerate}
\item \label{k=0} If $n\ge 2$ and $k=0$ then $\rk \mathcal H_{F_Y}(p)=1$.
\item \label{k=1}  If $n\ge 3$ and  $k=1$ then $\rk \mathcal H_{F_Y}(p) \ge 2$.
\item \label{k=2} If $n\ge 4$ and $k=2$ then, after a base change, we may write
$$
F_Y(x_0,\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\ldots,x_b) + x_0^{n-2}Q(x_1,\ldots,x_b) + F_X(x_1, \ldots, x_b)
$$
where $a \in \mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\rk \mathcal H_{F_Y}(p) \ge 2q$.

\end{enumerate}
\end{proposition}

In establishing the main theorem, we apply these blow-up computations to examine “the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\colon X\to Y$ to one of the finitely many rank-one Hessian classes provided by Theorem~\ref{thm:W_F}, yielding the finite list $e_1,\ldots,e_q$.

Question: Let $M$ be a closed (compact and without boundary) topological manifold of real dimension $2n$, and set $b=\dim_{\mathbb C} H^2(M,\mathbb C)$. For any smooth complex projective variety $X$ of complex dimension $n$ whose underlying topological space is $M$, identify $H^2(X,\mathbb C)$ with $H^2(M,\mathbb C)$. If $f\colon X\to Y$ is a divisorial contraction to a point, let $E$ denote its exceptional divisor and let $[E]\in H^2(M,\mathbb C)$ be its cohomology class under this identification. Which existence statement holds?

There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.

Choices:
B. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
C. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ for some finite $q$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
D. For every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, there exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]=e_i$.
E. There exist nonzero classes $e_1,\ldots,e_q\in H^2(M,\mathbb C)$ with $q\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\colon X\to Y$ to a point with exceptional divisor $E$, there is an index $i\in\{1,\ldots,q\}$ for which $[E]$ is proportional to $e_i$ by a nonzero scalar.

The stem sets up notation and context but does not explicitly or implicitly reveal which of the listed existence statements is correct. The correct choice is not singled out by wording in the prompt.

This is very close to a theorem-statement recognition item: the task is essentially to identify the exact valid formulation among nearby variants. It is not a pure verbatim restatement because the options differ in meaningful quantifiers and strength, but it remains only mildly non-tautological.

The question requires some careful reasoning about uniformity in X, the restriction to contractions to a point, exact equality versus proportionality, and the bound q ≤ b+1. However, it mainly tests precise statement discrimination rather than deeper mathematical generation or derivation.

The distractors are plausible and mathematically targeted: one removes the 'to a point' restriction, one weakens the bound, one alters quantifier uniformity, and one replaces equality by proportionality. These are distinct and reflect common theorem-statement confusion modes.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it primarily tests precise recall/parsing of a statement rather than deeper generative reasoning.


\label{thm:sos}
		For $f\in\cA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:sos:i}
			For any Hilbert space $\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \dots, Y_{\vg})\in\mathcal{B}(\mathcal{K})^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:ii}
			For any $n\in\mathbb N$ and any tuple of self-adjoint matrices
			$Y = (Y_{1}, \dots, Y_{\vg})\in M_n(\mathbb C)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:iii}
			There exist $r_1,\ldots,r_{N(d)}\in\cA_d$ s.t.
			\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}
		\end{enumerate}

		\noindent If $\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[label=\textup{(\roman*)}, resume]\itemsep=5pt
			\item \label{i:sos:iv}
			There exists $r\in\cA_d$ s.t.
			\begin{equation}\label{eq:1sosthm}
				f \ =\ r^*r.
			\end{equation}

		\end{enumerate}

\begin{remark}[What's new?]\rm
		\label{rem:whatsnew}
		The passage to operator coefficients necessitates several novel
		results and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\ref{thm:sos} and Theorem~\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).
		Namely, the Hahn-Banach theorem produces a separating linear functional $\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction
		based on $\varphi$ ultimately produces a tuple $Y$.
		Here we
		roughly follow the outline of \cite{MP05}.

		A key construction is that of a tuple of self-adjoint operators $A$
		based upon the left regular representation on Fock space; see Section~\ref{sec:fock}.
		We then show that, up to a universal constant, for a sum of squares polynomial $p,$
		the norm
		of $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\ref{prop:bounded}.
		This uniform bound is the main input in Section~\ref{sec:top} for proving that the cone $\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.

		On the GNS side, we introduce a new argument that exploits the WOT to associate to a
		separating linear functional a finite-rank psd noncommutative representing Hankel matrix and,
		on its range, construct the desired tuple~$Y$; see Section~\ref{sec:gns}. For the unitary result,
		Theorem~\ref{thm:usos}, we additionally modify the construction of Section~\ref{sec:fock} to produce a canonical
		tuple of unitary operators from the left-regular representation and adapt the GNS procedure
		to obtain a unitary tuple together with a representing vector that realizes the separating
		functional as a vector state; see Subsection~\ref{ssec:uGNS}.\qed
	\end{remark}

	\subsection{Reader's guide}\label{ssec:readersguide}

	The paper is structured as follows. The convex cone of sums of squares (making an appearance in \eqref{eq:sosthm} of Theorem~\ref{thm:sos}) is introduced and characterized in the next Section~\ref{sec:sos}. In Section~\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\cF_\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\ref{sec:top} we
	introduce a suitable topology on $\cA_\ad$ and
	collect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\ref{sec:gns}; see Proposition~\ref{prop:GNS}.
	Then Theorem~\ref{thm:sos} is proved in Section~\ref{sec:proof} and
	Theorem~\ref{thm:usos} is proved in Section~\ref{sec:uproof}.

	\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\ref{prop:closedcone}.

	\section{Convex Cone of (Sums of) Squares}\label{sec:sos}

	In this section a key player in the proof of Theorem~\ref{thm:sos}, the convex cone of sums of squares of polynomials, is introduced and
	studied. The main result in this section is Proposition~\ref{prop:closed under addition} (see also Remark \ref{rem:soscone}), which  gives a bound on the number of sums of squares needed to write a polynomial as a sum of squares.

	\begin{lemma}\label{lem:psd}
		If $T: \cH^{n} \to \cH^{n}$ be a psd linear map, then there exist linear maps $R_{i}: \cH \to \cH^{n},$ $i = 1, \dots, n,$ such that $T = \sum_{i=1}^{n} R_{i}R_{i}^{*}.$ Moreover, if $\cH$ is infinite-dimensional, then $T = R R^{*}$ for some $R : \cH \to \cH^{n}.$
	\end{lemma}
	\begin{proof}

		Since $T$ is psd, there exists a linear map $\tilde{R}: \cH^{n} \to \cH^{n}$ such that $T = \tilde{R} \tilde{R}^{*}.$ Write
		\[
		\tilde{R} \ =\  \begin{bmatrix}
			R_{1}, \dots, R_{n}
		\end{bmatrix}
		\]
		with respect to the orthogonal decomposition $\cH^{n} = \cH \oplus \dots \oplus \cH.$ The first part of the lemma follows by noting that each $R_{i}$ is a map from $\cH$ into $\cH^{n}.$ For the moreover part, let $U: \cH \to \cH^{n}$ be any unitary, and set $R = \tilde{R} U.$
	\end{proof}

	Index \df{$\mathbb{\cH}^{N(d)}$} and \df{$\cA_{d}^{N(d)}$} (the algebraic direct sum of $\cA_{d}$ with itself $N(d)$ times) by $\la x \ra_{d}.$
	Let $V_{d} \in \cA_{d}^{N}$ denote the \df{Veronese column vector} whose $w\in \la x \ra_{d}$ entry is $w$
	(adopting the  usual convention of viewing $w$ as the the $\cB(\cH)$-valued polynomial $I_\cH \, w$). For instance,
	if $\vg=2$ and $d=2$, then  \index{$V_d$}
	\[
	V_2\ =\  \text{col} \begin{pmatrix}
		1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2
	\end{pmatrix}.
	\]
	Let \df{$\cC_d$} denote the \df{cone of sums of squares} of polynomials of degree at most $d,$
	\begin{equation}\label{eq:sosdef}
		\cC_{d} \ :=\  \left\{ \sum\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \quad r_{i}\in \cA_{d},  \quad  i=1, \dots, N(d) \right\}
		\subseteq \cA_{2d}.
	\end{equation}
	Given $r\in \cA_d,$ the column vector  $R$  with $w$ entry $R_{w}^{*}$
	is called the \df{coefficient vector} of $r$ since  $r = R^{*} V_{d}.$ In particular,
	\[
	r^* r \ =\  V_d^* RR^* V_d
	\]
	so that  $r^*r$ has a representation  as  $V_d^* \SG V_d$ for a psd matrix $\SG.$

	\begin{prop} \label{prop:closed under addition}
		A polynomial $p\in \cA_{2d}$ is in $\cC_d$ if and only if there is a psd block matrix $\SG$ such that
		\begin{equation}
			\label{e:gram-rep}
			p \ =\  V_d^* \SG V_d.
		\end{equation}
		In fact, if $p=V_d^* \SG V_d,$ then factoring $\SG=\sum_{j=1}^{N(d)} R_jR_j^*$  with $R_j:\cH \to \oplus_{w\in \la x\ra_d}\cH$
		as in Lemma~\ref{lem:psd},
		setting $r_j = R_{j}^{*} V_{d} $
		gives,
		\[
		p \ =\ \sum_{j=1}^{N(d)} r_j^* r_j.
		\]

		In particular,  the set $\cC_d$  is a (convex) cone. 	\end{prop}

	We call any psd block matrix $\SG$ satisfying equation~\eqref{e:gram-rep} a \df{Gram representation} for $p.$

	\begin{proof}
		Given a sum of squares $p =  \sum_{i=1}^{N(d)} r_{i}^{*}r_{i},$ writing $r_j=R_j^{*} V_d$ gives
		$p = \sum_{i=1}^{N(d)} V_{d}^{*} R_{i} R_{i}^{*} V_{d},$ where $R_{i}$ is the coefficient vector corresponding to the polynomial $r_{i}.$  It follows that  $p = V_{d}^{*} \SG V_{d},$ where $\SG = \sum_{i=1}^{N(d)} R_{i} R_{i}^{*}.$ In particular, $\SG : \cH^{N(d)} \to \cH^{N(d)}$ is a psd linear map.

		Conversely, suppose there is a psd linear map $\SG:\oplus_{w\in \la x\ra_d} \cH \to \oplus_{w\in \la x \ra_d}\cH$ such that $p = V_d^* \SG V_d.$
		By Lemma~\ref{lem:psd}, there exist $R_j: \cH\to \oplus_{w\in \la x \ra_d}\cH$ such  that
		$\SG=\sum_{j=1}^{N(d)} R_j R_j^*.$ Setting
		$r_j =R_j^* V_d$, one obtains $p=\sum_{j=1}^{N(d)} r_j^*r_j.$

		By what has already been proved,
		if  $p,q \in \cC_{d},$ then there exist (psd) Gram representations  $p = V^{*} \SG_{p} V$ and $ q = V^{*} \SG_{q} V.$
		Now $p + q = V_{d}^{*} (\SG_{p} + \SG _{q}) V_{d} .$ Since $\SG_{p} + \SG_{q} : \cH^{N(d)}  \to \cH^{N(d)}$ is a psd linear map,
		what has already been proved shows
		$p + q \in \cC_{d}.$
	\end{proof}

	\begin{corollary}\label{cor:soscone}
		Letting   $V_{d} \in \cA_{d}^{N(d)}$ denote  the Veronese column vector,
		the convex cone of sums of squares of degree at most $2d$ is
		\[
		\cC_{d} = \left\{ V_{d}^{*} \SG V_{d}: \quad \SG =  [\SG_{v,w}]_{v,w\in \la x \ra_{d}} \in \cB(\cH)^{N(d) \times N(d)}, \quad \SG \succeq 0 \right\}.
		\]
	\end{corollary}

	\begin{remark}\rm\label{rem:soscone}
		It follows from the preceding discussion that the convex cone $\cC_{d}$ takes the form
		\[
		\cC_{d} \ =\ \{r^{*}r : \quad r \in \cA_{d}\}
		\]
		when the Hilbert space $\cH$ is infinite-dimensional.
		\qed
	\end{remark}

\label{ssec:uGNS}
	The only other point that needs attention is the proof of a suitable GNS construction as in Proposition~\ref{prop:GNS}.
	Since we cannot rely on non-cancellation of the highest orde

\begin{prop}\label{prop:closedcone}
		The convex cone $\cC_{d}$ is closed in $\cA_{\ad}$.
	\end{prop}

\begin{proof}
		Recall that we have endowed $\la x \ra$ with graded lexicographic order. If $\ad\ge |v|$ and  $v>w$, then
		\[
		\langle  A^{w} \Omega, e_{v} \rangle \ =\  0
		\]
		by Lemma~\ref{lem:faithful}. Hence, $\ME_\ad$ is upper triangular.
		Moreover, each diagonal entry is $1$ by Lemma~\ref{lem:faithful}. Thus, $\ME_{\ad}$ is invertible.
	\end{proof}

	\subsection{Extraction formula for coefficients}\label{ssec:coeff}
	Let $q = \sum Q_{w} w \in \cA_{\ad}.$ For $v\in \la x\ra_{\ad}$ 	define the linear functional
	$\Omega_{v} : \cB(\Ftgd) \to \mathbb{C}$ 	by
	\[
	\Omega_{{v}}(T) = \langle  T \Omega, e_{v} \rangle.
	\]
	The operator coefficients \(Q_v\) are obtained from $q(A)$  by solving the linear system
	\begin{align*}
		Z_{v}(q) \ :=&\ (\mathrm{id}_{\cB(\cH)} \otimes \Omega_{v}) q(A) \ =\   \sum\limits_{w} Q_{w} \otimes \Omega_{v} (A^{w}) \\
		\ =&\   \sum\limits_{w} \langle  A^{w} \Omega , e_{v} \rangle \, Q_{w} \ =\   \sum\limits_{w} [\ME_{\ad}]_{v,w} Q_{w},
	\end{align*}
	where $[\ME_{\ad}]_{v,w}$ is the $(v,w)$ entry of the matrix $\ME_{\ad}.$ In short, \index{$[\ME_d]$}
	\begin{equation}\label{eq:coeff2}
		Z(q) \ =\  \ME_{\ad} Q,
	\end{equation}
	where $Z(q)$ and $Q$ are column vectors with $Z_{v}(q)$ and $Q_{v}$  as the $v^{\rm th}$ entry of $Z$ and $Q,$ respectively.
	Since,  by Lemma~\ref{lem:invertible M}, $\ME_d$ is invertible,
	\begin{equation} \label{eq:coeff}
		Q \ =\  \ME_{\ad}^{-1} Z(q).
	\end{equation}
	{We refer to $\ME$ as the \df{extraction matrix}, and equation~\eqref{eq:coeff} as the \df{extraction formula} for the coefficients of $q.$ Note
		that this formula depends only upon $q(A);$ that is, the coefficients of $q$ are determined uniquely
		by $q(A).$}

	It follows from equation~\eqref{eq:coeff}  that  there exists a positive constant $\lambda_{\ad}$ (independent of $q$) such that
	\begin{equation} \label{eq:Coeff bound}
		\|Q_{w}\| \ \leq\  \lambda_{\ad} \, \|q(A)\| \quad \text{ for all $w \in \la x\ra_{\ad}$.}
	\end{equation}

	\begin{prop} \label{prop:bounded}
		If  $p \in \cC_{d},$ then the set
		\[
		\Gamma_p \ =\  \{\, \SG \in \cB(\cH)^{N(d)\times N(d)} \ : \quad \SG \succeq 0, \quad   V_d^* \SG V_d = p \,\}
		\]
		is norm bounded $($with respect to the operator norm on $\cB(\cH^{N(d)})$$)$. More precisely,
		there exists a constant $\mu_{d}$ $($depending only on $d$ and $\vg$ and not on $p$$)$ such that,
		for all $\SG\in\Gamma_p,$
		\[
		\|\SG\| \ \leq\  \mu_{d} \; \|p(A)\| ,
		\]
		where the tuple $A=A^{(\ad)}$ is defined in \eqref{eq:symcreate}
		for any $\ad\geq 2d$.
	\end{prop}

	\begin{proof}
		Fix $p \in \cC_{d}$ and $\SG \in \Gamma_p.$ Thus $p=V_d^* \SG V_d.$ By
		Proposition~\ref{prop:closed under addition}, there exists $Q_j: \cH\to \oplus_{w\in\la x\ra_d} \cH$ such that
		\begin{equation}
			\label{e:bound:1}
			p \ =\  \sum\limits_{j=1}^{N(d)} q_{j}^{*}q_{j} \ =\  V_d^* \left [ \sum\limits_{j=1}^{N(d)} Q_{j} Q_{j}^* \right ] V_d,
		\end{equation}
		where
		\[
		q_j \ =\   Q_j^* V_d = \sum_{w\in\la x\ra_d} Q_{j,w} w.
		\]
		By equation~\eqref{eq:Coeff bound}, for $v\in\la x \ra_d,$
		\[
		\| Q_{j,v}\| \ \le\  \lambda_d\; \|q_j(A)\|.
		\]
		From equation~\eqref{e:bound:1},
		\[
		\|q_j(A)\|^2 \ =\  \|q_j(A)^* q_j(A)\| \ \le\  \|p(A)\|.
		\]
		Thus, again using equation~\eqref{e:bound:1},
		\[
		\sum_{u,v\in \la x \ra_d} \|\SG_{u,v}\| \ \le\   \sum_{u,v\in \la x \ra_d} \, \sum_{j=1}^{N(d)} \|Q_{j,u}Q_{j,v}^*\|
		\ \le\  N(d)^3 \lambda_d^{2} \; \|p(A)\|.
		\]
		It follows that
		$\|\SG\|\le \mu_d \, \|p(A)\|$ for $\mu_d = N(d)^3 \lambda_d^{2}.$
	\end{proof}

\begin{equation}\label{eq:poly}
		p \ =\  \sum_{w\in\la x\ra}^{\rm finite} P_{w} w,
	\end{equation}

\begin{equation}
		\label{d:evalPunitary}
		\sum_{u\in\freeg}^{\rm finite} P_u u \ \mapsto\  \sum_u P_u^* u^{-1}.
	\end{equation}
	If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
	\[
	p(X)^* \ =\   \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*}   \ =\  p^*(X)
	\]
	for all $p\in\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\ccA$ and we let,
	for positive integers $d,$ \index{$\ccA$} \index{$\ccA_d$}
	\[
	\ccA_d \ =\ \left\{
	\sum_{\substack{u\in\freeg\\ |u|\le d}} P_u u \, :\, P_u \in \cB(\cH)
	\right\}.
	\]
	The number of words in $\freeg$ of length $\le d$, $(\freeg)_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
	\[
	N_{\rm red}(d)
	\ =\
	1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
	\ =\  \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
	\]

	\subsection{Main results}\label{ssec:mainresults}
	We are now ready to state our main results. The first is an operator-valued version of the classical 	sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
	is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
	results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

	\begin{thm}\label{thm:sos}
		For $f\in\cA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:sos:i}
			For any Hilbert space $\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \dots, Y_{\vg})\in\cB(\cK)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:ii}
			For any $n\in\N$ and any tuple of self-adjoint matrices
			$Y = (Y_{1}, \dots, Y_{\vg})\in M_n(\C)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:iii}
			There exist $r_1,\ldots,r_{N(d)}\in\cA_d$ s.t.
			\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}

\label{ssec:notation}
	Fix a positive integer $\vg$. Let
	\df{$\la x \ra$}  denote the free monoid on the $\vg$ letters of the alphabet $x = \{ x_{1}, \dots, x_{\vg}\}.$
	Its multiplicative identity

Positivity and factorization lie at the heart of real algebraic geometry and operator theory. In the commutative setting, positivity certificates via sums of squares (sos) trace back to
	Hilbert's $17^{\rm th}$ problem in 1900; for classical results and modern treatments see \cite{BCR98,Mar08,Sc24}.\looseness=-1

	In the 21st century, motivated by developments in linear systems theory \cite{SIG98,dOHMP09}, quantum physics \cite{brunner}, and free probability \cite{MS17}, the free (noncommutative) counterpart has evolved into a broad program within noncommutative function theory \cite{KVV14,MS11,AM15,BMV16,PTD22}. This framework encompasses noncommutative factorizations and noncommutative Positivstellens\"atze. Early landmarks include Helton’s theorem that (scalar) positive noncommutative polynomials are sums of squares \cite{Hel02} and McCullough’s factorization theory for noncommutative polynomials \cite{McC01}; see also \cite{HM04,HMP04,Po95,JM12,JMS21} and the references therein for further developments.

	This paper establishes operator-valued analogs of these factorization theorems: every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^{*}r$, with an analogous result for operator-valued noncommutative trigonometric polynomials (elements of the free group algebra).

	Beyond the noncommutative positivity literature, our results resonate with classical and modern operator factorization themes, including canonical/state-space factorizations of Bart--Gohberg--Kaashoek and collaborators \cite{BGK79,BGKR10}, and the operator Fejér--Riesz and multivariable outer factorization lines \cite{DR10,DW05,GW05}. While our focus is the free (noncommutative) polynomial and free group contexts, the methods developed, such as the WOT-closure mechanism via Fock-space evaluations and the finite-rank Hankel realization, are of independent interest and may be useful in adjacent problems within free analysis and operator theory.

	\subsection*{Guide to the introduction.}
	Notation is introduced in Subsection~\ref{ssec:notation}. The main results are stated and their proofs outlined in Subsection~\ref{ssec:mainresults}, while Subsection~\ref{ssec:readersguide} provides a roadmap for the remainder of the paper.

	\subsection{Notation}\label{ssec:notation}
	Fix a positive integer $\vg$. Let
	\df{$\langle x \rangle$}  denote the free monoid on the $\vg$ letters of the alphabet $x = \{ x_{1}, \dots, x_{\vg}\}.$
	Its multiplicative identity is the empty word
	\df{$\varnothing$}.
	We endow $\langle x\rangle$ with the \df{graded lexicographic order}.
	The length of a word $w \in \langle x \rangle$ is denoted by $|w|.$ The
	set of all elements (words) of $\langle x \rangle$ of length (or degree) at most $d$
	is denoted \df{$\langle x \ra_{d}$}.
	The cardinality of $\langle x\ra_d$  is \index{$N(d)$}
	\[
	N(d) \ =\  \sum_{i=0}^d \vg^i \ =\  \frac{\vg^{d+1}-1}{\vg-1}.
	\]

	Let \df{$\mathcal{H}$} be a fixed complex Hilbert space.
	Let  \df{$\mathcal{B}(\mathcal{H})$} be the space of all bounded linear operators on $\mathcal{H}$, and
	let
	\df{$\mathcal{A} $} to be the free semigroup $\mathcal{B}(\mathcal{H})$-algebra on $x,$
	i.e., $\mathcal{A}=\mathcal{B}(\mathcal{H})\langle x\rangle$. An element $p$ of $\mathcal{A}$ takes the form,
	\begin{equation}\label{eq:poly}
		p \ =\  \sum_{w\in\langle x\rangle}^{\rm finite} P_{w} w,
	\end{equation}
	where
	$P_{w} \in \mathcal{B}(\mathcal{H}),$ and is referred to as an (operator-valued)
	\df{polynomial} in $x.$
	Let \df{$\cA_{d}$} denote the elements from $\mathcal{A}$ of degree at most $d$.

	Equip $\mathcal{A}$ with the involution \df{$^*$}: on letters, $x_{j}^{*} = x_{j},$ on a word $w = x_{i_1} \cdots x_{i_n} \in \langle x \rangle,$
	\[
	w^{*} \ =\  x_{i_n} \cdots x_{i_1};
	\]
	and, on a polynomial $p$ as in \eqref{eq:poly}, \index{$p^*$}
	\[
	p^{*} \ =\  \sum P_{w}^{*} w^{*},
	\]
	where $P_{w}^{*}$ is the adjoint of the operator $P_{w}$ in $\mathcal{B}(\mathcal{H}).$

	Let $\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \dots, X_{\vg})$ be a tuple of operators from $\mathcal{B}(\mathcal{K}).$  The
	\df{evaluation} of $p$ at $X$ is defined as
	\[
	p(X) \ =\  \sum P_{w} \otimes X^{w},
	\]
	where $X^{w} = X_{i_1} \dots X_{i_n}$ for $w = x_{i_1} \dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.

	\subsubsection{Trigonometric polynomials}

	We will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\mathbb F_{\vg}$ on the $\vg$  letters  $x_i$, $i=1,\ldots,\vg$.
	Elements of $\mathbb F_{\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.

	Let $\mathscr{A}$ be the algebra $\mathcal{B}(\mathcal{H})[\mathbb F_{\vg}] = \mathcal{B}(\mathcal{H})\otimes\mathbb C[\mathbb F_{\vg}]$. Its elements are called \df{trigonometric polynomials}, and $\mathscr{A}$
	is
	endowed with the involution
	\begin{equation}
		\label{d:evalPunitary}
		\sum_{u\in\mathbb F_{\vg}}^{\rm finite} P_u u \ \mapsto\  \sum_u P_u^* u^{-1}.
	\end{equation}
	If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
	\[
	p(X)^* \ =\   \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*}   \ =\  p^*(X)
	\]
	for all $p\in\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\mathscr{A}$ and we let,
	for positive integers $d,$ \index{$\mathscr{A}$} \index{$\ccA_d$}
	\[
	\ccA_d \ =\ \left\{
	\sum_{\substack{u\in\mathbb F_{\vg}\\ |u|\le d}} P_u u \, :\, P_u \in \mathcal{B}(\mathcal{H})
	\right\}.
	\]
	The number of words in $\mathbb F_{\vg}$ of length $\le d$, $(\mathbb F_{\vg})_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
	\[
	N_{\rm red}(d)
	\ =\
	1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
	\ =\  \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
	\]

	\subsection{Main results}\label{ssec:mainresults}
	We are now ready to state our main results. The first is an operator-valued version of the classical 	sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
	is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
	results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

\subsection{Notation}\label{ssec:notation}
 Fix a positive integer $\vg$. Let
 \df{$\langle x \rangle$} denote the free monoid on the $\vg$ letters of the alphabet $x = \{ x_{1}, \dots, x_{\vg}\}.$
 Its multiplicative identity is the empty word
 \df{$\varnothing$}.
 We endow $\langle x\rangle$ with the \df{graded lexicographic order}.
 The length of a word $w \in \langle x \rangle$ is denoted by $|w|.$ The
 set of all elements (words) of $\langle x \rangle$ of length (or degree) at most $d$
 is denoted \df{$\langle x \ra_{d}$}.
 The cardinality of $\langle x\ra_d$ is \index{$N(d)$}
 \[
 N(d) \ =\ \sum_{i=0}^d \vg^i \ =\ \frac{\vg^{d+1}-1}{\vg-1}.
 \]

Let \df{$\mathcal{H}$} be a fixed complex Hilbert space.
 Let \df{$\mathcal{B}(\mathcal{H})$} be the space of all bounded linear operators on $\mathcal{H}$, and
 let
 \df{$\mathcal{A} $} to be the free semigroup $\mathcal{B}(\mathcal{H})$-algebra on $x,$
 i.e., $\mathcal{A}=\mathcal{B}(\mathcal{H})\langle x\rangle$. An element $p$ of $\mathcal{A}$ takes the form,
 \begin{equation}\label{eq:poly}
 p \ =\ \sum_{w\in\langle x\rangle}^{\rm finite} P_{w} w,
 \end{equation}
 where
 $P_{w} \in \mathcal{B}(\mathcal{H}),$ and is referred to as an (operator-valued)
 \df{polynomial} in $x.$
 Let \df{$\cA_{d}$} denote the elements from $\mathcal{A}$ of degree at most $d$.

Let $\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \dots, X_{\vg})$ be a tuple of operators from $\mathcal{B}(\mathcal{K}).$ The
 \df{evaluation} of $p$ at $X$ is defined as
 \[
 p(X) \ =\ \sum P_{w} \otimes X^{w},
 \]
 where $X^{w} = X_{i_1} \dots X_{i_n}$ for $w = x_{i_1} \dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.

We will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\mathbb F_{\vg}$ on the $\vg$ letters $x_i$, $i=1,\ldots,\vg$.
 Elements of $\mathbb F_{\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.

Let $\mathscr{A}$ be the algebra $\mathcal{B}(\mathcal{H})[\mathbb F_{\vg}] = \mathcal{B}(\mathcal{H})\otimes\mathbb C[\mathbb F_{\vg}]$. Its elements are called \df{trigonometric polynomials}, and $\mathscr{A}$
 is
 endowed with the involution
 \begin{equation}
 \label{d:evalPunitary}
 \sum_{u\in\mathbb F_{\vg}}^{\rm finite} P_u u \ \mapsto\ \sum_u P_u^* u^{-1}.
 \end{equation}
 If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
 \[
 p(X)^* \ =\ \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*} \ =\ p^*(X)
 \]
 for all $p\in\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\mathscr{A}$ and we let,
 for positive integers $d,$ \index{$\mathscr{A}$} \index{$\ccA_d$}
 \[
 \ccA_d \ =\ \left\{
 \sum_{\substack{u\in\mathbb F_{\vg}\\ |u|\le d}} P_u u \, :\, P_u \in \mathcal{B}(\mathcal{H})
 \right\}.
 \]
 The number of words in $\mathbb F_{\vg}$ of length $\le d$, $(\mathbb F_{\vg})_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
 \[
 N_{\rm red}(d)
 \ =\
 1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
 \ =\ \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
 \]

\subsection{Main results}\label{ssec:mainresults}
 We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
 is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
 results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

\begin{thm}\label{thm:sos}
		For $f\in\cA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:sos:i}
			For any Hilbert space $\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \dots, Y_{\vg})\in\cB(\cK)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:ii}
			For any $n\in\N$ and any tuple of self-adjoint matrices
			$Y = (Y_{1}, \dots, Y_{\vg})\in M_n(\C)^\vg$, $f(Y)\succeq0$;
			\item \label{i:sos:iii}
			There exist $r_1,\ldots,r_{N(d)}\in\cA_d$ s.t.
			\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}
		\end{enumerate}

		\noindent If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[label=\textup{(\roman*)}, resume]\itemsep=5pt
			\item \label{i:sos:iv}
			There exists $r\in\cA_d$ s.t.
			\begin{equation}\label{eq:1sosthm}
				f \ =\ r^*r.
			\end{equation}

		\end{enumerate}
	\end{thm}

\begin{thm}\label{thm:usos}
		For $f\in\ccA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:usos:i}
			For any Hilbert space $\cK$ and any tuple of unitary operators  $U = (U_{1}, \dots, U_{\vg})\in\cB(\cK)^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:ii}
			For any $n\in\N$ and any tuple of unitary matrices
			$U = (U_{1}, \dots, U_{\vg})\in M_n(\C)^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:iii}
			There exist $r_1,\ldots,r_{N_{\rm red}(d)}\in\ccA_d$ s.t.
			\begin{equation*}\label{eq:usosthm}
				f\ =\ \sum_{i=1}^{N_{\rm red}(d)} r_i^*r_i.
			\end{equation*}
		\end{enumerate}
		If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item[\rm (iv)] \label{i:usos:iv}
			There exists $r\in\ccA_d$ s.t.
			\begin{equation*}
				f\ =\ r^*r.
			\end{equation*}
		\end{enumerate}
	\end{thm}

Let $\ccA$ be the algebra $\cB(\cH)[\freeg] = \cB(\cH)\otimes\C[\freeg]$. Its elements are called \df{trigonometric polynomials}, and $\ccA$
 is
 endowed with the involution
 \begin{equation}
 \label{d:evalPunitary}
 \sum_{u\in\freeg}^{\rm finite} P_u u \ \mapsto\ \sum_u P_u^* u^{-1}.
 \end{equation}
 If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so
 \[
 p(X)^* \ =\ \left ( \sum P_w \otimes X^w\right)^* \ =\ \sum P_w^* \otimes X^{w^*} \ =\ p^*(X)
 \]
 for all $p\in\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\ccA$ and we let,
 for positive integers $d,$ \index{$\ccA$} \index{$\ccA_d$}
 \[
 \ccA_d \ =\ \left\{
 \sum_{\substack{u\in\freeg\\ |u|\le d}} P_u u \, :\, P_u \in \cB(\cH)
 \right\}.
 \]
 The number of words in $\freeg$ of length $\le d$, $(\freeg)_d$, is denoted by \df{$N_{\rm red}(d)$} and equals
 \[
 N_{\rm red}(d)
 \ =\
 1+\sum_{k=1}^{d} 2\vg\,(2\vg-1)^{k-1}
 \ =\ \frac{\vg (2\vg-1)^{d}-1}{\vg-1}.
 \]

\subsection{Main results}\label{ssec:mainresults}
 We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \cite{Hel02} and McCullough \cite{McC01}, Theorem~\ref{thm:sos}. The second, Theorem~\ref{thm:usos},
 is a factorization result for positive operator-valued trigonometric polynomials extending a long list of
 results pertaining to scalar-valued noncommutative trigonometric polynomials \cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \succeq 0$ means that the operator $T$ is positive semidefinite (psd).

\noindent If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
 \begin{enumerate}[label=\textup{(\roman*)}, resume]\itemsep=5pt
 \item \label{i:sos:iv}
 There exists $r\in\cA_d$ s.t.
 \begin{equation}\label{eq:1sosthm}
 f \ =\ r^*r.
 \end{equation}

\begin{remark} \rm
 Several remarks related to Theorem~\ref{thm:sos} are in order. \begin{enumerate}[\rm(a)]\itemsep=5pt
 \item Item~\ref{i:sos:iii} can also be phrased as a factorization result.
 Letting $r=\text{col} \begin{pmatrix} r_1 & \cdots & r_{N(d)}\end{pmatrix} \in
 \cA^{N(d)}=
 \cB(\cH,\cH^{N(d)})\la x\ra$, \eqref{eq:sosthm} simply states
 \[
 f\ =\ r^*r.
 \]
 We refer to \cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.
 \item
 That item~\ref{i:sos:iii} implies
 item~\ref{i:sos:i} implies item~\ref{i:sos:ii} is trivial. The main content of Theorem~\ref{thm:sos} is that item~\ref{i:sos:ii}
 implies item~\ref{i:sos:iii}. A routine argument shows the equivalence between \eqref{eq:sosthm} and \eqref{eq:1sosthm} in the infinite-dimensional case, see Remark~\ref{rem:soscone}.
 \item
 Our proof yields no bound on the size $n$ of matrices needed in item~\ref{i:sos:ii}.
 \item
 From Theorem~\ref{thm:sos} one can easily
 deduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification
 $z_j\mapsto \real{z_j}= \frac{z_j+z_j^*}2$ and hence
 $z_j^*\mapsto \imag{z_j}= \frac{z_j-z_j^*}{2i}$.
 \qed
 \end{enumerate}
 \end{remark} The following result is the unitary version of Theorem~\ref{thm:sos}.

\begin{thm}\label{thm:usos}
 For $f\in\ccA_{2d}$ the following are equivalent:
 \begin{enumerate}[\rm(i)]\itemsep=5pt
 \item \label{i:usos:i}
 For any Hilbert space $\cK$ and any tuple of unitary operators $U = (U_{1}, \dots, U_{\vg})\in\cB(\cK)^\vg$, $f(U)\succeq0$;
 \item \label{i:usos:ii}
 For any $n\in\N$ and any tuple of unitary matrices
 $U = (U_{1}, \dots, U_{\vg})\in M_n(\C)^\vg$, $f(U)\succeq0$;
 \item \label{i:usos:iii}
 There exist $r_1,\ldots,r_{N_{\rm red}(d)}\in\ccA_d$ s.t.
 \begin{equation*}\label{eq:usosthm}
 f\ =\ \sum_{i=1}^{N_{\rm red}(d)} r_i^*r_i.
 \end{equation*}
 \end{enumerate}
 If $\cH$ is infinite-dimensional, then the above statements are also equivalent to
 \begin{enumerate}[\rm(i)]\itemsep=5pt
 \item[\rm (iv)] \label{i:usos:iv}
 There exists $r\in\ccA_d$ s.t.
 \begin{equation*}
 f\ =\ r^*r.
 \end{equation*}
 \end{enumerate}
 \end{thm}

Index \df{$\mathbb{\cH}^{N(d)}$} and \df{$\cA_{d}^{N(d)}$} (the algebraic direct sum of $\cA_{d}$ with itself $N(d)$ times) by $\la x \ra_{d}.$
 Let $V_{d} \in \cA_{d}^{N}$ denote the \df{Veronese column vector} whose $w\in \la x \ra_{d}$ entry is $w$
 (adopting the usual convention of viewing $w$ as the the $\cB(\cH)$-valued polynomial $I_\cH \, w$). For instance,
 if $\vg=2$ and $d=2$, then \index{$V_d$}
 \[
 V_2\ =\ \text{col} \begin{pmatrix}
 1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2
 \end{pmatrix}.
 \]
 Let \df{$\cC_d$} denote the \df{cone of sums of squares} of polynomials of degree at most $d,$
 \begin{equation}\label{eq:sosdef}
 \cC_{d} \ :=\ \left\{ \sum\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \quad r_{i}\in \cA_{d}, \quad i=1, \dots, N(d) \right\}
 \subseteq \cA_{2d}.
 \end{equation}
 Given $r\in \cA_d,$ the column vector $R$ with $w$ entry $R_{w}^{*}$
 is called the \df{coefficient vector} of $r$ since $r = R^{*} V_{d}.$ In particular,
 \[
 r^* r \ =\ V_d^* RR^* V_d
 \]
 so that $r^*r$ has a representation as $V_d^* \SG V_d$ for a psd matrix $\SG.$

\begin{lemma}\label{lem:uweak2WOT}
 If $\varphi:\mathcal A_{2d}\to\mathbb C$ be an ultraweak continuous linear functional that
 separates the cone $\cC_{d}$ from a fixed polynomial $p$ in $\cA_{2d},$ that is,
 \[
 \varphi(r^{*}r) \geq 0 \quad \text{for all $r \in \cA_{d}$} \quad \text{and} \quad \varphi(p+p^{*}) <0,
 \]
 then there exists a WOT continuous linear functionl $\tilde{\varphi} : \cA_{2d} \to \mathbb{C}$ such that
 \[
 \tilde{\varphi}(r^{*}r) \geq 0 \quad \text{for all $r \in \cA_{d}$} \quad \text{and} \quad \tilde{\varphi}(p+p^{*}) <0.
 \]
 \end{lemma}
 \begin{proof}
 Since $\varphi$ is ultraweak continuous, there exist trace class operators $S_{w}$ $(w \in \la x\ra_{2d})$ in $\cB(\cH)$ such that
 \[
 \varphi(q) = \sum\limits_{w \in \la x\ra_{2d}} {\rm Tr}\, (S_{w} Q_{w}),
 \]
 where $q = \sum_{w \in \la x \ra_{2d}} Q_{w} w.$ For $r,r^\prime \in \cA_{d}$, \[
 \varphi(r^{*}r^\prime) \ =\ \sum\limits_{u,v\in \la x \ra_{d}} {\rm Tr}\, (S_{u^{*}v} R_{u}^{*}R^\prime_{v}),
 \]
 where $r = \sum_{u \in \la x \ra_{d}} R_{u} u$
 and $r^\prime = \sum_{v \in \la x \ra_{d}} R^\prime_{v} v.$ Denote by $S$ the $N(d) \times N(d)$ block operator matrix whose $(u,v)$ entry is $S_{v^{*}u}.$

\begin{prop}\label{prop:GNS}
 If $\varphi:\mathcal A_{2d+2}\to\mathbb C$ is a continuous linear functional such that
 \[
 \varphi(p^ *p)\ \ge\ 0 \]
 for all $p\in \cA_{d+1},$
 then there exist a finite-dimensional Hilbert space $\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\dots,Y_\vg)$ on $\cE$, and a vector $\gamma\in \cH\otimes \cE$ such that
 \[
 \varphi(q^ *p)\ =\ \big\langle p(Y)\gamma,\ q(Y)\gamma\big\rangle_{\cH\otimes \cE} \qquad\text{for all }p \in \cA_{d+1},\ q\in\mathcal A_{d}.
 \]
 Therefore, for all $p\in\mathcal A_{2d+1}$,
 \[
 \varphi(p)\ =\ \langle p(Y)\gamma,\ \gamma\rangle.
 \]
 \end{prop}

\begin{remark} \rm
		Several remarks related to Theorem~\ref{thm:sos} are in order. 		\begin{enumerate}[\rm(a)]\itemsep=5pt
			\item Item~\ref{i:sos:iii} can also be phrased as a factorization result.
			Letting $r=\text{col} \begin{pmatrix} r_1 & \cdots & r_{N(d)}\end{pmatrix} \in
			\mathcal{A}^{N(d)}=
			\mathcal{B}(\mathcal{H},\mathcal{H}^{N(d)})\langle x\rangle$, \eqref{eq:sosthm} simply states
			\[
			f\ =\ r^*r.
			\]
			We refer to \cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.
			\item
			That item~\ref{i:sos:iii} implies
			item~\ref{i:sos:i}  implies item~\ref{i:sos:ii}  is trivial. The main content of Theorem~\ref{thm:sos} is that item~\ref{i:sos:ii}
			implies item~\ref{i:sos:iii}. A routine argument shows the equivalence between \eqref{eq:sosthm} and \eqref{eq:1sosthm}  in the infinite-dimensional case, see Remark~\ref{rem:soscone}.
			\item
			Our proof yields no bound on the size $n$ of matrices needed in item~\ref{i:sos:ii}.
			\item
			From Theorem~\ref{thm:sos} one can easily
			deduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification
			$z_j\mapsto \operatorname{real}{z_j}= \frac{z_j+z_j^*}2$ and hence
			$z_j^*\mapsto \operatorname{imag}{z_j}= \frac{z_j-z_j^*}{2i}$.
			\qed
		\end{enumerate}
	\end{remark} The following result is the unitary version of Theorem~\ref{thm:sos}.

	\begin{thm}\label{thm:usos}
		For $f\in\ccA_{2d}$ the following are equivalent:
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item \label{i:usos:i}
			For any Hilbert space $\mathcal{K}$ and any tuple of unitary operators  $U = (U_{1}, \dots, U_{\vg})\in\mathcal{B}(\mathcal{K})^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:ii}
			For any $n\in\mathbb N$ and any tuple of unitary matrices
			$U = (U_{1}, \dots, U_{\vg})\in M_n(\mathbb C)^\vg$, $f(U)\succeq0$;
			\item \label{i:usos:iii}
			There exist $r_1,\ldots,r_{N_{\rm red}(d)}\in\ccA_d$ s.t.
			\begin{equation*}\label{eq:usosthm}
				f\ =\ \sum_{i=1}^{N_{\rm red}(d)} r_i^*r_i.
			\end{equation*}
		\end{enumerate}
		If $\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to
		\begin{enumerate}[\rm(i)]\itemsep=5pt
			\item[\rm (iv)] \label{i:usos:iv}
			There exists $r\in\ccA_d$ s.t.
			\begin{equation*}
				f\ =\ r^*r.
			\end{equation*}
		\end{enumerate}
	\end{thm}

	\begin{remark}[What's new?]\rm
		\label{rem:whatsnew}
		The passage to operator coefficients necessitates several novel
		results and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\ref{thm:sos} and Theorem~\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).
		Namely, the Hahn-Banach theorem produces a separating linear functional $\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction
		based on $\varphi$ ultimately produces a tuple $Y$.
		Here we
		roughly follow the outline of \cite{MP05}.

		A key construction is that of a tuple of self-adjoint operators $A$
		based upon the left regular representation on Fock space; see Section~\ref{sec:fock}.
		We then show that, up to a universal constant, for a sum of squares polynomial $p,$
		the norm
		of $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\ref{prop:bounded}.
		This uniform bound is the main input in Section~\ref{sec:top} for proving that the cone $\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.

		On the GNS side, we introduce a new argument that exploits the WOT to associate to a
		separating linear functional a finite-rank psd noncommutative representing Hankel matrix and,
		on its range, construct the desired tuple~$Y$; see Section~\ref{sec:gns}. For the unitary result,
		Theorem~\ref{thm:usos}, we additionally modify the construction of Section~\ref{sec:fock} to produce a canonical
		tuple of unitary operators from the left-regular representation and adapt the GNS procedure
		to obtain a unitary tuple together with a representing vector that realizes the separating
		functional as a vector state; see Subsection~\ref{ssec:uGNS}.\qed
	\end{remark}

	\subsection{Reader's guide}\label{ssec:readersguide}

	The paper is structured as follows. The convex cone of sums of squares (making an appearance in \eqref{eq:sosthm} of Theorem~\ref{thm:sos}) is introduced and characterized in the next Section~\ref{sec:sos}. In Section~\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\cF_\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\ref{sec:top} we
	introduce a suitable topology on $\cA_\lcal{d}$ and
	collect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\ref{sec:gns}; see Proposition~\ref{prop:GNS}.
	Then Theorem~\ref{thm:sos} is proved in Section~\ref{sec:proof} and
	Theorem~\ref{thm:usos} is proved in Section~\ref{sec:uproof}.

	\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\ref{prop:closedcone}.

The post-theorem discussion says the nontrivial direction in Theorem~\ref{thm:sos} is item~\ref{i:sos:ii} $\Rightarrow$ item~\ref{i:sos:iii} (while item~\ref{i:sos:iii} $\Rightarrow$ item~\ref{i:sos:i} $\Rightarrow$ item~\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\varphi$ to produce a tuple $Y$.

Key intermediate steps highlighted are:
\begin{itemize}
\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\ref{sec:fock}).
\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\|p(A)\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$ (Proposition~\ref{prop:bounded}).
\item Use this bound (Section~\ref{sec:top}) to show the sos cone $\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.
\item On the GNS side (Section~\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”
\end{itemize}
It also notes that in the infinite-dimensional case “a routine argument shows the equivalence between \eqref{eq:sosthm} and \eqref{eq:1sosthm}.”

The post-theorem discussion says the nontrivial direction in the main theorem is item~\ref{i:sos:ii} $\Rightarrow$ item~\ref{i:sos:iii} (while item~\ref{i:sos:iii} $\Rightarrow$ item~\ref{i:sos:i} $\Rightarrow$ item~\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\varphi$ to produce a tuple $Y$.

Key intermediate steps highlighted are:
\begin{itemize}
\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\ref{sec:fock}).
\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\|p(A)\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$. We first record the needed proposition.

\begin{prop} \label{prop:bounded}
		If  $p \in \cC_{d},$ then the set
		\[
		\Gamma_p \ =\  \{\, \SG \in \cB(\cH)^{N(d)\times N(d)} \ : \quad \SG \succeq 0, \quad   V_d^* \SG V_d = p \,\}
		\]
		is norm bounded $($with respect to the operator norm on $\cB(\cH^{N(d)})$$)$. More precisely,
		there exists a constant $\mu_{d}$ $($depending only on $d$ and $\vg$ and not on $p$$)$ such that,
		for all $\SG\in\Gamma_p,$
		\[
		\|\SG\| \ \leq\  \mu_{d} \; \|p(A)\| ,
		\]
		where the tuple $A=A^{(\ad)}$ is defined in \eqref{eq:symcreate}
		for any $\ad\geq 2d$.
	\end{prop}

\item Use this bound (Section~\ref{sec:top}) to show the sos cone $\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.
\item On the GNS side (Section~\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”
\end{itemize}
It also notes that in the infinite-dimensional case “a routine argument shows the equivalence between
\begin{equation}\label{eq:sosthm}
				f \ =\ \sum_{i=1}^{N(d)} r_i^*r_i.
			\end{equation}
and
\begin{equation}\label{eq:1sosthm}
				f \ =\ r^*r.
			\end{equation}
.”

Question: Fix a positive integer $\vg$ and a complex Hilbert space $\mathcal H$. Let $\mathcal A=\mathcal B(\mathcal H)\langle x_1,\dots,x_{\vg}\rangle$ be the algebra of operator-valued noncommutative polynomials in free variables $x_1,\dots,x_{\vg}$, and let $\mathcal A_d$ denote the polynomials of degree at most $d$. For a Hilbert space $\mathcal K$ and a self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in \mathcal B(\mathcal K)^{\vg}$, $f(Y)$ denotes the evaluation of $f$ at $Y$. Also set $N(d)=\sum_{i=0}^d \vg^i$. Given $f\in \mathcal A_{2d}$, which existence statement holds precisely under the positivity assumption that $f(Y)\succeq 0$ for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y\in\mathcal B(\mathcal K)^{\vg}$ (equivalently, for every $n\in\mathbb N$ and every self-adjoint tuple $Y\in M_n(\mathbb C)^{\vg}$)?

For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\in\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.

Choices:
B. For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\in\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(2d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(2d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.
C. If there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$, then for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$. Moreover, if $\mathcal H$ is infinite-dimensional and $f=r^*r$ for some $r\in\mathcal A_d$, then the same positivity conclusion holds.
D. For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\in\mathbb N$ and every tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.
E. For $f\in\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in\mathcal B(\mathcal K)^{\vg}$, one has $f(Y)\succeq 0$; (ii) for every $n\le N(d)$ and every self-adjoint tuple $Y=(Y_1,\dots,Y_{\vg})\in M_n(\mathbb C)^{\vg}$, one has $f(Y)\succeq 0$; (iii) there exist $r_1,\dots,r_{N(d)}\in\mathcal A_d$ such that $f=\sum_{i=1}^{N(d)} r_i^*r_i$. If $\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\in\mathcal A_d$ such that $f=r^*r$.

dropped converse/equivalence, keeping only sos-implies-positivity

self-adjointness hypothesis in evaluation

The stem does not explicitly reveal the correct option. It states the positivity hypothesis and asks for the precise corresponding existence/equivalence statement, but the exact sum-of-squares form, sharp bound N(d), and infinite-dimensional single-factor refinement are not given away.

This is largely a theorem-identification item: the correct answer is essentially the exact Positivstellensatz statement with small perturbations in the alternatives. It is not a pure verbatim restatement, since the student must distinguish nearby variants, but it still tests recall of a known theorem more than independent conclusion-drawing.

Moderate reasoning is needed to reject plausible variants: N(d) versus N(2d), self-adjoint versus arbitrary tuples, all matrix sizes versus only n ≤ N(d), and equivalence versus a one-way implication. Still, the task is mainly precision matching of a theorem statement rather than deeper generative mathematical reasoning.

The distractors are strong and mathematically meaningful. They target realistic failure modes: weakening an equivalence, changing the sharp Gram bound, dropping self-adjointness, and incorrectly restricting matrix sizes. They are distinct, plausible, and well-aligned with the content.

A solid theorem-discrimination MCQ with high-quality distractors and little answer leakage, but it mainly assesses precise recall/recognition of a theorem rather than substantial generative reasoning.


\label{thm:main}
    There is a polynomial time algorithm that, given a matching covered graph $G$, finds a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.

Consider a graph $G=(V,E)$ with an even number of vertices. A \emph{perfect matching} is a set of edges $M\subseteq E$ incident with each vertex exactly once. $G$ is \emph{matching covered} if every edge of $G$ belongs to some perfect matching.
The theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\'asz. An object of particular interest is the \emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \begin{equation*}
 \mathcal L(G): = \lat(\mathcal{M})= \{\sum_{M \in \mathcal{M}}k_M \mathbf{1}_M: k_M \in \mathbb{Z}\},
\end{equation*} where $\mathcal{M}$ denotes the set of all perfect matchings of $G$.
Lov\'asz \cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \emph{bricks} and \emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings.
A natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:

\section{Introduction}

The main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \emph{bipartite relaxation}:\begin{equation}\label{eq:pm-bipartite}
 P(G):=\left\{x\in \R^A_{+}: x(\delta(v))= 1 \forall v\in V \right\}.
\end{equation}
Matching covered graphs for which $PM(G)=P(G)$ are called \emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\zL(G)$ either finds a set of $\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.

Since we are not interested in edges that do not belong to any perfect matching, it is convenient to delete from $G$ all edges that are not in any perfect matching, thus making the graph matching covered.
The number of bricks in a tight cut decomposition of a matching covered graph affects the dimension of its perfect matching polytope. In particular, the dimension of the perfect matching polytope is \begin{equation}\label{eq:pm-dimension}
 \dim(PM(G)) = |E'|-|V|+1-b,
\end{equation} where $b$ denotes the number of bricks in a tight cut decomposition of $G$ and $E'$ is the set of edges that belong to a perfect matching \cite{edmonds1982brick}, with $E=E'$ for matching covered $G$. Furthermore, the dimension of the perfect matching lattice and its linear hull is \begin{equation}\label{eq:pm-lat-dimension}
 \dim (\lat(G)) = |E'|-|V|+2-b.
\end{equation}
We will call \emph{near-bricks} all matching covered graphs $G$ with exactly one brick.
The following fact due to Lov\'asz connects the linear hull of perfect matchings with $\zL(G)$: \begin{theorem}[Theorem $6.3$ in \cite{lovaszPM}]\label{thm:lovasz}
 In a matching covered graph $G=(V,E)$, let $\mathcal{M}$ be the set of perfect matchings. Then any $x \in \lin(\mathcal{M})\cap \Z^E$ satisfies $2x\in \zL(G)$. Furthermore, if $G$ has no Petersen bricks then $\lin(\mathcal{M})\cap \Z^E=\zL(G)$.
\end{theorem}

\begin{theorem}\label{thm:algo-correct}
 The result of Algorithm \ref{algorithm:basis-bvn} applied to a matching covered BvN graph is a set $\zB$ of linearly independent indicator vectors of perfect matchings that forms an integral basis for $\zL(G)$.
\end{theorem}
\begin{proof}
 Because $G$ is BvN it implies that any face of $P(G)$ has integral vertices. Therefore, all $y(e)$ found as a corner solution to \ref{LP2_match} will indeed be incidence vectors of some perfect matchings of $G$ (in fact, of $G[E_t]$).
 It is clear that the final set $\zB$ will contain exactly $\dim(PM(G))$ perfect matchings with the property that each one uses an edge not used in the previous ones. This means that the elements of $\zB$ are linearly independent and moreover, create an integral basis for $\lin(\zB)$.
\end{proof}

If, given any brick $G$, the above process terminates in $|E|-|V|+1$ steps and all the output vectors are integral, then they correspond to a basis for $\zL(G)$.
Therefore, the only algorithm obstruction in the non-BvN case is \ref{LP2_match} having a fractional solution $x^*$. This means that $x^*$ is a fractional vertex of $P(G)$ itself, which gives a certificate that $G$ is non-BvN.
As $x^*$ has to be half-integral, its support consists of vertex-disjoint edges of weight $1$ and odd cycles of weight $1/2$. Picking any of the odd cycles, we can consider the odd cut $C'$ whose shore is exactly the vertices of this cycle. Inevitably, we have $x^*(C')=0$.
In \Cref{sec:sep-find}, we describe a combinatorial algorithm to find an odd cut $C$ such that the inequality $x(C)\geq 1$ is violated by $x^*$ with more specific properties:
\begin{theorem}\label{thm:find-robust}
 Let $G$ be a brick that is non-BvN. Let $x^*$ be a fractional vertex of $P(G)$. There is a polynomial time algorithm that finds a separating cut $C$ of $G$ such that $x^*(G)<1$ and both $C$-contractions of $G$ are near-bricks whose brick is not the Petersen brick.
\end{theorem}

As in Section \ref{subsec:non-bvn-detect}, consider a non-BvN brick $G$ for which the BvN algorithm fails. Theorem \ref{thm:find-robust} guarantees the existence of a separating cut $C$ both of whose contractions, call them $G_1$ and $G_2$, have exactly one brick that is not the Petersen brick.
Since neither $G_1$ nor $G_2$ contain the Petersen graph, we can construct lattice bases $\zB_1$ and $\zB_2$ for the corresponding graphs. Since both contractions do not contain Petersen bricks, $\lin(\zB_i)\cap \Z^{E_i} = \zL(G_i)$ and hence by \Cref{thm:basis-comb} the set $\zB:=\zB_1 \odot \zB_2$ will be an integral basis for its linear hull. However, all elements of $\zB$ satisfy $x^T \mathbf{1}_C=1$, so to make it a basis for $\zL(G)$ we need to add at least one perfect matching $M$ with $|M\cap C|>1$.
We use the following fact shown in \cite{camposlucchesi}.
\begin{theorem}\label{thm:char-three}
 In a non-BvN brick that is not Petersen, every separating cut has a perfect matching intersecting it three times.
\end{theorem}
Such a perfect matching $M^*$ can be found in polynomial time by considering all possible intersections $M^*\cap C$ of size three. Finally, we can use $M^*$ to increment the basis. We can now prove that such a set $\zB \cup \{\mathbf{1}_{M^*}\}$ will be a basis for $\zL(G)$.
\begin{theorem}
 Let $G=(V,E)$ be a matching covered graph and let $C$ be a non-tight separating cut such that both $C$-contractions are near-bricks whose brick is not the Petersen brick. Let $M^*$ be a perfect matching with $|M^*\cap C|=3$. Then $\zB \cup \{\mathbf{1}_{M^*}\}$ is an integral basis for $\zL(G)$ consisting of perfect matchings.
\end{theorem}
\begin{proof}
 By Theorem \ref{thm:fdi-charact}, such $C$ defines a facet of $PM(G)$. Hence, using part $(iii)$ of Theorem \ref{thm:basis-comb}, $|\zB|=\dim(\zL(G))-1$. Thus, adding $M^*$ to $\zB$ makes it a basis for the linear hull of $\zL(G)$, so it suffices to prove that it will be an integral basis.
 Indeed, consider any $y \in \zL(G)$. We also know it is in the linear hull of $\zB\cup\{\mathbf{1}_{M^*}\}$. Therefore, we can write it as follows \begin{equation}\label{eq:linear-comb}
 y = \sum_{M\in \mathcal{M}} \alpha_M \mathbf{1}_M = \sum_{z^i\in \zB} \beta_i z^i+\beta_* \mathbf{1}_{M^*},
 \end{equation} where all $\alpha_M$ are integers and the goal is to prove that $\beta_i$ are all integer.
 The first equality of (\ref{eq:linear-comb}) when multiplied by $\textbf{1}_{E}$ gives \begin{equation*}
 \frac{1}{n/2} \textbf{1}^T_{E} y = \frac{1}{n/2}\sum_{M\in \mathcal{M}} \alpha_M \textbf{1}^T_{E}\mathbf{1}_M=\sum_{M\in \mathcal{M}} \alpha_M \in \Z.
 \end{equation*} Similarly, multiplying it by $\textbf{1}_C$ gives \begin{equation*}
 \textbf{1}^T_{C} y = \sum_{M\in \mathcal{M}} \alpha_M \textbf{1}^T_{C}\mathbf{1}_M\equiv \sum_{M\in \mathcal{M}} \alpha_M \mod 2.
 \end{equation*} Applying the same idea to the last expression of equality (\ref{eq:linear-comb}) we obtain \begin{equation*}
 \frac{1}{n/2} \textbf{1}^T_{E} y =\sum_{\mathbf{z}^i \in \zB} \beta_i + \beta_*, \textbf{1}^T_{C} y = \sum_{\mathbf{z}^i \in \zB} \beta_i + 3\beta_*.
 \end{equation*} Therefore, $\beta_*=\frac{1}{2} (\textbf{1}^T_{C} y-\frac{1}{n/2} \textbf{1}^T_{E} y)$, which is integral by the above.
 Finally, consider $y-\beta_* \mathbf{1}_{M^*} \in \lin(\zB)$. Since we already showed that $\zB$ is an integral basis for its linear hull, all the remaining coefficients $\beta_i$ are also integer.
\end{proof}

The main contribution of this paper is a new approach to basis construction for the perfect matching lattice based on polyhedral theory. In addition to the algorithmic result, we identify a class of graphs where a lattice basis can be obtained by a simple inductive process. Moreover, we connect the combinatorial and polyhedral properties of the perfect matchings, bridging our approach and structural methods of de Carvalho, Lucchesi, and Murty. More broadly, our polyhedral techniques offer new tools that may prove valuable in tackling longstanding open problems in matching theory.
To begin with, the \emph{perfect matching polytope} $PM(G)$ is the convex hull of its perfect matching incidence vectors and is described by the following set of inequalities (Edmonds, \cite{edmonds1965maximum}):
\begin{equation}\label{eq:pm-general}
    PM(G)=\left\{x\in \mathbb{R}^A_{+}: \begin{array}{cc}
    x(\delta(v))= 1 &  \forall v\in V\\
     x(C)\geq 1 &  \forall C \textmd{ odd cut}
\end{array}\right\}
\end{equation}
Here, a \emph{cut} $C = \delta(U)$ for some $U\subseteq V$ is the set of edges with exactly one endpoint in $U$; the sets $U$ and its complement $V\backslash U$ are the \emph{shores} of $C$. A cut is \emph{odd} if both of its shores have an odd cardinality.

The main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \emph{bipartite relaxation}:\begin{equation}\label{eq:pm-bipartite}
    P(G):=\left\{x\in \mathbb{R}^A_{+}: x(\delta(v))= 1  \forall v\in V \right\}.
\end{equation}
Matching covered graphs for which $PM(G)=P(G)$ are called \emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\mathcal L(G)$ either finds a set of $\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.

The paper is organized as follows. In Section \ref{sec:prelim} we discuss a way to decompose the graph into simple pieces and show how to reduce the basis construction for $\mathcal L(G)$ to these pieces. In Section \ref{sec:bvn} we describe a basis construction algorithm for \emph{Birkhoff von Neumann (BvN)} graphs based on polyhedral techniques and discuss what happens when applying this algorithm to a non-BvN graph. In Section \ref{sec:put-together} we show how to decompose non-BvN graphs into BvN ones using a specific type of odd cuts in a process similar to the tight cut decomposition. In Section \ref{sec:sep-find} we provide an algorithm to find an appropriate cut to be used in the above mentioned decomposition. Finally, in Appendix \ref{sec:classical} we draw connections between our polyhedral view of the basis and the classical approach based on \emph{ear decompositions}.

To prove Theorem~\ref{thm:main}, the paper’s approach is “based on polyhedral theory” via the perfect matching polytope $PM(G)$ (Edmonds’ description \eqref{eq:pm-general}) and its “bipartite relaxation” $P(G)$ in \eqref{eq:pm-bipartite}. The “main idea … is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its bipartite relaxation,” i.e., to reduce to Birkhoff--von Neumann (BvN) graphs where $PM(G)=P(G)$. The algorithm for a basis of $\mathcal L(G)$ then “either finds a set of $\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$ induced by an odd cut constraint $x(C)\ge 1$”; in the latter case, it “use[s] this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.” The overall proof/argument is organized by: reducing basis construction to decomposed “simple pieces” (Section~\ref{sec:prelim}); giving a basis construction algorithm for BvN graphs and analyzing applying it to non-BvN graphs (Section~\ref{sec:bvn}); decomposing non-BvN graphs into BvN ones “using a specific type of odd cuts” (Section~\ref{sec:put-together}); and providing an algorithm to find the needed cut (Section~\ref{sec:sep-find}).

Question:
Let $G=(V,E)$ be a matching covered graph, meaning that $G$ has a perfect matching and every edge of $G$ belongs to some perfect matching. For a perfect matching $M$, let $\mathbf{1}_M\in\{0,1\}^E$ denote its incidence vector, and let the perfect matching lattice be
\[
\mathcal L(G)=\left\{\sum_{M} k_M\mathbf{1}_M : k_M\in\mathbb Z,\ M\text{ a perfect matching of }G\right\}.
\]
A lattice basis for $\mathcal L(G)$ means a set of vectors in $\mathcal L(G)$ that is integer-linearly independent and whose integer span is exactly $\mathcal L(G)$. Which existence statement holds for such graphs?

There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.

Choices:
B. There exists a polynomial-time algorithm that, given any graph $G$ with a perfect matching, outputs a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.
C. For every matching covered graph $G$, there exists a lattice basis for $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.
D. There exists a single polynomial-time algorithm that, given any matching covered graph $G$, outputs a basis of the real linear span of $\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.
E. There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs all incidence vectors of perfect matchings of $G$, and these vectors form a lattice basis for $\mathcal L(G)$.

The stem defines the relevant objects but does not state or strongly hint at the algorithmic existence claim in choice A. The correct answer is not leaked explicitly or implicitly.

The item is close to theorem recognition: the correct option appears to be essentially the target theorem statement. However, it is not a pure restatement because the alternatives vary meaningfully in graph class, algorithmic strength, and type of basis.

Some reasoning is needed to distinguish the precise valid claim from nearby overstatements and weakenings, especially B, C, and D. Still, it mainly tests recognition of the exact theorem rather than substantial derivation or synthesis.

The distractors are mostly plausible and target natural failure modes: overgeneralizing from matching covered to all graphs with a perfect matching (B), confusing mere existence with polynomial-time constructibility (C), confusing lattice basis with real-span basis (D), and overclaiming that all perfect matchings are output and form a basis (E).

A solid theorem-discrimination MCQ with no answer leakage and generally strong distractors, but it leans toward recalling the exact theorem statement rather than eliciting deep generative reasoning.