\begin{theorem}\label{thm:main}
		Let $\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \in \N$ with $n \ge 2$. A map $\phi : M_n(\F) \to M_n(\F)$
		satisfies the mixed Jordan--power identity
		\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
		if and only if one of the following holds:
		\begin{itemize}
			\item[(a)] $\phi$ is constant, taking a fixed $(k+1)$-potent value;
			\item[(b)] there exist an invertible matrix $T \in M_n(\F)$, a ring monomorphism $\omega : \F \to \F$, and a $k$-th root of unity $\varepsilon \in \F$ such that
			\begin{equation}\label{eq:form of phi}
				\phi(X) = \varepsilon\, T\, \omega(X)\, T^{-1}
				\quad \text{ or } \quad
				\phi(X) = \varepsilon\, T\, \omega(X)^t\, T^{-1}.
			\end{equation}
		\end{itemize}
		In particular, every nonconstant solution of \eqref{eq:a certain product} is additive.
	\end{theorem}

\begin{remark}\label{rem:k-potents not diagonalizable}
		Suppose that $n \ge 2$. All matrices in $\pot_k(M_n)$ are diagonalizable if and only if $\chr(\F)$ does not divide $k-1$. Indeed, suppose that $\chr(\F)$ does not divide $k-1$, and let $P \in \pot_k(M_n)$ be arbitrary. We have
		$$P^{k} = P \, \implies \, P(P^{k-1}-I) = 0,$$
		which implies that the minimal polynomial $m_P \in \F[x]$ of $P$ necessarily divides
		$$q(x):=x(x^{k-1}-1).$$
		Since $\F$ is algebraically closed, to show that $P$ is diagonalizable in $M_n$, it suffices to show that $q$ (and consequently $m_P$) is square-free, meaning that it decomposes into a product of distinct linear factors. We have
		$q'(x) = kx^{k-1}-1$. Assuming that $x_0 \in \F$ satisfies $q(x_0) = q'(x_0) = 0$ would lead to $k-1=0$, which is a contradiction.

		Conversely, suppose that $\chr(\F) \mid (k-1)$. For $n=2$, the matrix
		$$
		A := \begin{bmatrix}
			1  & 1 \\ 0 & 1
		\end{bmatrix} \in M_2
		$$
		satisfies $$A^{k-1} = \begin{bmatrix}
			1  & k-1 \\ 0 & 1
		\end{bmatrix} = I \implies A^k = A$$
		so $A$ is a $k$-potent in $M_2$. On the other hand, the minimal polynomial of $A$ is $m_A(x) = (x-1)^2 \in \F[x]$, which is not square-free in $\F[x]$. Therefore, $A$ is not diagonalizable in $M_2$. When $n > 2$, a similar conclusion follows by considering the matrix $\diag(A,I_{n-2}) \in M_n$.

		Additionally, the matrix $A$ demonstrates other interesting properties of the order $\preceq$. It is not difficult to show directly that any matrix $B \in M_2$ satisfies $AB=BA=B^2$ if and only if $B \in \{0,A\}$. In particular, $A$ is a minimal element of the poset $(\pot_k(M_2)\setminus\{0\}, \preceq)$, even though its rank is $2$. It is also a consequence that $A$ \emph{cannot} be written in the form $A = P+Q$ for some nonzero mutually orthogonal $k$-potents $P,Q \in \pot_k(M_2)$. On the other hand, by Remark \ref{re:maximal element}, $A$ is also a maximal element of the poset $(\pot_k(M_2), \preceq)$.
	\end{remark}

The study of multiplicative maps between rings and algebras has a long and substantial history, centered on the question of when multiplicativity alone enforces stronger algebraic behavior, most notably additivity. Foundational contributions were made by Rickart~\cite{Rickart} in 1948 and Johnson~\cite{Johnson} in 1958, who established positive results under suitable structural assumptions and highlighted the connection with the rigidity of the additive structure of rings. A major advance was achieved by Martindale~\cite{Martindale} in 1969, who proved that every bijective multiplicative map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is automatically additive. In the same year, Jodeit and Lam~\cite{JodeitLam} classified all nondegenerate multiplicative self-maps of the matrix rings $M_n(\mathcal{R})$ over a principal ideal domain~$\mathcal{R}$, where nondegeneracy means that the map is not identically zero on the set of matrices with zero determinant. Specifically, for each such map $\phi: M_n(\ca{R}) \to M_n(\ca{R})$, either there exists a nonzero idempotent matrix $P \in M_n(\ca{R})$ such that $\phi - P$ is multiplicative and degenerate, or there exists an invertible matrix $T \in M_n(\ca{R})$ and a ring endomorphism $\omega$ of $\ca{R}$ such that $\phi$ takes one of the following two distinct forms:
	$$
	\phi(X) = T \, \omega(X) \, T^{-1} \quad \text{ or } \quad \phi(X) = T \, \omega(X)^* \, T^{-1},
	$$
	where $\omega(X)$ denotes the matrix obtained by applying $\omega$ entrywise, and $(\cdot)^*$ is the corresponding cofactor matrix. In particular, all bijective multiplicative self-maps on $M_n(\mathcal{R})$ are automatically additive and hence ring automorphisms. Taken together, these results expose a fundamental phenomenon: multiplicativity, when coupled with bijectivity, interacts strongly with the ambient ring structure, leaving little room for pathological nonadditive behavior. This rigidity principle later emerged as a central theme in preserver theory, especially in the context of matrix and operator algebras.

	This viewpoint was later extended from ordinary ring homomorphisms to \emph{Jordan homomorphisms}, which originate as homomorphisms in the category of \emph{Jordan algebras}, introduced by Jordan, von Neumann, and Wigner \cite{Jordan} in the 1930s as algebraic models for quantum observables.
	A \emph{Jordan algebra} is a nonassociative algebra $\ca{A}$ over a field equipped with a commutative multiplication $\diamond$ satisfying the \emph{Jordan identity}
	$$
	a^2 \diamond (a \diamond b) = a \diamond (a^2 \diamond b), \quad \text{ for all } a,b \in \ca{A}.
	$$
	Jordan rings are similarly defined. Every associative algebra (or ring) $\mathcal{A}$ naturally becomes a Jordan algebra (ring) when equipped with the \emph{Jordan product}
	$$
	a \diamond b := ab + ba.
	$$
	A Jordan algebra that is isomorphic to a Jordan subalgebra of an associative algebra is called a \emph{special Jordan algebra} (see e.g. \cite[Section~2.3]{HancheStormer}). Not all Jordan algebras are special; the Albert algebra~\cite{Albert} provides the classical example of an \emph{exceptional Jordan algebra}.

	In an associative setting, a linear (or additive) map $\phi: \ca{A} \to \ca{B}$ between associative algebras (or rings) $\ca{A}$ and $\ca{B}$ is called a \emph{Jordan homomorphism} if it satisfies
	$$
	\phi(a \diamond b) = \phi(a) \diamond \phi(b), \quad \text{ for all } a, b \in \ca{A}.
	$$
	If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\phi$ preserves squares with respect to the underlying product of $\mathcal{A}$, that is,
	$$
	\phi(a^2) = \phi(a)^2, \quad \text{ for all } a \in \ca{A}.
	$$
	Moreover, if $\ca{A}$ is an algebra over a field $\F$ with $\chr(\F) \ne 2$, the Jordan structure is commonly (esp.\ in quantum mechanics contexts) induced by the \emph{normalized Jordan product}
	\begin{equation}\label{eq:normJprodukt}
		a\circ b :=\frac12(ab+ba),
	\end{equation}
	and Jordan homomorphisms are equivalently defined as $\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.

	The problem of automatic additivity for \emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\circ$ or $\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\'ar \cite{Molnar}, who characterized $\circ$-preserving bijections $\phi:\mathcal{A} \to \mathcal{B}$ between standard operator algebras $\mathcal{A}$ and $\mathcal{B}$ with $\dim(\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\'ar's result  asserts that any $\circ$-preserving bijection $\phi : M_n(\C)\to M_n(\C)$, $n \ge 2$, takes the form
	\begin{equation}\label{eq:glavnieq}
		\phi(X)=T\,\omega(X)\,T^{-1} \quad \text{ or } \quad \phi(X)=T\,\omega(X)^t\,T^{-1},
	\end{equation}
	where $T\in M_n(\C)$ is invertible, $\omega$ is a ring automorphism of $\C$ applied entrywise, and $(\cdot)^t$ denotes matrix transposition. Lu \cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\diamond$- or $\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \cite{Ji} showed that if $\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\phi:\mathcal{A}\to \mathcal{B}$ to another Jordan algebra $\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.

	In the setting of full matrix algebras, our recent work \cite{GogicTomasevic-AM} provides a complete classification of all  $\circ$-preserving maps $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$, where $\chr(\mathbb{F}) \neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\eqref{eq:glavnieq}, where $T \in M_n(\mathbb{F})$ is invertible and $\omega : \mathbb{F} \to \mathbb{F}$ is a ring monomorphism \cite[Theorem~1.1]{GogicTomasevic-AM}.   This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\phi : \mathcal{A} \to \mathcal{B}$ is any $\diamond$-preserving map between $\mathbb{F}$-algebras (with $\chr(\mathbb{F}) \neq 2$), then the simple normalization
	$$
	\psi(x) := 2\,\phi\left(\frac{x}{2}\right), \quad x \in \mathcal{A},
	$$
	transforms $\phi$ into a $\circ$-preserving map $\psi$.  This reduction allows us to obtain a corresponding classification of $\diamond$-preserving maps on $M_n(\mathbb{F})$,
	with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \in M_n(\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\mathbb{R}$ and $\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \emph{structural matrix algebras}, i.e.\ subalgebras of $M_n(\mathbb{F})$ containing all diagonal matrices \cite{vanWyk} (see also \cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \cite{Coelho, GogicTomasevic-LAA}.

	A substantial body of literature also investigates multiplicative maps with respect to products other than the Jordan product, particularly various symmetrized or triple-product variants such as $(a,b) \mapsto aba$, $(a,b,c) \mapsto abc + cba$, and many others (see, e.g., \cite{AnHou, Kuzma, LesnjakSze, Lu2, Molnar2} and references therein). More recently, preservers of the product
	\begin{equation}\label{eq:the product}
		a^{2} \circ b = \frac12(a^{2} b + b a^{2})
	\end{equation}
	were studied by Ghorbanipour and Hejazian~\cite{GhorbanipourHejazian}, who showed that any bijection $\phi : \mathcal{A} \to \mathcal{A}$ preserving \eqref{eq:the product}, where $\mathcal{A}$ is a real standard Jordan operator algebra acting on a Hilbert space of dimension at least $2$, is of the form $\phi = \pm \psi$, where $\psi$ is a Jordan automorphism of $\mathcal{A}$.

	The product $(a,b)\mapsto a^2 \circ b$ plays an important role in the theory of JB-algebras. Recall that a \emph{JB-algebra} is a (typically real) Jordan algebra $(\mathcal{A},\circ)$ equipped with a complete submultiplicative norm $\|\cdot\|$  (that is, a Jordan Banach algebra) which additionally satisfies
	$$
	\|a\|^2 \le \|a^2 + b^2\|, \quad \text{ for all }  a,b \in \mathcal{A}.
	$$
	A central feature of JB-algebras is that many aspects of their structure, such as order, spectral theory, and the theory of ideals, are encoded in the associated quadratic operators (see, for example, \cite{Battaglia}). In particular, the fundamental quadratic mapping $U_a : \mathcal{A} \to \mathcal{A}$,
	$$
	U_{a}(b) := 2(a \circ b) \circ a - a^{2} \circ b,
	$$
	plays a key role in the analysis of orthogonality, annihilators, and quadratic ideals. Moreover, the product $(a,b)\mapsto a^2 \circ b$ was used in \cite{Battaglia} to define orthogonality in JB-algebras: elements $a,b \in \mathcal{A}$ are said to be orthogonal if $a^{2} \circ b = 0$. In the setting of special JB-algebras, such as the self-adjoint part of a $C^{*}$-algebra, the Jordan product is given by \eqref{eq:normJprodukt}. In this case, the expression $a^{2} \circ b$ agrees exactly with the symmetrized product appearing in \eqref{eq:the product}.

	\smallskip

	This motivates our study of \emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product
	$$
	a^k \circ b= \frac{1}{2}(a^k b + b a^k),
	$$
	for arbitrary $k \in \N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.

In an associative setting, a linear (or additive) map $\phi: \ca{A} \to \ca{B}$ between associative algebras (or rings) $\ca{A}$ and $\ca{B}$ is called a \emph{Jordan homomorphism} if it satisfies
 $$
 \phi(a \diamond b) = \phi(a) \diamond \phi(b), \quad \text{ for all } a, b \in \ca{A}.
 $$
 If the algebras (rings) are $2$-torsion-free, this is equivalent to requiring that $\phi$ preserves squares with respect to the underlying product of $\mathcal{A}$, that is,
 $$
 \phi(a^2) = \phi(a)^2, \quad \text{ for all } a \in \ca{A}.
 $$
 Moreover, if $\ca{A}$ is an algebra over a field $\F$ with $\chr(\F) \ne 2$, the Jordan structure is commonly (esp.\ in quantum mechanics contexts) induced by the \emph{normalized Jordan product}
 \begin{equation}\label{eq:normJprodukt}
 a\circ b :=\frac12(ab+ba),
 \end{equation}
 and Jordan homomorphisms are equivalently defined as $\circ$-multiplicative linear maps. The study of Jordan homomorphisms in associative settings is well established and constitutes a major research direction in abstract algebra, operator theory, and the theory of preservers.

The problem of automatic additivity for \emph{Jordan multiplicative maps} has been studied extensively since the early 2000s. This line of research focuses on maps that preserve the Jordan product, $\circ$ or $\diamond$, viewed as a natural analogue of the classical automatic additivity problem for multiplicative maps. A seminal contribution was made by Moln\'ar \cite{Molnar}, who characterized $\circ$-preserving bijections $\phi:\mathcal{A} \to \mathcal{B}$ between standard operator algebras $\mathcal{A}$ and $\mathcal{B}$ with $\dim(\mathcal{A})>1$, that is, subalgebras of bounded linear operators on a complex Banach space containing all finite-rank operators. In particular, all such maps are automatically additive. The finite-dimensional variant of Moln\'ar's result asserts that any $\circ$-preserving bijection $\phi : M_n(\C)\to M_n(\C)$, $n \ge 2$, takes the form
 \begin{equation}\label{eq:glavnieq}
 \phi(X)=T\,\omega(X)\,T^{-1} \quad \text{ or } \quad \phi(X)=T\,\omega(X)^t\,T^{-1},
 \end{equation}
 where $T\in M_n(\C)$ is invertible, $\omega$ is a ring automorphism of $\C$ applied entrywise, and $(\cdot)^t$ denotes matrix transposition. Lu \cite{Lu} subsequently extended this result to additional classes of associative algebras. Subsequently, An and Hou \cite{AnHou} studied Jordan multiplicative bijections on the real Jordan algebra of all self-adjoint operators on a complex Hilbert space of dimension greater than $1$, proving that any $\diamond$- or $\circ$-preserving bijection is automatically additive and implemented via unitary or anti-unitary conjugation. The problem of automatic additivity has also been investigated in the context of abstract Jordan algebras. In particular, Ji \cite{Ji} showed that if $\mathcal{A}$ is a Jordan algebra containing an idempotent satisfying suitable Peirce-type structural conditions, then any bijection $\phi:\mathcal{A}\to \mathcal{B}$ to another Jordan algebra $\mathcal{B}$ that preserves the Jordan product is necessarily additive, thereby unifying a range of operator-algebraic and ring-theoretic results within a single framework.

In the setting of full matrix algebras, our recent work \cite{GogicTomasevic-AM} provides a complete classification of all $\circ$-preserving maps $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$, where $\chr(\mathbb{F}) \neq 2$. We show that every such map is either constant (and hence equal to a fixed idempotent) or a Jordan ring monomorphism of the form~\eqref{eq:glavnieq}, where $T \in M_n(\mathbb{F})$ is invertible and $\omega : \mathbb{F} \to \mathbb{F}$ is a ring monomorphism \cite[Theorem~1.1]{GogicTomasevic-AM}. This generalizes the finite-dimensional version of Molnár's theorem. Moreover, if $\phi : \mathcal{A} \to \mathcal{B}$ is any $\diamond$-preserving map between $\mathbb{F}$-algebras (with $\chr(\mathbb{F}) \neq 2$), then the simple normalization
 $$
 \psi(x) := 2\,\phi\left(\frac{x}{2}\right), \quad x \in \mathcal{A},
 $$
 transforms $\phi$ into a $\circ$-preserving map $\psi$. This reduction allows us to obtain a corresponding classification of $\diamond$-preserving maps on $M_n(\mathbb{F})$,
 with the only difference that the constant map now takes the value $P/2$ for some idempotent $P \in M_n(\mathbb{F})$. In particular, this result shows that Jordan multiplicative maps on $M_n(\mathbb{F})$ are even more rigid than general multiplicative maps: every nonconstant map is necessarily additive. For the fields $\mathbb{R}$ and $\mathbb{C}$, this classification, together with the Jodeit--Lam theorem, was further extended in \cite{GogicTomasevic-LAMA} to injective multiplicative and Jordan multiplicative maps on \emph{structural matrix algebras}, i.e.\ subalgebras of $M_n(\mathbb{F})$ containing all diagonal matrices \cite{vanWyk} (see also \cite[Proposition~3.1]{GogicTomasevic-LAA}). Further details on structural matrix algebras, their automorphisms, and their Jordan monomorphisms can be found in \cite{Coelho, GogicTomasevic-LAA}.

\smallskip

This motivates our study of \emph{mixed Jordan--power preservers}, namely, maps that preserve a more general Jordan-type product
 $$
 a^k \circ b= \frac{1}{2}(a^k b + b a^k),
 $$
 for arbitrary $k \in \N$. Our main result provides a complete classification of mixed Jordan--power preservers on full matrix algebras.

\smallskip

\section{Notation and Preliminaries}\label{sec:prel}
 We begin by introducing notation that will be used throughout the paper. Let $\F$ be a fixed algebraically closed field of characteristic different from $2$. We denote by $\mathbb{F}^\times$ the multiplicative group of nonzero elements of $\mathbb{F}$, and by $\mathbb{F}[x]$ the polynomial algebra in one variable over $\mathbb{F}$.

\begin{lemma}\label{le:h exists}
 Let $\phi : M_n \to M_n$ be a nonzero map which satisfies \eqref{eq:a certain product} and $\phi(0) = 0$. There exists a unique multiplicative map $\omega : \F \to \F$ such that
 \begin{equation}\label{eq:homogeneity}
 \phi(\lambda X) = \omega(\lambda) \phi(X), \quad \text{ for all } \lambda \in \F \text{ and } X \in M_n.
 \end{equation}
 Moreover, there exists an invertible matrix $T \in M_n$ and $\varepsilon\in\Sph^1_k$ such that the map $$\psi:= T^{-1}\phi(\cdot)T$$ satisfies $$\psi(E_{jj}) = \varepsilon E_{jj}, \quad \text{ for each } j \in [n].$$
 Further, fix some distinct $i,j \in [n]$. Then $\psi(E_{ij})$ is a nonzero scalar multiple of $E_{ij}$ or $E_{ji}$. If $\psi(E_{ij}) = \beta_{ij} E_{ij}$ for some $\beta_{ij} \in \F^\times$, then for each $x,y \in \F$ we have
 \begin{equation}\label{eq: Eii+Eij 1}
 \psi(E_{ii} + x E_{ij}) = \varepsilon E_{ii} + \omega(x) (\beta_{ij} E_{ij}), \qquad \psi(E_{jj} + y E_{ij}) =
 \varepsilon E_{jj} + \omega(y) (\beta_{ij} E_{ij}).
 \end{equation}
 If, on the other hand, $\psi(E_{ij}) = \gamma_{ij} E_{ji}$ for some $\gamma_{ij} \in \F^\times$, then
 \begin{equation}\label{eq: Eii+Eij 2}
 \psi(E_{ii} + x E_{ij}) = \varepsilon E_{ii} + \omega(x) (\gamma_{ij} E_{ji}), \qquad \psi(E_{jj} + y E_{ij}) =
 \varepsilon E_{jj} + \omega(y) (\gamma_{ij} E_{ji}).
 \end{equation}
 Finally, if $n \ge 2$, the map $\omega : \F \to \F$ is a ring monomorphism. In particular, $\phi$ is $\K$-homogeneous.
 \end{lemma}
 \begin{proof}
 In view of Lemma \ref{le:basic properties II} (c) and (e), $\phi(E_{11}), \ldots ,\phi(E_{nn})$ are mutually orthogonal rank-one $(k+1)$-potents and therefore can be simultaneously diagonalized (see e.g.\ \cite[Theorem~8 of \S6.5]{HoffmanKunze}). Hence, by passing to map $T^{-1}\phi(\cdot)T$, for a suitable invertible matrix $T \in M_n$, without loss of generality we can assume that
 \begin{equation}\label{eq:phi fiksira dijagonalne}
 \phi(E_{jj}) = \varepsilon_j E_{jj}, \quad \text{ for some $\varepsilon_j \in \Sph^1_k$ for each } j \in [n].
 \end{equation}
 Note that by Lemma \ref{le:basic properties II} (f), for each diagonal $(k+1)$-potent $D \in \ca{D}_n$ we have
 \begin{equation}\label{eq:diagonal tripotents}
 \phi(D) =
 \sum_{(j,j) \in \supp D} \varepsilon_j E_{jj}.
 \end{equation}
 By Lemma \ref{le:zero map}, we have
 \begin{equation}\label{eq:lambda times ejj}
 \phi(\lambda E_{jj}) \ne 0, \quad\text{ for all } j \in [n] \text{ and } \lambda \in \F^\times.
 \end{equation} Further, for each $X \in M_n$ and $S \subseteq [n]$ we have
 \begin{equation}\label{eq:preserves support}
 \supp X \subseteq S \times S \implies \supp \phi(X) \subseteq S \times S.
 \end{equation}
 Indeed, denote the diagonal idempotent
 $P := \sum_{j \in [n]\setminus S} E_{jj}$ and note that a matrix $X\in M_n$ is supported in $S \times S$ if and only if $XP=PX=0$. In that case, obviously $X \circ P^k = 0$, so
 $$0 = \phi(X \circ P^k) = \phi(X) \circ \phi(P)^k \stackrel{\eqref{eq:diagonal tripotents}}= \phi(X) \circ P$$
 and hence Lemma \ref{le:Jordan product calculations} (a) implies the claim.
 \smallskip

Fix $(i,j), (j,p) \in [n]^2 \setminus \Delta_n$. If $i \ne p$, then by Lemma \ref{le:h exists} we have
 \begin{align*}
 \frac12 g(i,p)E_{ip} &= \phi\left(\frac12 E_{ip}\right) =\phi((E_{ii}+E_{ij})^k \circ E_{jp}) = \phi(E_{ii}+E_{ij})^k \circ \phi(E_{jp}) \\
 &\leftstackrel{\eqref{eq: Eii+Eij 1}}= (E_{ii}+g(i,j)E_{ij})^k \circ (g(j,p)E_{jp})\\
 &= \frac12 g(i,j)g(j,p)E_{ip},
 \end{align*}
 which implies $g(i,p) = g(i,j)g(j,p)$.
 On the other hand, if $i = p$, then by Lemma \ref{le:h exists} we have
 \begin{align*}
 \frac{1}{2}\phi\left(E_{ii} + E_{jj} + E_{ji}\right) &= \phi\left(\frac{1}{2}\left(E_{ii} + E_{jj} + E_{ji}\right)\right) = \phi((E_{ii}+E_{ij})^k \circ E_{ji}) \\
 &= \phi(E_{ii}+E_{ij})^k \circ \phi(E_{ji}) \\
 &\leftstackrel{\eqref{eq: Eii+Eij 1}}= (E_{ii}+ g(i,j)E_{ij})^k \circ (g(j,i)E_{ji})\\
 &= \frac12 g(j,i) \Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\Big).
 \end{align*}
 Furthermore, we have
 \begin{align*}
 E_{ii} +\frac12g(j,i) E_{ji} \,\, &\leftstackrel{\eqref{eq: Eii+Eij 1}}= \phi\left(E_{ii} +\frac12 E_{ji}\right) =\phi\left((E_{ii} + E_{jj} + E_{ji}) \circ E_{ii}^k\right) \\
 &= \phi\left(E_{ii} + E_{jj} + E_{ji}\right) \circ \phi(E_{ii})^k\\
 &= g(j,i) \Big(g(i,j)(E_{ii}+E_{jj}) + E_{ji}\Big) \circ E_{ii} \\
 &= g(j,i)\left(g(i,j) E_{ii} +\frac12 E_{ji}\right)
 \end{align*}
 We obtain
 $$
 g(i,j)g(j,i) = 1,
 $$
 as desired. Following \cite{Coelho}, denote by $$g^* : M_n \to M_n, \qquad g^*(E_{ij}) := g(i,j)E_{ij}$$
 the induced algebra automorphism of $M_n$. Note that $g^*$ is implemented via conjugation by the invertible diagonal matrix
 $$
 D := \operatorname{diag}\big(g(1,1), g(2,1), \ldots, g(n,1)\big) \in \ca{D}_n,
 $$
 i.e.\ $g^*(\cdot)=D(\cdot)D^{-1}$. Hence, by passing to the map $(g^*)^{-1}(\phi(\cdot)) = D^{-1}\phi(\cdot)D$, without loss of generality we can assume that
 $$\phi(E_{ij}) = E_{ij}, \quad \text{ for all } (i,j) \in [n]^2.$$
 In view of Lemma \ref{le:h exists}, denote by $\omega :\F \to \F$ the induced ring monomorphism that satisfies \eqref{eq:homogeneity}. We claim that
 \begin{equation}\label{eq:phi = omega}
 \phi(X) = \omega(X), \quad \text{ for all } X \in M_n.
 \end{equation}
 First assume that $X=[x_{ij}] \in M_n$ is of the form $X = Y^k$ for some $Y \in M_n$. Fix $(i,j) \in [n]^2$ and denote $\phi(X)=[\phi(X)_{ij}]$. If $i = j$, then by Lemma \ref{le:preserves triple product} we have
 \begin{align*}
 \omega(x_{ii})E_{ii} &= \phi(x_{ii}E_{ii}) = \phi(E_{ii}Y^k E_{ii}) = \phi(E_{ii})\phi(Y)^k \phi(E_{ii}) = E_{ii}\phi(X) E_{ii} \\
 &= \phi(X)_{ii}E_{ii},
 \end{align*}
 which shows that
 \begin{equation}\label{eq:Yii}
 \phi(X)_{ii} = \omega(x_{ii}).
 \end{equation} Now assume $i \ne j$. Denote $P:= E_{ii}+E_{ji}$ and note that
 $$
 PAP = (a_{ii}+a_{ij})P, \quad \text{ for all matrices $A=[a_{ij}] \in M_n$}.
 $$
 By \eqref{eq: Eii+Eij 1} we have $\phi(P) = P$ and hence
 \begin{align*}
 \omega(x_{ii}+x_{ij})P &=\phi((x_{ii}+x_{ij})P) = \phi(PXP) \stackrel{\text{Lemma }\ref{le:preserves triple product}}= \phi(P)\phi(X)\phi(P) \\
 &= P\phi(X)P = (\phi(X)_{ii}+\phi(X)_{ij})P.
 \end{align*}
 Therefore, by the additivity of $\omega$ and \eqref{eq:Yii} we obtain
 $$\omega(x_{ii})+\omega(x_{ij}) = \phi(X)_{ii}+\phi(X)_{ij} = \omega(x_{ii})+\phi(X)_{ij}.$$
 This implies $\phi(X)_{ij} = \omega(x_{ij})$, which shows \eqref{eq:phi = omega} for all matrices $X \in M_n(\F)$ which are a $k$-th power of some matrix. Using the notation of Lemma \ref{le:iterated R}, this means
 $$\phi(X) = \omega(X), \quad \text{ for all } X \in \ca{R}_0^n.$$
 We now extend this conclusion to all matrices in $M_n$. Note that if $A,B \in M_n$ satisfy $\phi(A) = \omega(A)$ and $\phi(B) = \omega(B)$, then clearly
 $$\phi(A^k \circ B) = \phi(A)^k \circ \phi(B) = \omega(A)^k \circ \omega(B) = \omega(A^k \circ B).$$
 Again returning to the notation of Lemma \ref{le:iterated R}, this inductively implies that $$\phi(X) = \omega(X), \quad \text{ for all } X\in \ca{R}^n.$$
 By the same lemma we have $\ca{R}^n = M_n$, which completes the proof of the theorem.
 \end{proof}
 We conclude the paper with the following generalization of \cite[Corollary~3.5]{GogicTomasevic-AM}, which follows immediately from the first part of the proof of Theorem~\ref{thm:main}.
 \begin{corollary}
 Let $m < n$. A map $\phi : M_n \to M_m$ satisfies \eqref{eq:a certain product} if and only if it is constant and equal to a fixed $(k+1)$-potent.
 \end{corollary}

This result extends all previously known Jordan-type additivity results on $M_n(\F)$, at least for algebraically closed fields with $\operatorname{char}(\F) \neq 2$. The proof of Theorem~\ref{thm:main}, presented in Section~\S\ref{sec:main}, follows the general strategy of \cite[Theorem~1.1]{GogicTomasevic-AM}. The classification relies on the fact that maps satisfying \eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements. These notions will be introduced and discussed in the next section. A notable difference from the case $k=1$, which corresponds to idempotents, is that when $\operatorname{char}(\F)$ is positive and divides $k$, $(k+1)$-potents may fail to be diagonalizable (see Remark~\ref{rem:k-potents not diagonalizable}). This phenomenon requires more refined arguments in order to handle the mixed Jordan--power preservers effectively. We also note that linear and closely related variants of $k$-potent preservers on matrix and operator algebras have been studied in papers such as \cite{HouHou,SongCao,YouWang}.

The proof of Theorem~\ref{thm:main} (given in Section~\S\ref{sec:main}) “follows the general strategy of \cite[Theorem~1.1]{GogicTomasevic-AM}.” The classification “relies on the fact that maps satisfying \eqref{eq:a certain product} preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\operatorname{char}(\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”

The proof of the main theorem (given later) “follows the general strategy of Gogic--Tomasevic, Theorem~1.1 (GogicTomasevic-AM).” The classification “relies on the fact that maps satisfying
\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
preserve $(k+1)$-potents along with their ranks and natural partial order, and that they are orthoadditive on these elements.” A key complication relative to the case $k=1$ (idempotents) is that if $\operatorname{char}(\F)$ is positive and divides $k$, then “$(k+1)$-potents may fail to be diagonalizable,” which “requires more refined arguments in order to handle the mixed Jordan--power preservers effectively.”

		\label{thm:main}
		Let $\F$ be an algebraically closed field of characteristic different from $2$, and let $n,k \in \N$ with $n \ge 2$. A map $\phi : M_n(\F) \to M_n(\F)$
		satisfies the mixed Jordan--power identity
		\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
		if and only if one of the following holds:
		\begin{itemize}
			\item[(a)] $\phi$ is constant, taking a fixed $(k+1)$-potent value;
			\item[(b)] there exist an invertible matrix $T \in M_n(\F)$, a ring monomorphism $\omega : \F \to \F$, and a $k$-th root of unity $\varepsilon \in \F$ such that
			\begin{equation}\label{eq:form of phi}
				\phi(X) = \varepsilon\, T\, \omega(X)\, T^{-1}
				\quad \text{ or } \quad
				\phi(X) = \varepsilon\, T\, \omega(X)^t\, T^{-1}.
			\end{equation}
		\end{itemize}
		In particular, every nonconstant solution of the mixed Jordan--power identity
		\begin{equation}\label{eq:a certain product}
			\phi(A^k \circ B) = \phi(A)^k \circ \phi(B), \qquad \text{for all } A,B \in M_n(\F),
		\end{equation}
		is additive.,

Biconditional or Equivalence
Classification or Bijection

Question:
Let \(\mathbb F\) be an algebraically closed field with \(\operatorname{char}(\mathbb F)\neq 2\), let \(n\ge 2\) and \(k\in\mathbb N\), and write \(A\circ B:=\tfrac12(AB+BA)\) for the Jordan product on \(M_n(\mathbb F)\). Which maps \(\phi:M_n(\mathbb F)\to M_n(\mathbb F)\) satisfy
\[
\phi(A^k\circ B)=\phi(A)^k\circ \phi(B)\qquad\text{for all }A,B\in M_n(\mathbb F)?
\]

Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\) (that is, \(P^{k+1}=P\)); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1},
\]
where \(\omega(X)\) means applying \(\omega\) entrywise. In particular, every nonconstant such \(\phi\) is additive.

Choices:
B.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\) and a ring monomorphism \(\omega:\mathbb F\to\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=T\,\omega(X)^t\,T^{-1}.
\]
In particular, every nonconstant such \(\phi\) is additive.
C.
Every map \(\phi:M_n(\mathbb F)\to M_n(\mathbb F)\) of one of the following forms satisfies
\[
\phi(A^k\circ B)=\phi(A)^k\circ\phi(B)\qquad\text{for all }A,B\in M_n(\mathbb F):
\]
either \(\phi\) is constant with value a fixed \((k+1)\)-potent matrix \(P\), or there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1}.
\]
D.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed idempotent matrix \(P\) (that is, \(P^2=P\)); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a scalar \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1},
\]
where \(\omega(X)\) is applied entrywise. In particular, every nonconstant such \(\phi\) is additive.
E.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a field automorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}.
\]
In particular, every nonconstant such \(\phi\) is additive.

replaced classification iff by one-way sufficient condition

constant-value type and unrestricted scalar factor

transpose branch and monomorphism generality

The stem does not state or strongly hint at the exact classification in the correct option. It only gives the functional identity and asks for an equivalent characterization.

This is essentially a theorem-recall item: the correct option is the classification theorem itself, stated in near-final form. It does not meaningfully reframe the result into a new setting or application.

There is some reasoning pressure because the options differ in subtle structural details (monomorphism vs automorphism, k-th vs (k+1)-th conditions, global vs pointwise scalar factor). However, the task mainly tests precise recall/discrimination rather than genuine generative mathematical reasoning.

The distractors are mathematically plausible and target realistic failure modes: wrong potent class, overly strong automorphism assumption, loss of the scalar factor, and incorrect pointwise dependence. They are distinct and nontrivial.

A technically strong but theorem-recall-heavy MCQ. It avoids answer leakage and has excellent distractors, but it is largely a direct restatement test rather than a genuinely generative reasoning problem.


Question:
Let \(\mathbb F\) be an algebraically closed field with \(\operatorname{char}(\mathbb F)\neq 2\), let \(n\ge 2\) and \(k\in\mathbb N\), and write \(A\circ B:=\tfrac12(AB+BA)\) for the Jordan product on \(M_n(\mathbb F)\). Which maps \(\phi:M_n(\mathbb F)\to M_n(\mathbb F)\) satisfy
\[
\phi(A^k\circ B)=\phi(A)^k\circ \phi(B)\qquad\text{for all }A,B\in M_n(\mathbb F)?
\]

Choices:
B.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\) and a ring monomorphism \(\omega:\mathbb F\to\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=T\,\omega(X)^t\,T^{-1}.
\]
In particular, every nonconstant such \(\phi\) is additive.
C.
Every map \(\phi:M_n(\mathbb F)\to M_n(\mathbb F)\) of one of the following forms satisfies
\[
\phi(A^k\circ B)=\phi(A)^k\circ\phi(B)\qquad\text{for all }A,B\in M_n(\mathbb F):
\]
either \(\phi\) is constant with value a fixed \((k+1)\)-potent matrix \(P\), or there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1}.
\]
D.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed idempotent matrix \(P\) (that is, \(P^2=P\)); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a ring monomorphism \(\omega:\mathbb F\to\mathbb F\), and a scalar \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}\qquad\text{or}\qquad \phi(X)=\varepsilon T\,\omega(X)^t\,T^{-1},
\]
where \(\omega(X)\) is applied entrywise. In particular, every nonconstant such \(\phi\) is additive.
E.
Exactly those maps \(\phi\) for which either: (i) \(\phi\) is constant, with value a fixed \((k+1)\)-potent matrix \(P\); or (ii) there exist an invertible matrix \(T\in M_n(\mathbb F)\), a field automorphism \(\omega:\mathbb F\to\mathbb F\), and a \(k\)-th root of unity \(\varepsilon\in\mathbb F\) such that for every \(X\in M_n(\mathbb F)\),
\[
\phi(X)=\varepsilon T\,\omega(X)\,T^{-1}.
\]
In particular, every nonconstant such \(\phi\) is additive.

The stem does not disclose the classification or directly hint at the correct structural form of the maps. The correct answer is not leaked by wording in the prompt.

This is essentially an exact-classification recall question: the correct option states the theorem itself. It does not ask the student to derive a new consequence from the hypothesis, but to recognize the theorem statement.

There is some reasoning pressure because the options differ in subtle but important ways (e.g. iff vs one-way implication, presence of transpose, monomorphism vs automorphism, k-th root of unity factor). However, the item mainly tests theorem recognition rather than substantial generative reasoning.

The distractors are mathematically plausible and target realistic confusions: omitting the root-of-unity factor, weakening an iff classification to a sufficient condition, replacing (k+1)-potent by idempotent and allowing arbitrary scalars, or excluding the transpose/monomorphism cases. They are distinct and well aligned with common failure modes.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it is largely a direct restatement of the classification result rather than a genuine generative-reasoning task.


\begin{theorem}\label{EucTrees}
 In the Euclidean setting, with edges labeled randomly by $n-1$ colors, we have $Z_{\MST} = \Theta(\sqrt n)$ w.h.p.\ and $Z_{\MST}-Z^*_{\MST}=\Theta(\sqrt{n})$ w.h.p.\end{theorem}

The minimum length of structures like spanning trees and Hamilton cycles among random points in Euclidean space has been a topic of interest for the better part of a century. Beardwood, Halton and Hammersley \cite{BHH} showed that the minimum cost of a tour through $n$ random points in the unit square is asymptotic to $\b_{TSP} n^{1/2}$ with high probability (we say an event $\cE_n$ occurs {\em with high probability} or w.h.p.\ if $\mathbb{P}(\cE_n) \rightarrow 1$ as $n \rightarrow \infty$). Here $\b_{TSP}$ is a positive constant, still unknown after more than 60 years. Steele \cite{Steele1} generalised this result to {\em Euclidean Functionals} and showed that the minimum cost of a spanning tree on $n$ random points is asymptotic to $\b_{MST} n^{1/2}$, w.h.p. In both cases, the cost of edges is determined by the Euclidean distance between points. Steele's work shows that there are many instances of optimisation problems where the expected minimum cost grows like $\beta n^{1/2}$ for unknown constants $\beta$. Frieze and Pegden \cite{FP1} showed the constants for lengths of lower bound structures for the tour (like the minimum spanning tree, or twice a matching) really are distinct from the constant for the TSP, even though the best known explicit bounds on the constants generally overlap.

When the costs are generated independently, the situation becomes much clearer. In the case of spanning trees where the edges of the complete graph $K_n$ are given independent uniform $[0,1]$ random costs, Frieze \cite{F1} showed that $\mathbb{E}(L_n)\sim\zeta(3)=\sum_{k=1}^\infty\frac{1}{k^3}$, where $L_n$ denotes the (random) minimum cost of a spanning tree.

The expected cost of the minimum cost Hamilton cycle in the complete graph $K_{n}$ is asymptotic to $\tau=\frac12\int_{x=0}^\infty y(x)dx$ where $y$ is the positive solution to $\brac{1+\frac{x}{2}}e^{-x}+\brac{1+\frac{y}{2}}e^{-y}=1$. This was proved by Wastl\"und \cite{Wast1}. Frieze \cite{F2} showed that the expected minimum cost of a Hamilton cycle is asymptotically equal to the expected minimum cost of a 2-factor.

If we orient the edges then it follows from the work of Karp \cite{K1} and Aldous \cite{A1} that the expected minimum cost of a directed Hamilton cycle is asymptotic to $\zeta(2)=\sum_{k=1}^\infty\frac{1}{k^2}=\frac{\pi^2}{6}$.

In this work we add a spot of color to the problems. We assume that the edges of the graphs in question have been given uniformly random colors and ask for minimum cost rainbow Hamilton cycles or spanning trees. In this context, a set $S$ of edges is {\em rainbow} colored if all the colors in $S$ are distinct. We do this within two contexts: either we have $n$ random points in $[0,1]^2$ and the cost of an edge is the Euclidean distance between its endpoints, or else we have independent uniform $[0,1]$ random costs.

In the first context we let ${\mathcal X}=\left\{\bx_1,\bx_2,\ldots,\bx_n\right\}$ where the $\bx_i$ are independently chosen uniformly from the unit square $[0,1]^2$. The cost of edge $\left\{\bx_i,\bx_j\right\}$ is the Euclidean length $|\bx_i-\bx_j|$. Our first results concern the length $Z_{\mathrm{MST}}$ of the minimum rainbow spanning tree in this model, when we randomly color labeled from a set of $n-1$ colors. (Note that with fewer colors, no rainbow spanning tree would exist. Also, even if we have enough colors there is always some small probability that there is no rainbow spanning tree at all, in which case $Z_{\mathrm{MST}}$ is undefined.) We let $Z^*_{\mathrm{MST}}$ denote the length of the minimum spanning tree (without the rainbow constraint); recall that $Z^*_{\mathrm{MST}}\sim \beta_{\mathrm{MST}} \sqrt n$ w.h.p., for some constant $\beta_{\mathrm{MST}}$.

\begin{theorem}\label{EucTours}
In the Euclidean setting, with edges labeled randomly by $q=(1+\ep)n$ colors, we have (a) $Z_{\TSP}=O(n^{1/2}\log n)$, and (b) $Z_{\TSP}-Z_{\TSP}^*=\Omega(\sqrt{n})$ w.h.p. Indeed, (a) holds for any set of $n$ points with randomly colored edges.
\end{theorem}
Here like before the starred version $Z^*_{\TSP}$ represents the minimum length in the absence of the rainbow condition.

We define a weighted bipartite (multi-)graph $\G$ with bipartition $\cX_1 \cup C$ where $C=[n-1]$ is our set of colors. The edge set will be the union $E_1\cup E_2$ of sets which we define next; for both sets $E_1$ and $E_2$, an edge $(\bx,c)$ will correspond to some edge $(\bx,\bx')$ with $\xi_2(\bx) < \xi_2(\bx')$ in the embedded graph that has been assigned color $c$. This last property guarantees that a perfect matching in $\G$ corresponds to a rainbow spanning tree in the embedded graph. So to prove the first assertion in Theorem~\ref{EucTrees}, it suffices to show that w.h.p.
\begin{equation}\label{match}
\text{$\G$ contains a perfect matching,}
\end{equation}
and
\begin{equation}\label{wt}
\text{the total weight of all edges in $\G$ is $O(n^{1/2})$.}
\end{equation}

\begin{claim}\label{clm:3}
 If the event $\cG$ holds (where $\cG$ is defined above \eqref{negG}), then for any fixed $S \subseteq A$ of size $s$, (under the probability of choosing random colors for each edge)
\[
 \Pr(\cE_c(S)\mid A) \leq \bfrac{s}{|A|}^{\sqrt{K}} + O(n^{-\sqrt{K} + 2}\log^{8\sqrt{K}} n).
\]
 \end{claim}
 \begin{proof}
 Fix $\bx\in A$ and recall the definition of $\nu_\bx$ above \eqref{negG}. Let $\cA$ be the event that for each $\bx \in A$, each color appears fewer than $\sqrt{K}$ times on edges of length at most $2Bn^{-1/2}\log^4 n$. The probability of this event failing for fixed $\bx \in A$ is at most
 \[|C| {\nu_\bx \choose \sqrt{K}}|C|^{-\sqrt{K}},\]
 and taking a union bound over $A$ and using \eqref{negG} gives the second bound in the claim.

We begin with a simple lemma.
\begin{lemma}\label{repeat}
Let $S$ be a set of size $\a=\Theta(n)$ that is randomly colored using $\b=\Theta(n)$ colors. Then w.h.p. there are $\Theta(n)$ colors that are used more than once.
\end{lemma}
\begin{proof}
Let $Z$ denote the number of colors that appear more than once. Then
\[
\E(Z)=\b\brac{1-\brac{1-\frac{1}{\b}}^{\a}-\a \cdot \frac{1}{\b}\brac{1-\frac{1}{\b}}^{\a-1}}=\Theta(n).
\]
Now changing the color of one edge changes $Z$ by at most one. So, applying McDiarmid's inequality \cite{McD} we have
\[
\Pr\brac{Z\leq \E(Z)/2}\leq \exp\set{-\frac{\E(Z)^2}{2\a}}=e^{-\Omega(n)}.
\]
\end{proof}
Now back to the main argument. For notational convenience, in this section we scale $\cX$ so that we are instead working with a set $\cY_n$ of $n$ random points in a square of side length $\sqrt{n}$; note that there is thus one point on average, per unit of area.

Let $X$ be a finite set and let $\cH$ be an $r$-bounded hypergraph on $X$. For $k \geq r$, let $X^* = X \times [k]$. Let $\cH^*$ denote the family consisting of all possible rainbow copies of $\cH$ using the color set $[k]$. Let $\xi_x\,(x\in X)$ be independent random variables, each uniform from $[0,1]$, and set
\[
\xi_{\cH}=\min_{S\in \cH^*}\sum_{x\in S}\xi_x.
\]
We prove
\begin{theorem}\label{rweight}
Let $\cH$ be a $\k$-spread, $r$-uniform hypergraph on vertex set $X$. Suppose that (i) $\k=\Omega(r)$ and (ii) $|X|\leq \k^2r/\log^5r$. If we randomly color the elements of $\cH$ with $k\geq r$ colors, then $\xi_\cH\leq 3C_0r/\k$ w.h.p. (assuming $r\to\infty$) for some absolute constant $C_0>0$.
\end{theorem}
(The paper \cite{HY} removes conditions (i) and (ii), but the proof doesn't adapt well to deal with the costs, as far as we can see.)
\begin{proof}
We let $\ell_0$ be the least $i$ such that $r/2^i<\log r$, and let $\ell$ be the least $i>\ell_0$ such that $r/2^i<1$. We sort the elements of $X=\set{x_1,x_2,\ldots,x_N}$ so that $\xi_{x_1}<\xi_{x_2}<\cdots<\xi_{x_N}$. We then let $p_i=C_i/\k$ where
\[
C_i=\begin{cases}C_0&i\leq \ell_0.\\\frac{\log r}{\log\log r}&\ell_0<i\leq \ell.\end{cases}
\]
Here $C_0$ is a large constant. Let $L_i=Np_i$ and define $n_0$ to be the smallest integer such that $2^{-n_0}\leq \log r$ and let $n$ be the least $i>n_0$ such that $r/2^i<1$. Then for $i\in[n]$ we let
\[
a_i=\sum_{j=1}^iL_i\text{ and }\e_i=\frac{2a_i}{N}.
\]
Then let $W^*_1=\set{x_1,x_2,\ldots,x_{a_1}},\,W^*_2=\set{x_{a_1+1},x_{a_1+2},\ldots,x_{a_2}}$ etc. Note that $W^*_1,W^*_2,\ldots,W^*_n$ are randomly colored random sets of size $L_i$ . The Chernoff bounds imply that
\begin{align}
\Pr(\exists i:\max\set{\xi_x:x\in W^*_i}\geq \e_i)&=\sum_{i=1}^n\Pr(Bin\brac{N,\e_i}\leq\e_i N)\nn\\
& \leq \sum_{i=1}^n e^{-\e_iN/8}\leq 2e^{-a_1/4}=2e^{-C_0N/\k}.\label{max}
\end{align}
In \cite{BFM} (as we will explain shortly), it is proved that with probability $1-\frac{\log\log r}{\log^{1/4}r}$, there is a rainbow colored edge $e$ such that $|e\setminus \bigcup_{j\leq i}W^*_j|\leq r/2^{i-1}$. We will use the main theorem (and its proof which we discuss below) from \cite{BFM}:
\begin{theorem}[Theorem 2 in \cite{BFM}]\label{thrainbow}
Let $\cH$ be an $r$-bounded, $\k$-spread hypergraph and let $X=V(\cH)$ be randomly colored from $Q=[q]$ where $q\ge r$. Suppose also that (i) $\k=\Omega(r)$, that is, there exists a constant $L>0$ such that $\kappa\ge Lr$ for all valid $r$, and that (ii) $N\leq \k^2r/\log^5r$. Then there is a constant $C>0$ such that if
\beq{mbounds}{
m\geq\frac{(C\log_2r)|X|}{\k}
}
then $X_m$ contains a rainbow colored edge of $\cH$ w.h.p.
\end{theorem}
 The proof of Theorem \ref{thrainbow} in \cite{BFM} proceeds by iteratively choosing random sets of colored elements $W^*_i$ (just as we have in our current proof). Theorem \ref{thrainbow} is proved by iteratively applying the lemma we state next. Here $\cH^*$ is the set of possible rainbow colored edges of $\cH$. For $H^* \in \cH^*$, $T^*(W^*,H^*)$ is $G^* \setminus W^*$ for some $G^* \in \cH^*$.
 \begin{lemma}[Lemma 5 in \cite{BFM}] \label{ML}
Let $H^*\in\cH^*$ be good with respect to $W^*$ if $H^*\sim W^*$ and $|T^*(W^*,H^*)|<r/2$. Let
\[
\text{$\cS$ be the event $|\set{H^*\in \cH^*:H^*\text{ is good}}|<(1-\e)|\cH^*|(1-p)^r$.}
\]
Then
\[
\Pr(\cS)\leq \exp\set{-\frac{\e^2\k^2q(1-p)}{16e^3N}} +\frac{2K}{C_0^{r/3}\e\brac{1-p}^r}
\]
\end{lemma}
In other words, so long as the unlikely event $\cS$ does not occur, we have that for each rainbow colored edge $H^*$, there is some rainbow colored edge $G^*$ that is mostly contained in $W^*$. The proof of Theorem \ref{thrainbow} proceeds by iteratively applying Lemma \ref{ML}, each time replacing $\cH^*$ with the hypergraph whose edges are the $T^*(W^*, H^*)$ (i.e. portions of edges of $\cH^*$ which are not contained in $W^*$). In \cite{BFM} they show that during all applications of the Lemma \ref{ML},
 the unlikely event $\cS$ never happens w.h.p. This means that the rainbow edge $H^*$ we obtain in the end was good for every application of Lemma \ref{ML}. In particular, we have
 \beq{forcost}{
|H^*\setminus \bigcup_{j\leq i}W^*_j|\leq \frac{r}{2^{i-1}}, \qquad \text{i=1, \ldots, $\ell$.}
}
In our setting, since $W^*_i$ was the set of vertices of cost at most $\e_i$, the rainbow edge $H^*$ we obtain from Theorem \ref{thrainbow} costs at most
\mults{
\sum_{i=1}^n\frac{r\e_i}{2^{i-1}}=\frac{r}{N}\sum_{i=1}^n\frac{1}{2^{i-1}}\sum_{j=1}^iNp_i\\
=r\brac{\sum_{i=1}^{\ell_0}\frac{C_i}{2^{i-1}\k}+\sum_{i=\ell_0+1}^\ell\frac{C_{\ell_0}+(i-\ell_0)\log r}{2^{i-1}\k}}\leq \frac{3C_0r}{\k}=O(1).
}

In particular, Theorem \ref{EucTrees} says that the minimum length doesn't change by more than a constant factor under a rainbow constraint, but that a constant-factor change does necessarily occur.  To put Theorem \ref{EucTrees} in context, there is an important sense in which it is much less obvious that the minimum rainbow spanning tree should have length $\Theta(\sqrt{n})$ than it is for the various optimization problems (matchings, 2-factors, linear programming relaxations of the TSP, etc) to which the general methods derived from Beardwood, Halton, and Hammersley and from Steele have been applied.  In particular, the rainbow condition seems to preclude a straightforward application of subadditivity, which explains, for example, why we do not yet know that $Z_{\mathrm{MST}}/\sqrt{n}$ has a limit.  In some sense, the success of subadditivity approaches to structure lengths in random point-sets reflects the sense in which the asymptotic structure lengths really depend only on the local geometry of the ambient space.  (For example, in the Beardwood-Halton-Hammersley theorem, the unit square can be replaced by any bounded open set of $\mathbb{R}^2$ of measure 1, and the asymptotic length of a tour through $n$ points remains the same, with the same unknown constant $\beta_{\mathrm{TSP}}$. But the rainbow condition we impose is a global constraint on the spanning tree that cannot be easily decomposed into noninteracting parts.

For the minimum length $Z_{\mathrm{TSP}}$ of a rainbow Hamilton cycle in the Euclidean model, we do not achieve an optimal result up to constant factors unless we allow a slight excess in the number of available colors.

\begin{theorem}\label{EucTours}
In the Euclidean setting, with edges labeled randomly by $q=(1+\varepsilon)n$ colors, we have (a) $Z_{\mathrm{TSP}}=O(n^{1/2}\log n)$, and (b) $Z_{\mathrm{TSP}}-Z_{\mathrm{TSP}}^*=\Omega(\sqrt{n})$ w.h.p. Indeed, (a) holds for any set of $n$ points with randomly colored edges.
\end{theorem}
Here like before the starred version $Z^*_{\mathrm{TSP}}$ represents the minimum length in the absence of the rainbow condition.

In the second scenario, $[0,1]^2$ is replaced by the complete graph $K_n$ and the costs $C(i,j)$ are independent uniform $[0,1]$ random variables.  We prove a general theorem (see Theorem \ref{rweight}) for cheap rainbow substructures in randomly colored structures with random costs. The proof is an adaptation of some of the recent work on thresholds in spread hypergraphs (see \cite{BFM}) based on Frankston, Kahn, Narayanan, and Park's recent proof of the fractional Kahn--Kalai conjecture \cite{FKNP}. Our Theorem \ref{rweight} implies the following:
\begin{theorem}\label{th2}
The minimum cost solution of the following problems have value $O(1)$ w.h.p. The proofs are non-constructive.
\begin{enumerate}[(a)]
\item Traveling Salesperson Problem in $K_n$ with at least $n$ colors for the edges.
\item Minimum cost perfect matching in complete $k$-uniform hypergraphs, $k\geq 2$ with at least $n/k$ colors for the edges.
\item Minimum cost classes of degree bounded spanning trees in $K_n$ with at least $n-1$ colors for the edges e.g. spanning binary trees.
\end{enumerate}
\end{theorem}
The rest of the paper is organized as follows. In Section \ref{sec:treeUB} we prove the first assertion of Theorem \ref{EucTrees}, i.e.\ that $Z_{\mathrm{MST}} = \Theta(\sqrt{n})$ w.h.p. In Section \ref{sec:tours} we prove the first assertion of Theorem \ref{EucTours}, i.e.\ that $Z_{\mathrm{TSP}}=O(\sqrt {n}\log n)$ for $n$ colors. In Section \ref{sec:separate} we prove the remaining assertions in Theorems \ref{EucTrees} and \ref{EucTours}, i.e.\ that $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt{n})$ for $n$ colors and that $Z_{\mathrm{TSP}}-Z_{\mathrm{TSP}}^*=\Theta(\sqrt{n})$ for $(1+\varepsilon)n$ colors. In Section \ref{sec:spread} we prove Theorem \ref{th2}. Finally, we make some observations and state open questions in Section \ref{sec:conclusion}.

The post-theorem discussion does not give a proof outline for Theorem~\ref{EucTrees} beyond noting why it is "much less obvious" (the rainbow condition is a "global constraint" that "seems to preclude a straightforward application of subadditivity"). The only structural indication is the paper roadmap: in Section~\ref{sec:treeUB} they "prove the first assertion of Theorem~\ref{EucTrees}, i.e.\ that $Z_{\mathrm{MST}}=\Theta(\sqrt n)$ w.h.p.," and in Section~\ref{sec:separate} they "prove the remaining assertions in Theorem~\ref{EucTrees}\ldots i.e.\ that $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt n)$ for $n$ colors."

\label{EucTrees}
 In the Euclidean setting, with edges labeled randomly by $n-1$ colors, we have $Z_{\mathrm{MST}} = \Theta(\sqrt n)$ w.h.p.\ and $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt{n})$ w.h.p.,

Question: Let ${\mathcal X}=\{\mathbf x_1,\dots,\mathbf x_n\}$ be points chosen independently and uniformly from the unit square $[0,1]^2$. Form the complete graph on ${\mathcal X}$, give each edge its Euclidean length, and label the edges randomly using exactly $n-1$ colors. A spanning tree is called rainbow if its $n-1$ edges all have distinct colors. Let $Z_{\mathrm{MST}}$ be the total Euclidean length of a minimum-length rainbow spanning tree, and let $Z^*_{\mathrm{MST}}$ be the total Euclidean length of the ordinary minimum spanning tree without the rainbow requirement. Here “with high probability” means with probability tending to $1$ as $n\to\infty$. Which statement holds with high probability in this model?

Both $Z_{\mathrm{MST}}=\Theta(\sqrt n)$ and $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt n)$.

Choices:
B. Both $Z_{\mathrm{MST}}=\Theta(\sqrt n)$ and $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=o(\sqrt n)$.
C. $Z_{\mathrm{MST}}=\Theta(\sqrt n)$.
D. Both $Z_{\mathrm{MST}}=\Theta(\sqrt n\log n)$ and $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=\Theta(\sqrt n)$.
E. There exists a constant $C>0$ such that $Z_{\mathrm{MST}}\le C\sqrt n$ w.h.p., but $Z_{\mathrm{MST}}-Z^*_{\mathrm{MST}}=O(1)$ w.h.p.

The stem defines the model and quantities but does not explicitly reveal the asymptotic behavior. There are no clear verbal cues that point directly to choice A over the others.

The item is close to a theorem-recall question: one option appears to restate a specific asymptotic result almost verbatim. It is not a pure tautology because the student must distinguish among nearby asymptotic alternatives, but it still mainly tests recognition of a known conclusion.

Some reasoning is needed to compare orders such as Θ(√n), o(√n), Θ(√n log n), and O(1). However, the question mostly rewards prior knowledge of the result rather than generating a conclusion from first principles.

Choices B, D, and E are plausible asymptotic confusions, but choice C is a weaker statement that would also be true if A is true. That makes the option set less clean for a single-correct MCQ and lowers distractor quality.

Moderately good but theorem-recall-heavy. It avoids direct answer leakage, yet it is somewhat close to restating a known result, and the weaker-true option creates ambiguity in the distractor set.


[{\cite[Theorem~5.14]{Guterman2025}}]\label{MainTheoremEvenIntro}Assume that $k \ge 4, n \ge k + 2, n + k$ is even and $|{\mathbb F}| > k$. Let $T\colon \M_{n\,k} ({\mathbb F}) \to \M_{n\,k} ({\mathbb F})$ be a linear map. Then $\det_{n\,k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\,k} ({\mathbb F})$ if and only if there exist $A \in \M_{n\,n}({\mathbb F})$ and $B \in \M_{k\,k}({\mathbb F})$ such that
\[
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\]
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and
\[
T(X) = AXB
\]
for all $X \in \M_{n\,k} ({\mathbb F}).$

The theory of linear maps preserving matrix invariants is well-known direction of research in Linear Algebra connected with other fields of mathematics. It has a long history which dates back to the beginning of 20th century in the works of Frobenius~\cite{GF} and continues at present time (see~\cite{LAMA199233} for the survey of the results until the end of 20th century and~\cite{hogben_handbook_2014} for a brief introduction).

The Cullis' determinant is an invariant of rectangular matrix generalizing the notion of the ordinary determinant of square matrix which was introduced by Cullis in~\cite{cullis1913} and studied later by Radi\'{c} in~\cite{radic1966,radic2005,radic1991,radic2008}, Makarewicz, Mozgawa, Pikuta and Sza\l{}kowski in~\cite{makarewicz2014,makarewicz2016,makarewicz2020}, Amiri, Fathy and Bayat in~\cite{amiri2010}, and Nakagami and Yanay in~\cite{NAKAGAMI2007422}. Thus, the question regarding the description of linear maps preserving the Cullis' determinant arises naturally. The detailed introduction to the subject could be found in~\cite{Guterman2025}.
In particular, \cite{Guterman2025} contains the description of linear maps preserving the Cullis determinant $\det_{n\,k}$ for the case when $ k \ge 4$, $n \ge k + 2$ and $n + k$ is even as it stated in the following theorem (all necessary definitions are given in Section~\ref{sec:prelim}).

The Cullis' determinant is an invariant of rectangular matrix generalizing the notion of the ordinary determinant of square matrix which was introduced by Cullis in~\cite{cullis1913} and studied later by Radi\'{c} in~\cite{radic1966,radic2005,radic1991,radic2008}, Makarewicz, Mozgawa, Pikuta and Sza\l{}kowski in~\cite{makarewicz2014,makarewicz2016,makarewicz2020}, Amiri, Fathy and Bayat in~\cite{amiri2010}, and Nakagami and Yanay in~\cite{NAKAGAMI2007422}. Thus, the question regarding the description of linear maps preserving the Cullis' determinant arises naturally. The detailed introduction to the subject could be found in~\cite{Guterman2025}.
In particular, \cite{Guterman2025} contains the description of linear maps preserving the Cullis determinant $\det_{n\,k}$ for the case when $ k \ge 4$, $n \ge k + 2$ and $n + k$ is even as it stated in the following theorem (all necessary definitions are given in Section~\ref{sec:prelim}).

The parity of $n + k$ is significant because if $n + k$ is odd, then there exist linear maps preserving the Cullis' determinant which do not have such form (Corollary~\ref{cor:AdditionLinearMapNPLusKOdd}).

\begin{theorem}[Theorem~\ref{thm:MainTheoremCullisNKOdd}]Assume that $|\F| > k \ge 4$, $n \ge k + 2$ and $n + k$ is odd. Let $T\colon \M_{n\,k} (\F) \to \M_{n\,k} (\F)$ be a linear map. Then $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\,k} (\F)$ if and only if there exist $A \in \M_{n\, n}(\F)$ and $B \in \M_{k\, k}(\F)$ such that
\[
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\]
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and a linear map\linebreak $\phi\colon \M_{n\,k}(\F) \to W_{n\,k}$ such that
\[
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\]
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and
\begin{equation*}
T(X) = AXB + \phi(X)
\end{equation*}
for all $X \in \M_{n\,k} (\F).$ Here $W_{n\,k} \subseteq \M_{n\,k}(\F)$ denotes the space of matrices, all rows of which are equal, being a radical of $\det_{n\,k}$.
\end{theorem}

\begin{lemma}[{\cite[Lemma~5.13]{Guterman2025}}]\label{lem:CullisDeg1Rank1}Assume that $|\F| > k \ge 4$, $n \ge k + 2$ and $n + k$ is even. Let $T\colon \M_{n\, k} (\F) \to \M_{n\, k} (\F)$ be a linear map such that $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\, k} (\F).$ Then $\rk (X) = 1$ implies $\rk (T(X)) = 1.$
\end{lemma}

We omit the subscripts for $\sgn_{[n]}$ and $\sgn_{n\,k}$ if this cannot lead to a misunderstanding.\begin{definition}[\cite{NAKAGAMI2007422}, Theorem 13]\label{def:CullisDet} Let $n \ge k$, $X \in \M_{n\,k} (\mathbb F)$. Then Cullis' determinant $\det_{n\, k}(X)$ of $X$ is defined to be the function:
$$
\det_{n\, k} (X) = \sum_{\sigma \in \mathcal C_{k}^{[n]}} \sgn_{n\,k} (\sigma) x_{\sigma(1)\, 1}x_{\sigma(2)\, 2}\ldots x_{\sigma(k)\, k}.
$$
We also denote $\det_{n\, k} (X)$ as follows
\[
\det_{n\,k}(X)=\begin{vmatrix}x_{1\,1} & \cdots & x_{1\,k}\\
\vdots & \cdots & \vdots\\
x_{n\,1} & \cdots & x_{n\,k}
\end{vmatrix}_{n\,k}.
\]
If $n = k$, then we also write $\det_{k}$ or $\det$ instead of $\det_{n\,k}$ because in this case $\det_{n\,k}$ is clearly equal to an ordinary determinant of a square matrix.
\end{definition}

\begin{lemma}[{\cite[Lemma~3.1]{Guterman2025}}]\label{lem:TwoSidedMulPreservesDet}Let $n \ge k$, $A \in \M_{n\,n}(\F)$, $B \in \M_{k\,k}(\F)$, and let $T\colon \M_{n\,k}(\F) \to \M_{n\,k}(\F)$ be a linear map defined by
\begin{equation}\label{eq:TwoSideMul}
T(X) = AXB
\end{equation}
for all $X \in \M_{n\,k}(\F).$ Then
\[
\det_{n\,k} (T(X)) = \det_{n\,k}(X)
\]
for all $X \in \M_{n\,k} (\F)$ if and only if
\begin{equation}\label{eq:TwoSideMul2}
\det_{n\,k} \Bigl(A(|d]\Bigr)\cdot \det_k \Bigl(B\Bigr) = \sgn (d)
\end{equation}
for all $d \in \binom{[n]}{k}.$
\end{lemma}

\begin{lemma}[{\cite[Theorem~5.14]{Guterman2025}}]\label{lem:MainTheoremEven}Assume that $|\F| > k \ge 4, n \ge k + 2$ and $n + k$ is even. Let $T\colon \M_{n\, k} (\F) \to \M_{n\, k} (\F)$ be a linear map. Then $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\, k} (\F)$ if and only if there exist $A \in \M_{n\, n}(\F)$ and $B \in \M_{k\, k}(\F)$ such that
\begin{equation}\label{thm:MainTheoremEvenKGe4:eq}
\det_{n\, k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \cdot \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\end{equation}
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and
\[
T(X) = AXB
\]
for all $X \in \M_{n\, k} (\F).$
\end{lemma}

\begin{theorem}\label{thm:MainTheoremCullisNKOdd}Assume that $|\F| > k \ge 4$, $n \ge k + 2$ and $n + k$ is odd. Let $T\colon \M_{n\,k} (\F) \to \M_{n\,k} (\F)$ be a linear map. Then $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\,k} (\F)$ if and only if there exist $A \in \M_{n\, n}(\F)$ and $B \in \M_{k\, k}(\F)$ such that
\begin{equation}\label{thm:MainTheoremCullisNKOdd:eqq}
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\end{equation}
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and a linear map\linebreak $\phi\colon \M_{n\,k}(\F) \to W_{n\,k}$ such that
\begin{equation}\label{thm:MainTheoremCullisNKOdd:eq}
T(X) = AXB + \phi(X)
\end{equation}
for all $X \in \M_{n\,k} (\F).$

\begin{corollary}\label{cor:AdditionLinearMapNPLusKOdd}Suppose that $n + k$ is odd. Let
$\phi \colon \M_{n\,k}(\F) \to W_{n\,k}$
be a linear map. Denote by $S$ a linear map $\M_{n}(\F) \to \M_{n\,k}(\F)$ defined by
\begin{equation}\label{eq:AddingKernel}
S(X) = X + \phi (X)
\end{equation}
for all $X \in \M_{n\,k}(\F).$ Then
\[
\det_{n\, k} (S(X)) = \det_{n\,k}(X)
\]
for all $X \in \M_{n\,k} (\F).$
\end{corollary}

\begin{theorem}\label{thm:MainTheoremCullisNKOdd}Assume that $|\F| > k \ge 4$, $n \ge k + 2$ and $n + k$ is odd. Let $T\colon \M_{n\,k} (\F) \to \M_{n\,k} (\F)$ be a linear map. Then $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\,k} (\F)$ if and only if  there exist $A \in \M_{n\, n}(\F)$ and $B \in \M_{k\, k}(\F)$ such that
\begin{equation}\label{thm:MainTheoremCullisNKOdd:eqq}
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\end{equation}
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and a linear map\linebreak $\phi\colon \M_{n\,k}(\F) \to W_{n\,k}$ such that
\begin{equation}\label{thm:MainTheoremCullisNKOdd:eq}
T(X) = AXB + \phi(X)
\end{equation}
for all $X \in \M_{n\,k} (\F).$

Here $W_{n\,k} \subseteq \M_{n\,k}(\F)$ is the space of matrices, all rows of which are equal, which is defined in Definition~\ref{def:WNK} and  is a radical of $\det_{n\,k}$ by Lemma~\ref{lem:NKOddRadDetNK}.
\end{theorem}

The parity of $n + k$ is significant because if $n + k$ is odd, then there exist linear maps preserving the Cullis' determinant which do not have such form (Corollary~\ref{cor:AdditionLinearMapNPLusKOdd}).

However, in this paper we show that the case $(n,k)$ with $n \ge k$ and $n + k$ odd reduces to the case $(n-1,k)$ (Lemma~\ref{lem:FromNKOddToNKEvenIfKerW}\ref{lem:FromNKOddToNKEvenIfKerW:part6}). Using this reduction we prove the main theorem of this paper providing the description of linear maps preserving $\det_{n\,k}$ if $n + k$ is odd.

\begin{theorem}[Theorem~\ref{thm:MainTheoremCullisNKOdd}]Assume that $|{\mathbb F}| > k \ge 4$, $n \ge k + 2$ and $n + k$ is odd. Let $T\colon \M_{n\,k} ({\mathbb F}) \to \M_{n\,k} ({\mathbb F})$ be a linear map. Then $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\,k} ({\mathbb F})$ if and only if  there exist $A \in \M_{n\, n}({\mathbb F})$ and $B \in \M_{k\, k}({\mathbb F})$ such that
\[
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\]
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and a linear map\linebreak $\phi\colon \M_{n\,k}({\mathbb F}) \to W_{n\,k}$ such that
\[
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\]
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and
\begin{equation*}
T(X) = AXB + \phi(X)
\end{equation*}
for all $X \in \M_{n\,k} ({\mathbb F}).$ Here $W_{n\,k} \subseteq \M_{n\,k}({\mathbb F})$ denotes the space of matrices, all rows of which are equal, being a radical of $\det_{n\,k}$.
\end{theorem}

Our proof of the main theorem relies on the notion of the radical of a function which was introduced and studied by Waterhouse~\cite{Waterhouse1983} (Definition~\ref{def:radical} below). The reduction of the case $(n,k)$ with $k \ge 4, n \ge k + 2$ and $n + k$ odd to the case $(n-1,k)$ with $k \ge 4, n \ge k + 2$ and $(n-1) + k$ even is done by finding the radical of $\det_{n\,k}$ explicitly and using its properties.

Thus, the results of the current paper and its predecessor~\cite{Guterman2025} provide the solution to linear preserver problem for the Cullis' determinant if $k \ge 4, n \ge k + 2$ and the ground field is large enough. The cases $k = 1$ and $k = 2$ are considered in~\cite{Guterman2025} as well. The case $n = k + 1$ is considered in~\cite{Guterman2024}, where the first attempt to find the description of linear maps preserving the Cullis' determinant is made. The remaining case $k = 3$ need a special approach because the proof for the case $k \ge 4$ relies on the following lemma and will be discussed separately. We remark that in this paper we apply the results obtained in~\cite{Guterman2025} and we do not use any results from~\cite{Guterman2024}.

\begin{lemma}[{\cite[Lemma~5.13]{Guterman2025}}]\label{lem:CullisDeg1Rank1}Assume that $|{\mathbb F}| > k \ge 4$, $n \ge k + 2$ and $n + k$ is even. Let  $T\colon \M_{n\, k} ({\mathbb F}) \to \M_{n\, k} ({\mathbb F})$ be a linear map such that  $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\, k} ({\mathbb F}).$ Then $\rk (X) = 1$ implies $\rk (T(X)) = 1.$
\end{lemma}

The example below shows that this lemma does not hold for $k = 3$.

\begin{example}[{\cite[Example~5.11]{Guterman2025}}]\label{cex:K3NKEvenInt}Let $B = \begin{psmallmatrix}
                                 1 & 0 & 0 & 0 & -1 & 0 & \cdots & 0\\
                                 0 & 1 & 0 & -1 & 0 & 0 & \cdots & 0\\
                                 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 0
                                        \end{psmallmatrix}^t \in \M_{n\,3}({\mathbb F}).$ Then $\rk (B) = 2$ and $\deg_{\lambda} (\det_{n\,k}(A + \lambda B)) \le 1$ for all $A \in \M_{n\,3}({\mathbb F})$.
\end{example}

Nevertheless, if $k = 3$, then linear maps preserving the Cullis determinant admit the same description as in the case when $k \ge 4$ and will be studied separately.\bigskip

The paper is organized as follows: in Section~\ref{sec:prelim} we provide the basic facts regarding $\det_{n\,k}$; in Section~\ref{sec:kernels} we find the radical of $\det_{n\,k}$ if $n + k$ is odd; in Section~\ref{sec:maintheorem} we reduce the linear preserver problem for $\det_{n\,k}$ and $n + k$ is odd to the linear preserver problem for $\det_{(n-1)\,k}$ and prove the main characterization theorem for $n + k$ odd.

The introduction explains that “the parity of $n+k$ is significant” since when $n+k$ is odd “there exist linear maps preserving the Cullis' determinant which do not have such form.” It then outlines a reduction argument: “the case $(n,k)$ with $n\ge k$ and $n+k$ odd reduces to the case $(n-1,k)$” (via Lemma~\ref{lem:FromNKOddToNKEvenIfKerW}…), and “using this reduction we prove the main theorem … for $n+k$ … odd.”

It further specifies what drives this reduction: “Our proof of the main theorem relies on the notion of the radical of a function … (Definition~\ref{def:radical} below). The reduction … is done by finding the radical of $\det_{n\,k}$ explicitly and using its properties.” In particular, for $n+k$ odd one identifies the radical space $W_{n\,k}$ (“the space of matrices, all rows of which are equal, being a radical of $\det_{n\,k}$”), which leads to the possibility of an additive radical term $\phi(X)$ in the odd-parity characterization.

The introduction explains that “the parity of $n+k$ is significant” since when $n+k$ is odd “there exist linear maps preserving the Cullis' determinant which do not have such form.” It then outlines a reduction argument: “the case $(n,k)$ with $n\ge k$ and $n+k$ odd reduces to the case $(n-1,k)$” (via the following lemma).

\begin{lemma}\label{lem:FromNKOddToNKEvenIfKerW}Assume that  $|\F| > k$, $n > k$ and $n + k$ is odd. Let $T\colon \M_{n\, k} (\F) \to \M_{n\, k} (\F)$ be a linear map such that $\det_{n\, k} (T(X)) = \det_{n\,k}(X)$ and $\Img(T) = \M^0_{n\,k}(\F)$. Then the following statements hold.
\begin{enumerate}[label=(\alph*), ref=(\alph*)]
\item\label{lem:FromNKOddToNKEvenIfKerW:part4} $\Ker(T) = \rad(\det_{n\,k})$.
\item\label{lem:FromNKOddToNKEvenIfKerW:part5} $L^{+}\circ L^{-} \circ T = T$.
\item\label{lem:FromNKOddToNKEvenIfKerW:part6} $T \circ L^{+} \circ L^{-} = T$.
\item\label{lem:FromNKOddToNKEvenIfKerW:part2} If a linear map $S$ is defined by
\begin{equation}\label{lem:FromNKOddToNKEvenIfKerW:eq}
S = L^{-} \circ T \circ L^{+},
\end{equation}
then $T = L^{+} \circ S \circ L^{-}$.
\end{enumerate}
\end{lemma}

Using this reduction we prove the main theorem for $n+k$ odd.

It further specifies what drives this reduction: “Our proof of the main theorem relies on the notion of the radical of a function …” where the radical is defined as follows.

\begin{definition}[Cf.~{\cite[text at the beginning of Section~1]{Waterhouse1983}}]\label{def:radical}Let $\F$ be a field, $V$ be a finite-dimensional vector space over $\F$, and let $f$ be a function from $V$ to $\F$. The \emph{radical} of $f$, denoted by $\rad (f)$ is a subset of $V$ defined by
\[
\rad (f) = \{\mathbf w \mid f(\mathbf v + \lambda \mathbf w) = f(\mathbf v)\;\;\mbox{for all}\;\; \mathbf v \in V,\;\; \lambda \in \F\}.
\]
\end{definition}

The reduction is done by finding the radical of $\det_{n\,k}$ explicitly and using its properties. In particular, for $n+k$ odd one identifies the radical space $W_{n\,k}$ (“the space of matrices, all rows of which are equal, being a radical of $\det_{n\,k}$”), which leads to the possibility of an additive radical term $\phi(X)$ in the odd-parity characterization.

[{Guterman2025, Theorem~5.14} ]\label{MainTheoremEvenIntro}Assume that $k \ge 4, n \ge k + 2, n + k$ is even and $|{\mathbb F}| > k$. Let $T\colon \M_{n\,k} ({\mathbb F}) \to \M_{n\,k} ({\mathbb F})$ be a linear map. Then $\det_{n\,k} (T(X)) = \det_{n\,k}(X)$ for all $X \in \M_{n\,k} ({\mathbb F})$ if and only if there exist $A \in \M_{n\,n}({\mathbb F})$ and $B \in \M_{k\,k}({\mathbb F})$ such that
\[
\det_{n\,k} \Bigl(A(|i_1,\ldots, i_k]\Bigr) \det_k \Bigl(B\Bigr) = (-1)^{i_1 + \ldots + i_k - 1 - \ldots - k}
\]
for all increasing sequences $1 \le i_1 < \ldots < i_k \le n $ and
\[
T(X) = AXB
\]
for all $X \in \M_{n\,k} ({\mathbb F}).$

Biconditional or Equivalence
Existential–Universal

Question: Let \(\M_{n,k}(\mathbb F)\) be the space of \(n\times k\) matrices over a field \(\mathbb F\), and let \(\det_{n,k}\) denote the Cullis determinant on \(\M_{n,k}(\mathbb F)\), a determinant-like invariant for rectangular matrices that agrees with the usual determinant when \(n=k\). Assume \(k\ge 4\), \(n\ge k+2\), \(n+k\) is even, and \(|\mathbb F|>k\). Let \(T\colon \M_{n,k}(\mathbb F)\to \M_{n,k}(\mathbb F)\) be linear. Which statement characterizes exactly when \(T\) preserves the Cullis determinant, that is, when \(\det_{n,k}(T(X))=\det_{n,k}(X)\) for every \(X\in \M_{n,k}(\mathbb F)\)?

The equality \(\det_{n,k}(T(X))=\det_{n,k}(X)\) for all \(X\in \M_{n,k}(\mathbb F)\) holds if and only if there exist \(A\in \M_{n,n}(\mathbb F)\) and \(B\in \M_{k,k}(\mathbb F)\) such that, for every increasing sequence \(1\le i_1<\cdots<i_k\le n\),
\[
\det_{n,k}\bigl(A(|i_1,\ldots,i_k]\bigr)\,\det_k(B)=(-1)^{i_1+\cdots+i_k-(1+\cdots+k)},
\]
where \(A(|i_1,\ldots,i_k]\) is the \(n\times k\) submatrix of \(A\) formed by columns \(i_1,\ldots,i_k\), and
\[
T(X)=AXB
\]
for all \(X\in \M_{n,k}(\mathbb F)\).

Choices:
B.
The equality \(\det_{n,k}(T(X))=\det_{n,k}(X)\) for all \(X\in \M_{n,k}(\mathbb F)\) holds if and only if there exist \(A\in \M_{n,n}(\mathbb F)\), \(B\in \M_{k,k}(\mathbb F)\), and a linear map \(\phi\colon \M_{n,k}(\mathbb F)\to W_{n,k}\), where \(W_{n,k}\subseteq \M_{n,k}(\mathbb F)\) is the subspace of matrices whose rows are all equal, such that for every increasing sequence \(1\le i_1<\cdots<i_k\le n\),
\[
\det_{n,k}\bigl(A(|i_1,\ldots,i_k]\bigr)\,\det_k(B)=(-1)^{i_1+\cdots+i_k-(1+\cdots+k)},
\]
and
\[
T(X)=AXB+\phi(X)
\]
for all \(X\in \M_{n,k}(\mathbb F)\).
C.
If there exist \(A\in \M_{n,n}(\mathbb F)\) and \(B\in \M_{k,k}(\mathbb F)\) such that, for every increasing sequence \(1\le i_1<\cdots<i_k\le n\),
\[
\det_{n,k}\bigl(A(|i_1,\ldots,i_k]\bigr)\,\det_k(B)=(-1)^{i_1+\cdots+i_k-(1+\cdots+k)},
\]
and
\[
T(X)=AXB
\]
for all \(X\in \M_{n,k}(\mathbb F)\), then \(\det_{n,k}(T(X))=\det_{n,k}(X)\) for all \(X\in \M_{n,k}(\mathbb F)\).
D.
The equality \(\det_{n,k}(T(X))=\det_{n,k}(X)\) for all \(X\in \M_{n,k}(\mathbb F)\) holds if and only if there exist \(A\in \M_{n,n}(\mathbb F)\) and \(B\in \M_{k,k}(\mathbb F)\) such that, for some increasing sequence \(1\le i_1<\cdots<i_k\le n\),
\[
\det_{n,k}\bigl(A(|i_1,\ldots,i_k]\bigr)\,\det_k(B)=(-1)^{i_1+\cdots+i_k-(1+\cdots+k)},
\]
and
\[
T(X)=AXB
\]
for all \(X\in \M_{n,k}(\mathbb F)\).
E.
The equality \(\det_{n,k}(T(X))=\det_{n,k}(X)\) for all \(X\in \M_{n,k}(\mathbb F)\) holds if and only if there exist invertible matrices \(A\in \M_{n,n}(\mathbb F)\) and \(B\in \M_{k,k}(\mathbb F)\) such that
\[
\det(A)\det(B)=1
\]
and
\[
T(X)=AXB
\]
for all \(X\in \M_{n,k}(\mathbb F)\).

dropped the converse direction of the characterization

universality over all increasing index sequences

specific minor-sign constraint replaced by ordinary determinant condition

The stem does not explicitly reveal the correct characterization; it only asks for the exact preservation criterion. Although the wording signals that an 'if and only if' theorem is sought, it does not itself disclose the content of the answer.

This is essentially a direct theorem-recall item: the correct option is the full preservation characterization itself. The task mainly asks the student to recognize the exact statement of the theorem rather than derive or apply it.

Some reasoning is needed to distinguish the exact characterization from a weaker one-way implication, a quantifier-weakened variant, and a superficially similar determinant condition. However, the item still primarily tests recognition/recall of the theorem rather than genuine generative mathematical reasoning.

The distractors are mathematically meaningful and target plausible failure modes: adding an extra radical-like term, dropping the converse direction, weakening a universal quantifier to an existential one, and confusing the rectangular invariant with the ordinary determinant-preserver condition.

A technically solid but theorem-recall-heavy MCQ. It avoids direct answer leakage and has strong distractors, but it is largely tautological and only mildly pressures genuine reasoning.