\begin{thm}\label{thm: explicit ribbon expansion in full generality}
  Let $R^{(1)}, \ldots, R^{(k)}$ be ribbons, where each
  \( R^{(i)} = \lambda^{(i)}/\mu^{(i)} \) satisfies
  \( \ell(\lambda^{(i)})\le n \). Then for any $I \subseteq [n-1]$, we
  have
\begin{align*}
    &\imm_{\T{I}} \left( \tJT_{R^{(1)}}( \vec{x}^{(1)})
    *\cdots *
    \tJT_{R^{(k)}}(\vec{x}^{(k)}) \right) \\
    &= \sum_{\nu_1 \vdash m_1, \, \ldots, \, \nu_k \vdash m_k}
    \left(
    \sum_{
    \substack{
    I_i \subseteq [a_i,b_i-1] \\
    I_1 \cap \cdots \cap I_k = I
    }} \,
    \prod_{i=1}^k
    f^{\nu_i} \left( \NDes(R^{(i)}) \cup d(I_i, R^{(i)}) \right)
    \right)
    s_{\nu_1}(\vec{x}^{(1)})
    \cdots
    s_{\nu_k}(\vec{x}^{(k)}),
\end{align*}
where $m_i$ is the size of $R^{(i)}$ and $[a_i, b_i-1] = \{j:\lambda^{(i)}_j>\mu^{(i)}_j\}$.
\end{thm}

\begin{thm}\label{thm: representation theoretic model for Phi(s_R)}
  Let \( R \) be a ribbon of size \(n\), and consider the
  \(\mathfrak{S}_n\times \mathfrak{S}_n\)-module
\[
    \tilde{H}_{|\Des(R')|-1}((\tilde{B}_n)_{\Des(R')})
\]
as described above. Then we have
\[
\Phi_{\vec x, \vec y} (s_R)
=\omega\left(\Frob_{\vec x, \vec y}
\left(\tilde{H}_{|\Des(R')|-1}((\tilde{B}_n)_{\Des(R')})\right)\right).
\]
\end{thm}

\begin{prop}\label{prop: counterexample}
  Let $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$
  be a collection of connected skew shapes
  that is not essentially $(3 \times 2)$-avoiding.
  Then there exists a multinetwork $\vec{H}$ for this collection satisfying $ f_1(\beta(\vec{H})) = \sgn(\beta(\vec{H})) < 0$.
\end{prop}

A matrix is said to be \emph{totally positive} if all of its minors are nonnegative. Total positivity arises in various areas of mathematics and has wide-ranging applications. Originally studied by Gantmacher and Krein \cite{Gantmacher1935,Gantmakher1937} in the context of classical analysis and numerical interpolation, total positivity has since become a central topic in representation theory, geometry, mathematical physics, and algebraic combinatorics (see the survey by Fomin \cite{Fomin2010} and references therein).

By the Cauchy--Binet theorem, it is immediate that the product
\( MN \) of two totally positive matrices \( M \) and \( N \) is also
totally positive. However, their Hadamard (entrywise) product
\( M * N \) need not be totally positive in general.
Nevertheless, there exists a class of matrices whose total positivity is preserved under the Hadamard product.
A \textit{Toeplitz matrix} with \textit{Toeplitz sequence $(a_0,a_1,\ldots)$} is an infinite, upper-triangular matrix
of the form $(a_{j-i})_{i,j=0}^\infty$.
The \emph{Laguerre--Pólya class} is the class of entire functions that arise as uniform limits of
univariate polynomials with nonnegative coefficients and real roots.
Mal\'{o} \cite{Malo1895} proved that the Hadamard (entrywise) product of two totally positive Toeplitz matrices remains totally positive, provided that their Toeplitz sequences are the coefficient sequences of series in the Laguerre--Pólya class.

For an entire function \( p(t) = \prod_{i\ge0}(1 + \alpha_it) \) in the Laguerre--Pólya class, if we replace the nonnegative real numbers \( \alpha_i \ge 0 \) by variables \( x_i \), then the coefficient of $t^k$ in \( p(t) \) becomes the \emph{elementary symmetric function} \( e_k(\vec{x}) \) in the variables \( \vec x = (x_1, x_2 ,\dots) \). In this way, the Toeplitz matrix whose entries are given by the coefficients of \( p(t) \) can be regarded as a special case of the following matrix:
\[
   M(\vec{x}) = \bigl(e_{j-i}(\vec{x})\bigr)_{i,j \ge 0}.
\]
The minors of this matrix are determinants of the \emph{(dual) Jacobi--Trudi matrices}\footnote{Our convention differs from the standard one by a conjugation of the skew shape \( \lambda/\mu \); see \Cref{rmk: nonstandard convention}.}
\[
   \tJT_{\lambda/\mu}(\vec{x}) := \bigl(e_{\lambda_i - \mu_j - i + j}(\vec{x})\bigr)_{i,j=1}^{\ell(\lambda)}.
\]
Here \(\lambda/\mu\) is a skew shape, and \( \ell(\lambda) \) denotes the number of parts of the partition \( \lambda \). By the Jacobi--Trudi identity and the Littlewood--Richardson rule, every minor of \( M(\vec{x}) \) is Schur-positive, and hence monomial-positive. Schur positivity and monomial positivity are stronger notions of positivity than positivity of real numbers.
In connection with Mal\'{o}’s theorem, one may then ask whether the total positivity of the Hadamard product of two Toeplitz matrices \( M(\vec{x}) \) and \( M(\vec{y}) \), when \( x_i, y_j \ge 0 \), can be strengthened to the monomial positivity of each minor. Very recently, Sokal~\cite{Sokal2024} formulated the following conjecture on the monomial positivity of the Hadamard product of \( M(\vec{x}) \) and \( M(\vec{y}) \) for distinct sequences of variables \( \vec{x} = (x_1, x_2, \dots) \) and \( \vec{y} = (y_1, y_2, \dots) \).

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
  The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
  monomial positive. In other words, every minor
\[
    \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

In the present work, we study a strengthening of Sokal's conjecture along several axes. First, we allow products of Jacobi--Trudi matrices indexed by different skew shapes. In other words, we consider the Hadamard product of an arbitrary minor of $M(\vec{x})$ with an arbitrary minor of $M(\vec{y})$. Second, we consider Hadamard products of $k$ Jacobi--Trudi matrices, rather than just two. Lastly, we generalize the determinant to an arbitrary \textit{Temperley--Lieb immanant}, which is a certain generalization of the determinant introduced by Rhoades and Skandera \cite{Rhoades2005}.

\begin{conj}\label{conj: stronger sokal's conjecture}
  Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
  skew shapes
  $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
  multi-symmetric function
    \[\imm_\tau \left(
    \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
    * \cdots *
    \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
    \right)\]
    is monomial positive.
\end{conj}

While \Cref{conj: Sokal's conjecture} and our stronger \Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape not containing a $3 \times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \Cref{conj: stronger sokal's conjecture} is Schur positive.

By the Cauchy--Binet theorem, it is immediate that the product
\( MN \) of two totally positive matrices \( M \) and \( N \) is also
totally positive. However, their Hadamard (entrywise) product
\( M * N \) need not be totally positive in general.
Nevertheless, there exists a class of matrices whose total positivity is preserved under the Hadamard product.
A \textit{Toeplitz matrix} with \textit{Toeplitz sequence $(a_0,a_1,\ldots)$} is an infinite, upper-triangular matrix
of the form $(a_{j-i})_{i,j=0}^\infty$.
The \emph{Laguerre--Pólya class} is the class of entire functions that arise as uniform limits of
univariate polynomials with nonnegative coefficients and real roots.
Mal\'{o} \cite{Malo1895} proved that the Hadamard (entrywise) product of two totally positive Toeplitz matrices remains totally positive, provided that their Toeplitz sequences are the coefficient sequences of series in the Laguerre--Pólya class.

For an entire function \( p(t) = \prod_{i\ge0}(1 + \alpha_it) \) in the Laguerre--Pólya class, if we replace the nonnegative real numbers \( \alpha_i \ge 0 \) by variables \( x_i \), then the coefficient of $t^k$ in \( p(t) \) becomes the \emph{elementary symmetric function} \( e_k(\vec{x}) \) in the variables \( \vec x = (x_1, x_2 ,\dots) \). In this way, the Toeplitz matrix whose entries are given by the coefficients of \( p(t) \) can be regarded as a special case of the following matrix:
\[
 M(\vec{x}) = \bigl(e_{j-i}(\vec{x})\bigr)_{i,j \ge 0}.
\]
The minors of this matrix are determinants of the \emph{(dual) Jacobi--Trudi matrices}\footnote{Our convention differs from the standard one by a conjugation of the skew shape \( \lambda/\mu \); see \Cref{rmk: nonstandard convention}.}
\[
 \tJT_{\lambda/\mu}(\vec{x}) := \bigl(e_{\lambda_i - \mu_j - i + j}(\vec{x})\bigr)_{i,j=1}^{\ell(\lambda)}.
\]
Here \(\lambda/\mu\) is a skew shape, and \( \ell(\lambda) \) denotes the number of parts of the partition \( \lambda \). By the Jacobi--Trudi identity and the Littlewood--Richardson rule, every minor of \( M(\vec{x}) \) is Schur-positive, and hence monomial-positive. Schur positivity and monomial positivity are stronger notions of positivity than positivity of real numbers.
In connection with Mal\'{o}’s theorem, one may then ask whether the total positivity of the Hadamard product of two Toeplitz matrices \( M(\vec{x}) \) and \( M(\vec{y}) \), when \( x_i, y_j \ge 0 \), can be strengthened to the monomial positivity of each minor. Very recently, Sokal~\cite{Sokal2024} formulated the following conjecture on the monomial positivity of the Hadamard product of \( M(\vec{x}) \) and \( M(\vec{y}) \) for distinct sequences of variables \( \vec{x} = (x_1, x_2, \dots) \) and \( \vec{y} = (y_1, y_2, \dots) \).

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
 The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
 monomial positive. In other words, every minor
\[
 \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

In the present work, we study a strengthening of Sokal's conjecture along several axes. First, we allow products of Jacobi--Trudi matrices indexed by different skew shapes. In other words, we consider the Hadamard product of an arbitrary minor of $M(\vec{x})$ with an arbitrary minor of $M(\vec{y})$. Second, we consider Hadamard products of $k$ Jacobi--Trudi matrices, rather than just two. Lastly, we generalize the determinant to an arbitrary \textit{Temperley--Lieb immanant}, which is a certain generalization of the determinant introduced by Rhoades and Skandera \cite{Rhoades2005}.

\begin{conj}\label{conj: stronger sokal's conjecture}
 Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
 skew shapes
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
 multi-symmetric function
 \[\imm_\tau \left(
 \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
 * \cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
 \right)\]
 is monomial positive.
\end{conj}

While \Cref{conj: Sokal's conjecture} and our stronger \Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape not containing a $3 \times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \Cref{conj: stronger sokal's conjecture} is Schur positive.

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
  The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
  monomial positive. In other words, every minor
\[
    \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

\begin{conj}\label{conj: stronger sokal's conjecture}
  Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
  skew shapes
  $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
  multi-symmetric function
    \[\imm_\tau \left(
    \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
    * \cdots *
    \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
    \right)\]
    is monomial positive.
\end{conj}

\begin{remark}\label{rmk: nonstandard convention}
Our definition of dual Jacobi--Trudi matrices
differs from the standard convention by a conjugation of $\lambda / \mu$.
When using the standard convention, the reader should take this change into account.
For example, results stated in terms of $(3 \times 2)$-avoiding skew shapes
would instead concern $(2 \times 3)$-avoiding skew shapes.
\end{remark}

\begin{conj}\label{conj: stronger sokal's conjecture}
 Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
 skew shapes
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
 multi-symmetric function
 \[\imm_\tau \left(
 \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
 * \cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
 \right)\]
 is monomial positive.
\end{conj}

While \Cref{conj: Sokal's conjecture} and our stronger \Cref{conj: stronger sokal's conjecture} remain open, we prove that the conjectures hold in the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape not containing a $3 \times 2$ block of cells. In fact, in this scenario, we demonstrate that the expression in \Cref{conj: stronger sokal's conjecture} is Schur positive.

We focus on this restricted class of skew shapes for three key reasons. First, \Cref{conj: Sokal's conjecture} and \Cref{conj: stronger sokal's conjecture} will not always extend to Schur positivity if the skew shapes do not all avoid a $3 \times 2$ block. For example, the expression $\det(\tJT_{(2,2,2)}(\vec{x}) * \tJT_{(2,2,2)}(\vec{y}))$ is not Schur positive.

In \Cref{sec:step1 m expansion}, we prove \Cref{thm: TL pos for 3x2 avoiding}, i.e., that the Temperley--Lieb immanant
 \[\imm_\tau \left(
 \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
 * \cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
 \right)\]
is $s$-positive for any $\tau \in \K_n$, provided that each $\lambda^{(i)} / \mu^{(i)}$ is a skew shape that does not contain a $3 \times 2$ block of cells. We say that such shapes are $(3 \times 2)$-\emph{avoiding}.

\begin{thm}\label{thm: TL multinetwork expansion}
 Let \( \tau\in \K_n \) and suppose
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ are
 $(3 \times 2)$-avoiding skew shapes, with
 \( \ell(\lambda^{(i)})\le n \) for all \( i \). If $\tau=\T{I}$ for
 some $I \subseteq [n-1]$, we have
 \[
 \imm_{\tau} \left( \tJT_{\lambda^{(1)} /
 \mu^{(1)}}(\vec{x}^{(1)}) * \cdots * \tJT_{\lambda^{(k)} /
 \mu^{(k)}}(\vec{x}^{(k)}) \right) = \sum_{\vec{H}}
 2^{\epsilon(\vec{H})} \wt_{\underline{\vec x}}(\vec{H}),
 \]
 where the sum is over all $(\underline{\lambda}, \underline{\mu})$-multinetworks
 $\vec{H}$ with $\beta(\vec{H}) = 2^{\epsilon(\vec{H})} \B{I}$.
 If $\tau$ is not of this form, the immanant is $0$.

\begin{cor}\label{cor:all TL imms s positive}
 For any $\tau \in \K_n$ and any $(3 \times 2)$-avoiding skew shapes
 $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ with
 \( \ell(\lambda^{(i)})\le n \) for all \( i \), the immanant
 \[ \imm_\tau \left( \tJT_{\lambda^{(1)} /
 \mu^{(1)}}(\vec{x}^{(1)}) *\cdots *
 \tJT_{\lambda^{(k)} / \mu^{(k)}}(\vec{x}^{(k)})
 \right)
\] is Schur positive.
\end{cor}

\begin{proof}
 By \Cref{thm: TL multinetwork expansion}, we may assume that \( \tau=\tau(I) \)
 for some \( I\subseteq [n-1] \).
 Recall that a
 \((\underline{\lambda}, \underline{\mu})\)-multinetwork \(\vec{H}\)
 is a tuple \((H^{(1)}, \ldots, H^{(k)})\) in which each \(H^{(i)}\)
 is a \((\lambda^{(i)} ,\mu^{(i)})\)-subnetwork. By \Cref{lem: beta
 for multinetworks}, the condition
 \(\beta(\vec{H}) = 2^{\epsilon(\vec{H})}\B{I}\) is equivalent to
 \(I_1 \cap \cdots \cap I_k = I\), where
 \(\beta(H^{(i)})=2^{\epsilon(H^{(i)})}\B{I_i}\) for each \( i \).
 Therefore, the equation in \Cref{thm: TL multinetwork expansion} can be
 rewritten as
 \begin{multline}
 \imm_{\tau} \left( \tJT_{\lambda^{(1)} /
 \mu^{(1)}}(\vec{x}^{(1)}) * \cdots * \tJT_{\lambda^{(k)} /
 \mu^{(k)}}(\vec{x}^{(k)}) \right) \\
 = \sum_{\substack{I_1,\dots,I_k \subseteq [n-1] \\ I_1 \cap \cdots \cap I_k = I}} \,
 \prod_{i=1}^k
 \left( \sum_{\substack{H^{(i)} \\ \beta(H^{(i)}) = 2^{\epsilon(H^{(i)})}\B{I_i}}}
 2^{\epsilon(H^{(i)})} \wt_{\vec{x}^{(i)}}(H^{(i)}) \right) .
 \label{eq:network expansion verbose}
 \end{multline}
 By \Cref{thm: TL multinetwork expansion}, for each $i \in [n]$, we
 have
 \[
 \sum_{\substack{H^{(i)} \\ \beta(H^{(i)}) = 2^{\epsilon(H^{(i)})}\B{I_i}}}
 2^{\epsilon(H^{(i)})} \wt_{\vec{x}^{(i)}}(H^{(i)})
 = \imm_{\T{I_i}} \left( \tJT_{\lambda^{(i)} / \mu^{(i)}}(\vec{x}^{(i)}) \right).\]
By \cite[Proposition 3]{RHOADES2006793} and \cite[Proposition 5 and the paragraph above it]{RHOADES2006793}, each Temperley--Lieb immanant of a single Jacobi--Trudi matrix is Schur positive. Thus the expression in \eqref{eq:network expansion verbose} is Schur positive.
\end{proof}

The proof of \Cref{thm: TL multinetwork expansion} had two primary steps.
First, we expanded each Temperley--Lieb immanant as a sum over subnetworks, i.e., wrote
\[\imm_{\tau} \left( \tJT_{\lambda^{(1)} / \mu^{(1)}}(\vec{x}^{(1)}) *\cdots * \tJT_{\lambda^{(k)} / \mu^{(k)}}(\vec{x}^{(k)}) \right) = \sum_{\vec{H}} f_\tau(\beta(\vec{H})) \wt_{\underline{\vec x}}(\vec{H}).\]
Then, we argued that $f_\tau(\beta(\vec{H})) \geq 0$ whenever $\vec{H}$ is a multinetwork for a collection of $(3 \times 2)$-avoiding skew shapes.
The result below demonstrates that the proof method for \Cref{thm: TL multinetwork expansion}
will not work for collections of skew shapes that are not essentially $3 \times 2$ avoiding.
Note that if \( \tau=1 \), then \( f_\tau=f_1 \) is the sign character \( \sgn \).

\begin{conj}[Sokal's conjecture]\label{conj: Sokal's conjecture}
  The Hadamard product \( M(\vec x) * M(\vec y) \) is totally
  monomial positive. In other words, every minor
\[
    \det(\tJT_{\lambda/\mu}(\vec x) * \tJT_{\lambda/\mu}(\vec y))
\]
is a multi-symmetric function in the variables \( \vec x \) and \( \vec y \) with
nonnegative integer coefficients.
\end{conj}

\begin{conj}\label{conj: stronger sokal's conjecture}
  Given any Temperley--Lieb immanant $\imm_{\tau}$ and any family of
  skew shapes
  $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$, the
  multi-symmetric function
    \[\imm_\tau \left(
    \tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
    * \cdots *
    \tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)})
    \right)\]
    is monomial positive.
\end{conj}

\begin{thm}\label{thm: TL pos for 3x2 avoiding}
Suppose $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ is a collection of skew shapes each not containing a $3 \times 2$ block of cells. Then for any Temperley--Lieb immanant $\imm_{\tau}$, the multi-symmetric function
\[\imm_{\tau} \left(\tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
* \cdots *
\tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)}) \right)\]
is Schur positive.
\end{thm}

We focus on this restricted class of skew shapes for three key reasons. First, \Cref{conj: Sokal's conjecture} and \Cref{conj: stronger sokal's conjecture} will not always extend to Schur positivity if the skew shapes do not all avoid a $3 \times 2$ block. For example, the expression $\det(\tJT_{(2,2,2)}(\vec{x}) * \tJT_{(2,2,2)}(\vec{y}))$ is not Schur positive.

Second, the proof of \Cref{thm: TL pos for 3x2 avoiding} involves techniques introduced by Rhoades and Skandera in \cite{Rhoades2005}. Said techniques can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\"{o}m--Gessel--Viennot lemma.
However, we demonstrate that a similar sign-reversing involution argument cannot work to prove \Cref{conj: Sokal's conjecture} or \Cref{conj: stronger sokal's conjecture} when the collection of skew shapes is not \textit{essentially $(3 \times 2)$-avoiding}, a mild generalization of the condition in \Cref{thm: TL pos for 3x2 avoiding}. See \Cref{prop: counterexample} and the ensuing discussion.

Third, skew shapes avoiding a $3 \times 2$ block generalize the well-known \emph{ribbons}, which are skew shapes avoiding a $2 \times 2$ block. In the special case when each $\lambda^{(i)} / \mu^{(i)}$ is a ribbon, we can write the Schur expansion from \Cref{thm: TL pos for 3x2 avoiding} explicitly in terms of standard Young tableaux with restricted descent sets.
See \Cref{thm: explicit ribbon expansion in full generality}.

In particular, when taking the Hadamard product of a ribbon Jacobi--Trudi matrix with
itself, we obtain the following the following explicit formula.

\begin{thm}\label{thm: Schur expansion of JT_R*JT_R}
Let \( R \) be a ribbon of size \( m \). Then we have a manifestly positive Schur expansion
\[
    \det\left( \tJT_R(\vec x) * \tJT_R(\vec y) \right) = \sum_{\lambda, \mu \vdash m} \sum_{\substack{I, J \subseteq [m-1] \\ I \cap J = \Des(R')}} f^{\lambda}(I) f^{\mu}(J) \, s_\lambda(\vec x) s_\mu(\vec y),
  \]
  where \( s_\lambda \) is the Schur function, \( f^{\lambda}(I) \)
  is the number of standard Young tableaux of shape \( \lambda \)
  with descent set \( I \), and \( \Des(R) \) is the descent set of
  \( R \).
\end{thm}
We explore the latter result further by constructing an
\( \mathfrak{S}_n \times \mathfrak{S}_n \)-module whose Frobenius
image equals the determinant in \Cref{thm: Schur expansion of
  JT_R*JT_R}, thereby establishing Schur positivity from the
representation-theoretic perspective. See \Cref{thm: representation theoretic model for Phi(s_R)}.

The rest of this paper is organized as follows. In Section~\ref{Sec:
  Preliminaries}, we review background material on multi-symmetric
functions, lattice paths, immanants, and the representation theory of products of symmetric groups.
Section~\ref{Sec: main proofs for 3x2 avoiders} is devoted to the proofs of Theorems~\ref{thm: TL pos for 3x2 avoiding}~and~\ref{thm: Schur expansion of JT_R*JT_R}.
Section \ref{Sec: rep theory construction} concerns the construction of the \( \mathfrak{S}_n \times \mathfrak{S}_n \)-module that realizes \Cref{thm: Schur expansion of JT_R*JT_R}.
Finally, Section~\ref{Sec: Concluding remarks} discusses some open problems and
directions for future research.

The post-theorem introduction indicates that the proof of \Cref{thm: TL pos for 3x2 avoiding} "involves techniques introduced by Rhoades and Skandera in \cite{Rhoades2005}," which "can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\"{o}m--Gessel--Viennot lemma." No further step-by-step outline for proving \Cref{thm: TL pos for 3x2 avoiding} is given beyond this description (and the note that Section~\ref{Sec: main proofs for 3x2 avoiders} contains the proofs).

The post-theorem introduction indicates that the proof of the main theorem “involves techniques introduced by Rhoades and Skandera in \cite{Rhoades2005},” which “can be viewed as a generalization of the sign-reversing involution normally used to prove the Lindstr\"{o}m--Gessel--Viennot lemma.” No further step-by-step outline for proving the main theorem is given beyond this description (and the note that Section~\ref{Sec: main proofs for 3x2 avoiders} contains the proofs).

\label{thm: TL pos for 3x2 avoiding}
Suppose $\lambda^{(1)} / \mu^{(1)}, \ldots, \lambda^{(k)} / \mu^{(k)}$ is a collection of skew shapes each not containing a $3 \times 2$ block of cells. Then for any Temperley--Lieb immanant $\imm_{\tau}$, the multi-symmetric function
\[\imm_{\tau} \left(\tJT_{\lambda^{(1)} / \mu^{(1)}} (\vec{x}^{(1)})
* \cdots *
\tJT_{\lambda^{(k)} / \mu^{(k)}} (\vec{x}^{(k)}) \right)\]
is Schur positive.,

Question:
Let \(\lambda^{(1)}/\mu^{(1)},\ldots,\lambda^{(k)}/\mu^{(k)}\) be skew shapes, each of which does not contain a \(3\times 2\) block of cells. For each \(i\), let \(\vec{x}^{(i)}\) be a set of variables, let \(e_r(\vec{x}^{(i)})\) denote the elementary symmetric function of degree \(r\), and define the (dual) Jacobi--Trudi matrix by
\[
\operatorname{tJT}_{\lambda^{(i)}/\mu^{(i)}}(\vec{x}^{(i)})
:= \bigl(e_{\lambda^{(i)}_a-\mu^{(i)}_b-a+b}(\vec{x}^{(i)})\bigr)_{a,b=1}^{\ell(\lambda^{(i)})}.
\]
If \(*\) denotes the Hadamard (entrywise) product of matrices, and \(\operatorname{imm}_{\tau}\) is any Temperley--Lieb immanant (an immanant-type matrix functional generalizing the determinant), which of the following conclusions about
\[
\operatorname{imm}_{\tau}\!
\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is valid? Here, Schur positive means that when the resulting multi-symmetric function is expanded in the product Schur basis in the variable sets \(\vec{x}^{(1)},\ldots,\vec{x}^{(k)}\), all coefficients are nonnegative.

For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!
\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive.

Choices:
B.
For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive, provided that the collection of skew shapes does not contain a \(3\times 2\) block of cells in the union of all shapes.
C.
For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is monomial positive.
D.
There exists a Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), depending only on \(k\), such that for every collection of skew shapes \(\lambda^{(1)}/\mu^{(1)},\ldots,\lambda^{(k)}/\mu^{(k)}\) each not containing a \(3\times 2\) block of cells, the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive.
E.
For every Temperley--Lieb immanant \(\operatorname{imm}_{\tau}\), the multi-symmetric function
\[
\operatorname{imm}_{\tau}\!\left(
\operatorname{tJT}_{\lambda^{(1)}/\mu^{(1)}}(\vec{x}^{(1)})
* \cdots *
\operatorname{tJT}_{\lambda^{(k)}/\mu^{(k)}}(\vec{x}^{(k)})
\right)
\]
is Schur positive for arbitrary skew shapes \(\lambda^{(1)}/\mu^{(1)},\ldots,\lambda^{(k)}/\mu^{(k)}\), without any restriction excluding a \(3\times 2\) block of cells.

The stem does not explicitly reveal the correct option. It sets up the objects and asks for the valid conclusion; although the definition of Schur positivity makes that property salient, it does not directly identify choice A.

Choice A is essentially the theorem statement itself under the same hypotheses, so the item functions largely as theorem recall rather than requiring selection among genuinely competing derived conclusions.

There is some reasoning involved in distinguishing the exact theorem from stronger, weaker, or quantifier-altered variants, but the presence of an option that directly matches the theorem keeps generative pressure modest.

The distractors are mathematically plausible and target common failure modes: weakening the conclusion (monomial vs. Schur positivity), altering the hypothesis, changing quantifiers, and overgeneralizing by removing assumptions.

A mathematically well-constructed but theorem-recall-heavy MCQ. Distractors are strong, but the correct answer is too close to a direct restatement, so the item has limited generative reasoning value.


\label{main result}
Let $k$ be a positive integer such that $k\in \{7, 8, 10,$ $11,12,13\}.$ If $\{a, b, c, d\}$ is a $D(4)$-quadruple with $b=ka$, then it is regular. In other words, we have $d=d_{\pm}.$

\label{intr}

\begin{definition}Let $n\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.
\end{definition}
We research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \cite{blizfil} the nonexistence of $D(4)$-quintuples.

For a $D(4)$-triple $\{a,b,c\}, a<b<c$, we define
 \[ d_{\pm}(a,b,c)=a+b+c+\frac{1}{2}(abc \pm \sqrt{(ab+4)(ac+4)(bc+4)}).\]

$D(4)$-quadruple  $\{a,b,c,d_{+}\}$ is called a \textbf{regular quadruple}
 if $d_{-} \neq 0$, then $\{a,b,c,d_{-}\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.

In both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

In this paper, we study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$ are defined by following the methods in \cite{glavni}. So, the main theorem we prove is:

\label{intr}

\begin{definition}Let $n\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.
\end{definition}
We research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \cite{blizfil} the nonexistence of $D(4)$-quintuples.

$D(4)$-quadruple $\{a,b,c,d_{+}\}$ is called a \textbf{regular quadruple}
 if $d_{-} \neq 0$, then $\{a,b,c,d_{-}\}$ is a regular $D(4)$-quadruple with $d_{-}<c$. It is easy to verify that $c=d_+(a,b,d_-)$.

In both the classical case, $n=1$, and $n=4$, conjectures about the uniqueness of an extension of a triple to a quadruple with a larger element are still open.

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

In this paper, we study the extensibility of $D(4)$-pairs $\{a,b\}$, where $b = ka$ and $k \in \{7,8,10,11,12,13\}$ are defined by following the methods in \cite{glavni}. So, the main theorem we prove is:

\begin{conjecture}\label{conj}
Any $D(4)$-quadruple is regular.
\end{conjecture}

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10,$ $11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$.\ These examples support the Conjecture \ref{conj}.
\section{Extensions of pairs to triples}\label{section_2}

\begin{cor}\label{nova lema}
Let $\{a,ka,c\}$ be a $D(4)$-triple, $k \in \{10,11,12,13\}$. Then $c=c_{\nu}^{\pm}$.
\end{cor}
\begin{proof}
Let's show that $\{1,5,a,ka \}$ is not a $D(4)$-quadruple for $k \in \{10,11,12,13\}$. If $\{1,5,a,ka \}$ is a $D(4)$-quadruple, by using \cite[Lemma 6.1]{mbt}, Lemma \ref{lower_on_b}, and Lemma \ref{pomocna lema za lemu ispod}, we obtain $a=a_{\nu}^{\pm}$, $ka=d_+(1,5,a)$, and $d_+(1,5,a)=a_{\nu+1}^{\pm}$ for the same choice of $\pm$. We divide both sides of \begin{equation}\label{pomocna}
 ka = d_+(1,5,a)=1+5+a+\frac{1}{2}(5a+3\sqrt{5a+4}\sqrt{a+4})
\end{equation} by $a$ and, using the fact that $a \geq 12$, we obtain $k \leq 8$.
\end{proof}

Secondly, we observe the solutions of the system of equations (\ref{eqn:sol_pell_bc}) and (\ref{eqn:sol_pell_ab}) and determine the intersections $y=A_n=B_l$ of sequences $(A_n)_n$ and $(B_l)_l$ defined by
\begin{align}
 &A_0=y_1,\ A_1=\frac{1}{2}(ty_1+bz_1),\ A_{n+2}=tA_{n+1}-A_n,\label{seq:An}\\
 &B_0=y_2,\ B_1=\frac{1}{2}(ry_2+bx_2),\ B_{l+2}=rB_{l+1}-B_l, \ n,l\geq 0.\label{seq:Bl}
\end{align}
The initial terms of these sequences are described in the next lemma, whose proof follows \cite[Lemma 2.5]{glavni}.
\begin{lemma}\label{lem:lem_4}
Assume that $\{a, b, c', c\}$ is not a $D(4)$-quadruple for any $c'$ with $0<c'<c^{\pm}_{\nu-1}$ and $b\geq 556881$. Then, $A_{2n}=B_{2l+1}$ has no solution. Moreover, if $A_{2n} = B_{2l}$, then $y_2 = 2$. In other cases, we have $y_2 =\pm 2.$
\end{lemma}

For any nonzero algebraic number $\alpha$ of degree $d$ over $\QQ$ whose minimal polynomial over $\ZZ$ is $a_0 \prod_{j=1}^{d}(X-\alpha^{(j)}),$ we denote by
$$
 h(\alpha)= \frac{1}{d}\left(\log |a_0|+\sum_{j=1}^{d}\log\max\left(1,\left|\alpha^{(j)} \right| \right) \right)
$$
its absolute logarithmic height. We recall the following result due to Matveev \cite{Matveev:2000}.
\begin{lemma}\label{Matveev}
Denote by $\alpha_1,\ldots,\alpha_j$ algebraic numbers, not $0$ or $1,$ by $\log\alpha_1,\ldots,$ $\log\alpha_j$ determinations of their logarithms, by $D$ the degree over $\QQ$ of the number field $\KK=\QQ(\alpha_1,\ldots,\alpha_j),$ and by $b_1,\ldots,b_j$ integers. Define $B=\max\{|b_1|,\ldots,|b_j|\}$ and
$$A_i=\max\{Dh(\alpha_i),|\log\alpha_i|,0.16\}\;\; (1\leq i\leq j),$$
where $h(\alpha)$ denotes the absolute logarithmic Weil height of $\alpha.$
Assume that the number
$$
\Lambda= b_1\log\alpha_1+\cdots+b_n\log\alpha_j
$$
does not vanish. Then
$$
|\Lambda|\geq \exp\{-C(j,\chi)D^2A_1\cdots A_j\log(eD)\log(eB)\},
$$
where $\chi=1$ if $\KK \subset\RR$ and $\chi=2$ otherwise and
$$
C(j,\chi)=\min\left\{\frac{1}{\chi}\left( \frac{1}{2}ej\right)^{\chi}30^{j+3}j^{3.5},2^{6j+20}\right\}.
$$
\end{lemma}
By applying this result we obtain upper bound for $l$ and $m$ in terms of $a$.
\begin{proposition}\label{proposition-1}
Assume that $c\in \{c_1^+, c_2^\pm\}.$ If $P_l=Q_m,$ with $m>1$ then
\begin{align*}
\frac{l}{\log(el)}&< 3.34\cdot 10^{13}\cdot\log^2(8.09c^2),\quad \text{with solutions of Type $1$},\; \\
\frac{l}{\log(el)}&< 6.63\cdot 10^{13}\cdot\log^2(8.09c^2), \quad \text{with solutions of Type $2$}.
\end{align*}
If $Q_m=P_l,\ l\geq1$ with $c=c_1^-,$ then we get
\begin{align*}
\frac{m}{\log(em)}< 13.36\cdot 10^{13}\cdot\log^2(21.3a),\quad \text{for}\; k=7,8,10,11,12,13.
\end{align*}
\end{proposition}
\begin{proof}
 We will demonstrate the proof for the case $c=c_1^-$ since the proof for the other cases closely follows \cite[Proposition 2.9]{glavni}. We apply Lemma \ref{proposition-1}
 with $j=3$ and $\chi=1$ to the linear form \ref{Lambda'} and take \[D=4,\ b_1=m,\ b_2=-l,\ b_3=1,\ \alpha_1=\beta,\ \alpha_2=\alpha,\ \alpha_3=\gamma'.\]
 Since $l\leq m$, we can take $B=m$. Also, we have
 \[h(\alpha_1)=\frac{1}{2}\log\beta,\ h(\alpha_2)=\frac{1}{2}\log\alpha.\]
 Since $\gamma'=\gamma^{-1}$, then $h(\gamma)=h(\gamma')$ and from \cite[Proposition 2.9]{glavni} we have
 \[h(\gamma)<\frac{1}{4}\log\left[\frac{2^4r^4c^4(1+\sqrt{k})^4}{(c-a)^2}\right].\]
 We have \begin{align*}(c_1^--a)^2&=a^2(2\sqrt{k+\frac{4}{a^2}}-k)^2>a^2(2\sqrt{k}-k)^2,\\
 r&<a\sqrt{1.1k},\\ c=c_1^-&=a(k+1-2\sqrt{k+\frac{4}{a^2}})<a(\sqrt{k}-1)^2,\end{align*} and now it follows that
 \[h(\gamma')=h(\gamma)<\frac{1}{4}\log(93309063a^6)<\frac{3}{2}\log(21.3a).\]
 By applying Lemma \ref{proposition-1} with
 \[A_1=2\log\beta,\ A_2=2\log\alpha,\ A_3=6\log(21.3a)\]
 we get
 \begin{align}\label{Matveev_0}
\log|\Lambda'|>-1.3901&\cdot 10^{11}\cdot16\cdot2 \cdot\log\beta\\&\cdot2\log\alpha\cdot6\log(21.3a)\cdot\log(4e)\cdot\log(em).\nonumber
\end{align}
From $1)$ of Lemma \ref{bound-Lambda} and using $l\geq1$, it is easy to conclude that $$m\log\beta<l\log\alpha+2.6\alpha^{-2l}-\log\gamma'<2l\log\alpha.$$
Also,
$$\log|\Lambda'|<-1.9069l\log\alpha$$
and
$$\log\alpha=\log\left[a\left(\sqrt{k+\frac{4}{a^2}}+\sqrt{k}\right)\right]<\log(7.212a)<\log(21.3a).$$
Now, by combining everything, we get
\[\frac{m}{\log(em)}<1.336\cdot10^{14}\cdot\log^2(21.3a),\ \text{for}\ k=7,8,10,11,12,13. \qedhere\]
\end{proof}

In this section, we examine the equation \[x=Q_m=P_l.\] Firstly, we state a useful result from \cite{glavni} and then apply it to obtain lower bounds for the indices $m$ and $l$ in terms of $a$.
\begin{lemma}\cite[Lemma 3.1]{glavni}\label{lem_6} If $a$ is odd, then
\begin{align}
\label{cong_1} Q_{2m}&\equiv x_0+\dfrac{1}{2} a(cx_0m^2+sz_0m)\pmod{a^2},\\
P_{2l}&\equiv x_2+\dfrac{1}{2} a(bx_2l^2+ry_2l)\pmod{a^2} .
\end{align}
If $a$ is even, then
\begin{align}
 Q_{2m}&\equiv x_0+\dfrac{1}{2} a(cx_0m^2+sz_0m)\pmod{\dfrac{1}{2} a^2},\\
P_{2l}&\equiv x_2+\dfrac{1}{2} a(bx_2l^2+ry_2l)\pmod{\dfrac{1}{2} a^2} .
\end{align}
\end{lemma}
\begin{lemma}\label{lower-Type-a} \hfill
\begin{enumerate}[1)]
 \item Assume that $b=ka,\ k\in \{7,8,10,11,12,13\}$ and $c=c_1^-$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have
$$
m \geq \frac{1}{4}\left(-2+\sqrt{4+\sqrt{a}}
\right).$$
 \item Assume that $b=ka,\ k\in \{7,8,10,11,12,13\}$ and $c \in \{c_1^+,c_2^{\pm}\}$. If the equation $P_l=Q_m$ has a solution $(l,m)$ (of Type $1$), then we have
$$
l \geq \frac{1}{12}\left(-2+\sqrt{4+3\sqrt{a}}
\right).$$
\end{enumerate}
\end{lemma}
\begin{proof}
 Proof follows \cite[Lemma 3.2]{glavni} and here we will demonstrate the proof of $1)$. From the recurrent sequence (\ref{sequence_s_nu}), we have $s \equiv 2,r\ (\bmod\ a)$. Also, $b=ka \equiv 0\ (\bmod\ a)$ and $c=c_{1}^{-}\equiv -2 r\ (\bmod\ a)$. Using the previous Lemma, with solutions in Type $1$, we get
 $$
- 4 r m^2\pm 4m\equiv \pm2rl\pmod{a},\quad \mbox{if}\quad s\equiv 2\pmod{a}
$$
and
$$
\ - 4m^2\pm 2m\mp2l\equiv 0\pmod{\dfrac{a}{\gcd (a,r)}},\quad \mbox{if}\quad s\equiv r\pmod{a}.
$$
In the first case, we multiply the congruence by $r$ and since $r^2 \equiv 4\ (\bmod\ a)$ we have $r \equiv \pm2\ (\bmod\ a')$ for some $a'$ that divides $a$ and $a' \geq \sqrt{a}$. So we get
$$-16m^2\pm8m\mp8l \equiv 0\ (\bmod\ a').$$
Using $l\leq m$ it follows that
$$16m^2+16m \geq |-16m^2\pm8m\mp8l|\geq a'\geq \sqrt{a},$$
which implies
\begin{align}\label{1}
 m \geq \frac{1}{4}\left(-2 +\sqrt{4+\sqrt{a}} \right).\end{align}
In the second case, since $\gcd(a,r)\leq2$ and $l \leq m$, we have
$$4m^2+4m \geq |- 4m^2\pm 2m\mp2l|\geq\frac{a}{\gcd(a,r)}\geq\frac{a}{2},$$
which implies
\begin{align}\label{2}
 m \geq \frac{1}{4}\left(-2+\sqrt{4+2a}\right).
\end{align}
Combining (\ref{1}) and (\ref{2}) we obtain the desired inequality.
\end{proof}

The cases $k \in \{2,3,5,6\}$ have already been observed in \cite{glavni} and \cite{Filipin}, and it is easy to verify that pairs $\{a,4a\}$ and $\{a,9a\}$ cannot satisfy the $D(4)$ property.

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10,$ $11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3-5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$.\ These examples support the Conjecture \ref{conj}.

Firstly, in Section $2$ we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10, 11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then in Sections $3$--$5$, we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$ (i.e., $d=d_{\pm}$).

Firstly, we show that $D(4)$-pairs $\{a,ka\}$, $k\in \{7, 8, 10, 11,12,13\}$ can be extended to a $D(4)$-triple only with an element $c$, which is a member of a family of positive integers depending on $a$. Then we prove that these triples can only be extended to regular $D(4)$-quadruples $\{a, ka, c, d\}$ (i.e., $d=d_{\pm}$).

Question: Let \(k\) be a positive integer with \(k\in\{7,8,10,11,12,13\}\). Suppose \(\{a,b,c,d\}\) is a set of four distinct positive integers with the property \(D(4)\), meaning that for any two distinct elements \(x,y\in\{a,b,c,d\}\), the number \(xy+4\) is a perfect square, and assume moreover that \(b=ka\). Which conclusion about \(\{a,b,c,d\}\) is valid under these hypotheses?

The quadruple \(\{a,b,c,d\}\) is regular; equivalently, the fourth element satisfies \(d=d_{\pm}\).

Choices:
B. The quadruple \(\{a,b,c,d\}\) is regular for every positive integer \(k\ge 7\) with \(b=ka\); equivalently, one always has \(d=d_{\pm}\).
C. There exists a regular \(D(4)\)-quadruple having the form \(\{a,ka,c,d\}\); in particular, for such a quadruple the fourth element satisfies \(d=d_{\pm}\).
D. The quadruple \(\{a,b,c,d\}\) can be extended from the pair \(\{a,ka\}\) only when the third element \(c\) belongs to one of the admissible families depending on \(a\); consequently, \(d\) need not equal \(d_{\pm}\).
E. For each \(k\in\{7,8,10,11,12,13\}\), every \(D(4)\)-quadruple \(\{a,b,c,d\}\) with \(b=ka\) is uniquely determined by the pair \(\{a,b\}\), so in particular there is exactly one possible value of \(d\), namely \(d=d_{+}\).

universal conclusion for every such quadruple

final step forcing regularity from admissible c-families

regularity versus uniqueness of extension and exclusion of d_-

The stem does not reveal the correct conclusion explicitly or by near-paraphrase. It states the hypotheses and asks which conclusion follows, without signaling that regularity is the intended answer.

The item is close to a theorem-recall question: the correct option essentially states the target theorem under the given hypotheses. However, it is not a pure tautology because the alternatives vary in strength, scope, and quantification.

Some reasoning is needed to distinguish the exact valid conclusion from stronger, weaker, or overgeneralized variants. Still, the item mainly tests precise theorem recognition rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: one overgeneralizes the range of k, one gives only a weaker existential claim, one suggests a tempting but false structural caveat, and one adds an unjustified uniqueness strengthening.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward recall of the exact result than toward deep generative reasoning.


Assume that all roots $\lambda_k$  of $p$ have rational real and imaginary parts.  If  $\varphi$  is holomorphic at the roots  and all values $\varphi^{(\nu)}(\lambda_k)$ have also rational real and imaginary parts then the series expansions of $f_k$ have coefficients with rational real and imaginary parts only.

\bigskip

Using {\it multicentric calculus} \cite{Ne12a} it is possible to get stable and convergent power series based representations for piecewise constant holomorphic functions.  When these series expansions are truncated,  we obtain polynomials which formally agree with  polynomials obtained as Hermite interpolants.  In this paper we  present  different ways of creating the multicentric representations.  Then we comment different  ways of creating Hermite interpolants  for the same purpose.  All these different representations can be done with exact arithmetics, but we shall then compare their performance under assumptions where  we allow in evaluations of variables tiny errors.  In the representations obtained from the multicentric form the errors do not accumulate essentially at all, unlike in the other expressions.

While with Hermite interpolation we may form a basis from polynomials,  of which the interpolative  polynomial can be obtained as linear combination, in multicentric representation we use a fixed Lagrangian interpolation basis  but   consider the coefficients as functions. And it is the series expansions of these "coefficients" which   when truncated yield  formally the Hermite interpolant.  This  is the cause for a very different order of steps in  evaluating the polynomials and is visible in  the sensitivity in error propagation.

\bigskip
Different approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\it taking  a polynomial as a new variable}.    We assume that a polynomial $p$ of degree $d$ with simple roots $\lambda_k$ has been selected and  the lemniscate set
$$
V(\rho) =\{ z \in \mathbb C: \ |p(z)| \le \rho \}
$$
has several components.  In {\it multicentric calculus} we represent  scalar  functions
$$
\varphi:  V(\rho) \rightarrow \mathbb C, \ \  z \mapsto \varphi(z)
$$
by vector valued functions
$$
f: B(\rho) \rightarrow \mathbb C^d,   \ \   w \mapsto f(w)
$$
so that  if $z \in V(\rho)$ then the new variable $w= p(z)$ is in the disc $B(\rho)$ and within the lemniscate
\[
\gamma_\rho = \{ z \in \mathbb C: \ |p(z)| = \rho \}.
\]
The original function can then be recovered by
\begin{equation}\label{multicentric}
\varphi(z) = \sum_{k=1}^d \delta_k(z) f_k(p(z))
\end{equation}
where the polynomial $\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\delta_k(\lambda_k)=1$, while $\delta_k(\lambda_j)=0$ for $j\not= k$.

 Jacobi \cite{Jac} considered (without the Cauchy integral) expansions of the form
 $$
 \varphi(z) \sim \sum_{j=0}^\infty c_j(z) p(z)^j
 $$
 where $c_j$ are polynomials of degree less than $d$,  see e.g.  \cite{Ki1906, Fe32, Wa69, Ma77}.   However, in this tradition  $w=p(z)$ was not  considered a truly new variable.
 Hence, the real potential of  utilising power series in discs for computational purposes was overlooked.  We call this multicentric calculus;  observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the "local centers" $\lambda_k$.     On this development see \cite{Ne12a, Ne12b, Ap16, Ne16, An19},  but we aim here to be sufficiently  self-contained.

 An important application of multicentric calculus  is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of  holomorphic functions $\varphi$ near the spectrum $\sigma(A)$ of a given bounded operator $A$.  When the spectrum  lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series
 $$ f_k(w)  = \sum_{j=0} ^\infty \alpha_{k,j} w^j
 $$
 available and converging fast, then only a moderate number of terms of the power series is needed to represent   $f_k(B)$, after which  an approximation of
 $$
 \varphi(A) = \sum_{k=1}^d \delta_k(A) f_k(B)
 $$
 can be computed.  Notice that if, say for a sparse large scale matrix $A$,  the  matrix $\varphi(A)$ is only needed at a vector $b$, then all  multiplications needed to create the approximate $\varphi(A)b$ can be obtained using matrix-vector multiplications only.

Here we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further  narrowing the focus on {\it functions which are identically $1$ in one component and  vanish in the others}  as the general case can be obtained by superposing.
In this approach we have approximation errors mainly  from the truncation of power series of the components of $f$  and numerical errors from  evaluation of the  truncated expansions at the  operator or matrix $B=p(A)$.  However,  it is important to notice that we  have the liberty to choose the local centers $\lambda_k$ to have rational real and imaginary parts as then  in  the multicentric representation the coefficients in the power series of  the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.

Using {\it multicentric calculus} \cite{Ne12a} it is possible to get stable and convergent power series based representations for piecewise constant holomorphic functions. When these series expansions are truncated, we obtain polynomials which formally agree with polynomials obtained as Hermite interpolants. In this paper we present different ways of creating the multicentric representations. Then we comment different ways of creating Hermite interpolants for the same purpose. All these different representations can be done with exact arithmetics, but we shall then compare their performance under assumptions where we allow in evaluations of variables tiny errors. In the representations obtained from the multicentric form the errors do not accumulate essentially at all, unlike in the other expressions.

While with Hermite interpolation we may form a basis from polynomials, of which the interpolative polynomial can be obtained as linear combination, in multicentric representation we use a fixed Lagrangian interpolation basis but consider the coefficients as functions. And it is the series expansions of these "coefficients" which when truncated yield formally the Hermite interpolant. This is the cause for a very different order of steps in evaluating the polynomials and is visible in the sensitivity in error propagation.

\bigskip
Different approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\it taking a polynomial as a new variable}. We assume that a polynomial $p$ of degree $d$ with simple roots $\lambda_k$ has been selected and the lemniscate set
$$
V(\rho) =\{ z \in \mathbb C: \ |p(z)| \le \rho \}
$$
has several components. In {\it multicentric calculus} we represent scalar functions
$$
\varphi: V(\rho) \rightarrow \mathbb C, \ \ z \mapsto \varphi(z)
$$
by vector valued functions
$$
f: B(\rho) \rightarrow \mathbb C^d, \ \ w \mapsto f(w)
$$
so that if $z \in V(\rho)$ then the new variable $w= p(z)$ is in the disc $B(\rho)$ and within the lemniscate
\[
\gamma_\rho = \{ z \in \mathbb C: \ |p(z)| = \rho \}.
\]
The original function can then be recovered by
\begin{equation}\label{multicentric}
\varphi(z) = \sum_{k=1}^d \delta_k(z) f_k(p(z))
\end{equation}
where the polynomial $\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\delta_k(\lambda_k)=1$, while $\delta_k(\lambda_j)=0$ for $j\not= k$.

Jacobi \cite{Jac} considered (without the Cauchy integral) expansions of the form
 $$
 \varphi(z) \sim \sum_{j=0}^\infty c_j(z) p(z)^j
 $$
 where $c_j$ are polynomials of degree less than $d$, see e.g. \cite{Ki1906, Fe32, Wa69, Ma77}. However, in this tradition $w=p(z)$ was not considered a truly new variable.
 Hence, the real potential of utilising power series in discs for computational purposes was overlooked. We call this multicentric calculus; observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the "local centers" $\lambda_k$. On this development see \cite{Ne12a, Ne12b, Ap16, Ne16, An19}, but we aim here to be sufficiently self-contained.

An important application of multicentric calculus is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of holomorphic functions $\varphi$ near the spectrum $\sigma(A)$ of a given bounded operator $A$. When the spectrum lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series
 $$ f_k(w) = \sum_{j=0} ^\infty \alpha_{k,j} w^j
 $$
 available and converging fast, then only a moderate number of terms of the power series is needed to represent $f_k(B)$, after which an approximation of
 $$
 \varphi(A) = \sum_{k=1}^d \delta_k(A) f_k(B)
 $$
 can be computed. Notice that if, say for a sparse large scale matrix $A$, the matrix $\varphi(A)$ is only needed at a vector $b$, then all multiplications needed to create the approximate $\varphi(A)b$ can be obtained using matrix-vector multiplications only.

Here we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further narrowing the focus on {\it functions which are identically $1$ in one component and vanish in the others} as the general case can be obtained by superposing.
In this approach we have approximation errors mainly from the truncation of power series of the components of $f$ and numerical errors from evaluation of the truncated expansions at the operator or matrix $B=p(A)$. However, it is important to notice that we have the liberty to choose the local centers $\lambda_k$ to have rational real and imaginary parts as then in the multicentric representation the coefficients in the power series of the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.

\bigskip
Different approaches to get the multicentric representations are surveyed in Section 2 but the point of departure is in {\it taking a polynomial as a new variable}. We assume that a polynomial $p$ of degree $d$ with simple roots $\lambda_k$ has been selected and the lemniscate set
$$
V(\rho) =\{ z \in \mathbb C: \ |p(z)| \le \rho \}
$$
has several components. In {\it multicentric calculus} we represent scalar functions
$$
\varphi: V(\rho) \rightarrow \mathbb C, \ \ z \mapsto \varphi(z)
$$
by vector valued functions
$$
f: B(\rho) \rightarrow \mathbb C^d, \ \ w \mapsto f(w)
$$
so that if $z \in V(\rho)$ then the new variable $w= p(z)$ is in the disc $B(\rho)$ and within the lemniscate
\[
\gamma_\rho = \{ z \in \mathbb C: \ |p(z)| = \rho \}.
\]
The original function can then be recovered by
\begin{equation}\label{multicentric}
\varphi(z) = \sum_{k=1}^d \delta_k(z) f_k(p(z))
\end{equation}
where the polynomial $\delta_k$ is, as in the Lagrange interpolation, the polynomial of degree $d-1$ such that $\delta_k(\lambda_k)=1$, while $\delta_k(\lambda_j)=0$ for $j\not= k$.

Jacobi \cite{Jac} considered (without the Cauchy integral) expansions of the form
 $$
 \varphi(z) \sim \sum_{j=0}^\infty c_j(z) p(z)^j
 $$
 where $c_j$ are polynomials of degree less than $d$, see e.g. \cite{Ki1906, Fe32, Wa69, Ma77}. However, in this tradition $w=p(z)$ was not considered a truly new variable.
 Hence, the real potential of utilising power series in discs for computational purposes was overlooked. We call this multicentric calculus; observe that $|p(z)|^{1/d}$ is the geometric average of distances from $z$ to the "local centers" $\lambda_k$. On this development see \cite{Ne12a, Ne12b, Ap16, Ne16, An19}, but we aim here to be sufficiently self-contained.

An important application of multicentric calculus is in holomorphic functional calculus, where one needs an accurate and and easily computable approximation of holomorphic functions $\varphi$ near the spectrum $\sigma(A)$ of a given bounded operator $A$. When the spectrum lies inside the components bounded by the lemniscate, then the spectrum of $B= p(A)$ lies inside the corresponding disc and if we have power series
 $$ f_k(w) = \sum_{j=0} ^\infty \alpha_{k,j} w^j
 $$
 available and converging fast, then only a moderate number of terms of the power series is needed to represent $f_k(B)$, after which an approximation of
 $$
 \varphi(A) = \sum_{k=1}^d \delta_k(A) f_k(B)
 $$
 can be computed. Notice that if, say for a sparse large scale matrix $A$, the matrix $\varphi(A)$ is only needed at a vector $b$, then all multiplications needed to create the approximate $\varphi(A)b$ can be obtained using matrix-vector multiplications only.

Here we concentrate on holomorphic functions which are piecewise constant. However, this can be done by further narrowing the focus on {\it functions which are identically $1$ in one component and vanish in the others} as the general case can be obtained by superposing.
In this approach we have approximation errors mainly from the truncation of power series of the components of $f$ and numerical errors from evaluation of the truncated expansions at the operator or matrix $B=p(A)$. However, it is important to notice that we have the liberty to choose the local centers $\lambda_k$ to have rational real and imaginary parts as then in the multicentric representation the coefficients in the power series of the components of $f$ are all rational, and the computations to create them can be done with exact arithmetic.

We use the following notation for truncating power series. If
\begin{equation} \label{katkaisu}
a(x) \sim \sum_{j=0} ^\infty c_j x^j , \ \ \text { then } \lfloor a \rfloor_n (x) = \sum_{j=0}^n c_j x^j.
\end{equation}
Thus, when we truncate the series for $f_k(w)$ we can write at $w=p(z)$
\begin{equation}\label{katkaisunmerkitys}
\lfloor f_k \rfloor_n (p(z)) = \sum_{j=0}^n \alpha_{k,j}\ p(z)^j.
\end{equation}

We decompose the Cauchy kernel $\frac{1}{\lambda - z}$ writing
\begin{equation}\label{decompCauchy}
\frac{1}{\lambda - z} = \frac{1}{p(\lambda)- p(z)} \ \sum_{k=1}^d p'(\lambda_k) \delta_k(\lambda) \delta_k(z).
\end{equation}
Denoting
\begin{equation}\label{decompkernel}
K_k(\lambda, w) = \frac{1}{\lambda - \lambda_k} \frac{p(\lambda)}{p(\lambda)-w}
\end{equation} we obtain for $f_k$ the expression
\begin{equation}\label{fexpressed}
f_k(w) = \frac{1}{2\pi i } \int_ {\gamma_\rho} K_k(\lambda, w) \varphi(\lambda)d\lambda,
\end{equation}
where $\varphi$ is holomorphic inside $\gamma_\rho$ and continuous up to boundary. Here we are interested in piecewise continuous $\varphi$ which means that if the lemniscate would pass through a critical point, then $\varphi$ would have to take the same value in each component touching this critical point. Without loss on generality we assume that $\rho$ is such that the lemniscate contains no critical points and in particular the lemniscate consists of smooth curves, each surrounding at least one $\lambda_k$. Then, if $|w| < |p(\lambda)| = \rho$, the series
$$
f_k(w) = \sum_{n=0} ^\infty \alpha_{k, n} w^n
$$
converges where
\begin{equation}\label{expressderivatives}
\alpha_{k, n} \ = \ \frac{1}{2\pi i} \int_{\gamma_\rho} \frac{1}{ \lambda - \lambda_k } \ \frac{\varphi(\lambda)} {p(\lambda)^n} \ d \lambda.
\end{equation}

If we differentiate the holomorphic $\varphi$ in (\ref{multicentric})
and assume the data $\varphi^{(\nu)} (\lambda_k)$ to be given, then an explicit recursion for $\alpha_{j, \nu} = f_j^{(\nu)}(0)/ \nu !$ is obtained from
\begin{equation}\label{explicitrecursion}
p'(\lambda_j)^{\nu} f_j^{(\nu)}(0) = \varphi^{(\nu)}(\lambda_j)- t_{j,\nu},
\end{equation}
where
$$
t_{j,\nu}= \sum_{k=1}^d \sum_{\mu=0}^{\nu-1 } \binom{\nu}{\mu} \delta_k^{\nu-\mu}(\lambda_j) \sum_{l=0}^{\mu} b_{\mu,l}(\lambda_j) f_k^{(l)}(0) + \sum_{l=0}^{\nu-1} b_{\nu,l}(\lambda_j) f_j^{(l)}(0)
$$
and the polynomials $b_{\nu,l}$ are obtained from a triangular table, see \cite{Ne12a, An22}.

We can use the integral representation from Section~\ref{cauchysection} to derive bounds for the accumulated errors.
Let $\varphi$ be holomorphic for $|p(z)| < r$ and denote for $\rho<r$
$$
M(\rho) = \max_{|p(z)| \le \rho} |\varphi (z)|.
$$
We assume here that the exact values of coefficients $\alpha_{k,j}$ can be used, as this is the case e.g. when the roots $\lambda_k$ have rational real and imaginary parts and $\varphi$ takes only integer values.

\begin{proof}
This follows immediately from the explicit recursion, see \cite{Ne12a, An19} or  (\ref{explicitrecursion}) below.
\end{proof}

We  use the following notation for truncating power series.  If
\begin{equation} \label{katkaisu}
a(x) \sim \sum_{j=0} ^\infty c_j x^j , \ \  \text { then }   \lfloor a \rfloor_n (x) = \sum_{j=0}^n c_j x^j.
\end{equation}
Thus, when we truncate  the series for $f_k(w)$ we can write at $w=p(z)$
\begin{equation}\label{katkaisunmerkitys}
\lfloor f_k \rfloor_n (p(z)) = \sum_{j=0}^n \alpha_{k,j}\  p(z)^j.
\end{equation}

\bigskip
Consider now the Hermite interpolation problem: at $d$ distinct  complex points $\lambda_k$  to find polynomial $P$ such that
\begin{equation}\label{hermiteyleinen}
P^{(\nu)}(\lambda_k) = a_{k,\nu} \ \text{ for } \nu= 0, \dots, n; \ \  k=1,\dots,d.
\end{equation}
\begin{proposition}
Let $\varphi$ be holomorphic at interpolation points $\lambda_k$. Then  the truncated polynomial of the multicentric representation  of $\varphi$
$$
P(z)= \sum_{k=1}^d \delta_k(z) \lfloor f_k\rfloor_n (p(z))
$$ is the unique polynomial of degree $N \le d(n+1)-1$ satisfying the interpolation problem (\ref{hermiteyleinen}) when $a_{k,\nu}= \varphi^{(\nu)}(\lambda_k)$.
\end{proposition}

\begin{proof}
Clearly, $P$ is of at most degree $d(n+1)-1$. Consider the difference $\varphi-P$. We have
$$
\varphi(z) - P(z) = p(z)^{n+1} \; \sum_{k=1}^d \delta_k(z) \negthickspace \sum_{j=n+1}^\infty \! \alpha_{k,j} \, p(z)^{j-n-1}
$$
which shows that the interpolation conditions are all satisfied.
\end{proof}

\bigskip

Denote by $\delta_{k,j,n}$  the solution to the special case of (\ref{hermiteyleinen} )  with $a_{k,j}=1$, while $a_{m, \nu} =0$ when $(m,\nu) \not=(k,j)$.  Then $(\delta_{k,j,n})$  is a basis  such that the general problem (\ref{hermiteyleinen}) is solved by
$$
P(z) = \sum_{k=1}^d \sum_{j=0}^n a_{k,j} \  \delta_{k,j,n}(z).
$$

We shall keep writing $\varphi$ and $f$ when we discuss general  formulae but use $\chi$ and $g$  in the  particular case where $\chi(z) = 1$ in the component containing the root $\lambda_1$  and vanishing in the components containing  other roots, so that
$$
\chi(z) = \sum_{k=1}^d \delta_k(z) g_k(p(z)).
$$
When we truncate   here the power series of $g_k$'s we  get the basis function $\delta_{1,0,n}$  which  has the multicentric representation
\begin{equation}\label{kantapolynom}
\delta_{1,0,n}(z) = \sum_{k=1}^d \delta_k(z) \lfloor g_k \rfloor_n (p(z)).
\end{equation}
This  representation of  the polynomial $\delta_{1,0,n}$  is later denoted by $M_n$, letter $M$ for multicentric.

In Section 3 we indicate different direct ways to evaluate this basis polynomial.  With $H_n$ we denote the direct representation  as powers of $z$,  $S_n$ shall denote a special representation of  the basis polynomial and $T_n$ is reserved for a particular formulation of the basis polynomial suitable for parallel computing.

In Section 4  we  consider the numerical stability of these different ways and in particular see that the order of computational steps obtained from truncation the multicentric representation is superior, as we let $n$  grow  while keeping the interpolation points, the roots of $p$ fixed.  More specifically,   we shall demonstrate their  different sensitivity using the following  two polynomials, one with real zeros symmetrically around he origin
  \begin{equation}\label{symmetric}
  p(z)= z (1-z^2)
  \end{equation}
  and  another, nonsymmetric one:
  \begin{equation}\label{nonsymmetric}
  p(z) = z( z^2+4z+5).
  \end{equation}
  In both polynomials we choose $\lambda_1=0$ to be the root  lying in the component where the piecewise holomorphic function would be identically 1.  Thus, the first one  models a situation where we would be using Riesz functional calculus to concentrate in the middle of the spectrum while the second could be thought as modelling a situation where one would like to identify the invariant subspace of an operator corresponding to the unstable part  of the spectrum. Notice that the behaviors are independent of translating and scaling of the  basic variable $z$.

This is stated to “follow[] immediately from the explicit recursion,” with references to \cite{Ne12a, An19} or “(\ref{explicitrecursion}) below.”

Question:
Let $p$ be a polynomial with simple roots $\lambda_1,\dots,\lambda_d$, and let $\delta_k$ be the Lagrange interpolation polynomials of degree $d-1$ characterized by $\delta_k(\lambda_k)=1$ and $\delta_k(\lambda_j)=0$ for $j\ne k$. Suppose a function $\varphi$ admits the multicentric representation
\[
\varphi(z)=\sum_{k=1}^d \delta_k(z)\,f_k(p(z)),
\]
where each component function $f_k$ has a power-series expansion at $0$ of the form
\[
f_k(w)=\sum_{n=0}^\infty \alpha_{k,n}w^n.
\]
Assume that every root $\lambda_k$ has rational real and imaginary parts, and that $\varphi$ is holomorphic in a neighborhood of each root $\lambda_k$ with the property that for every $k$ and every derivative order $\nu\ge 0$, the value $\varphi^{(\nu)}(\lambda_k)$ has rational real and imaginary parts. Which conclusion about the series expansions of the functions $f_k$ is valid under these hypotheses?

For every $k$ and every $n\ge 0$, the coefficient $\alpha_{k,n}$ has rational real and imaginary parts; equivalently, the power-series expansion of each $f_k$ has coefficients whose real and imaginary parts are all rational.

Choices:
B. For every $k$ and every $n\ge 0$, the coefficient $\alpha_{k,n}$ is an algebraic complex number; in general one cannot conclude that its real and imaginary parts are rational without additionally assuming that $\varphi$ is globally holomorphic on a lemniscate neighborhood.
C. For every $k$, the power-series expansion of $f_k$ at $0$ has coefficients whose real and imaginary parts are algebraic numbers.
D. For every fixed derivative order $\nu\ge 0$, there exists a choice of multicentric component functions $f_1,\dots,f_d$ such that the coefficients $\alpha_{k,n}$ have rational real and imaginary parts for all $k$ and all $0\le n\le \nu$.
E. For every $k$ and every $n\ge 0$, the coefficient $\alpha_{k,n}$ has rational real and imaginary parts provided only that each root $\lambda_k$ has rational real and imaginary parts and that the values $\varphi(\lambda_k)$ themselves have rational real and imaginary parts.

The stem does not explicitly reveal the correct option. It states hypotheses and asks for the valid conclusion; the correct answer is not embedded verbatim in the question.

This is essentially a direct theorem-recall item: the hypotheses are laid out in full and the correct choice restates the target conclusion almost exactly. It tests recognition of the theorem more than selection among genuinely competing mathematical outcomes.

There is some reasoning pressure because one distractor is a weaker true statement and others alter quantifiers or assumptions. However, the item mainly rewards recalling the exact theorem rather than generating a conclusion from first principles.

The distractors are plausible and mathematically meaningful: they target common confusions about rationality vs algebraicity, local vs global hypotheses, finite-order vs all-order conclusions, and zeroth-order data vs full derivative data.

A mathematically well-constructed but largely theorem-restatement MCQ: little answer leakage and strong distractors, but limited non-tautological or generative reasoning value.


\begin{theorem}\label{thm_specialcase}
    Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes.
\end{theorem}

\subsection{Background}

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

Elkies proved in \cite{MR903384} that an elliptic curve defined over a number field of odd degree has infinitely many primes of supersingular reduction, and extended the result in \cite{MR1030140} to elliptic curves over any number field with at least one real embedding. This has been generalized to various families of abelian varieties such as abelian surfaces with multiplication by the quaternion algebra of discriminant $6$, with field of moduli $\mathbb{Q}$ and which is a Jacobian in characteristic $2$ and $3$ in \cite{MR2414789}. For a detailed discussion of related works, see \cite{chen2025infinitelysupersingularprimesmumfords}.

\subsection{The main result}
We extend the result of \cite{MR2414789} to number fields with at least one real embedding, and weaken the local conditions at the bad reduction primes $2$ and $3$. We expect that similar methods can be applied to the settings in \cites{MR2704678, MR2705896, li2025infinitelyprimesbasicreduction,chen2025infinitelysupersingularprimesmumfords}.

A special case of our main theorem is as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
 and has field of moduli a number field $L$ with at least one real embedding.
 Write $j_0:=j(C)\in L$ and $\Nm_{L/\bQ}(j_0) = \frac{n}{d}$ with $n,d\in \bZ, (n,d,6) = 1, d>0$.
 Assume at least one of the following conditions:
 \begin{enumerate}
 \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\bQ]+v_3(d\Nm_{L/\bQ}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
 \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
 \item $[L:\bQ]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
 \end{enumerate}
 Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

In each case, we are going to find some discriminant $D$ with a rational prime $p\notin \Nm_{L/\bQ}(S)$ such that $v_p(P_{D}(j_0)) > 0$ and $\left(\frac{D}{p}\right) \neq 1$, then as in \cite{MR2414789}, there is a prime $\fp \notin S$ above $p$, such that the Jacobian of $C$ has supersingular, and hence superspecial reduction.
 \begin{enumerate}
 \item Choose a prime $l$ satisfying the conditions:
 \begin{itemize}
 \item $l\equiv 13 \pmod{24}$,
 \item $\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-4l}(x)$ has a real root $j(a_i)$ in a subinterval of $(-\frac{16}{27},0)$ to be specified later.
 \end{itemize}
 Let $D = -4l$. By \cite{MR2414789}*{Lemma 6} and \Cref{lem:red23}, \begin{align*}
 P_{-4l}(x) &\equiv x^{h'} \pmod{4}, \\
 P_{-4l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-4l}(x) &\equiv (27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \leq 0$ for all $\mathfrak{p}$ above $2,3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)$, then
 $N, dQ\in \bN$ and $(N, 6) = 1,\, (dQ,2) = 1$.
 Suppose $l\nmid N$, then \begin{align*}
 \left(\frac{-4l}{N}\right) &= \left(\frac{-1}{N}\right)\left(\frac{l}{N}\right)= \left(\frac{-1}{N}\right)\left(\frac{N}{l}\right) \\
 &= \left(\frac{-1}{N}\right)\left(\frac{dQ}{l}\right) = \left(\frac{-1}{N}\right)\left(\frac{l}{dQ}\right)\\
 &= \left(\frac{-1}{N}\right)\left(\frac{-1}{dQ}\right) \left(\frac{-l}{dQ}\right) \\ &= \left(\frac{-1}{s(d\Nm_{L/\bQ} j_0)^{h'}}\right)\left(\frac{-1}{s'3^{[L:\bQ]}d\Nm_{L/\bQ}(j_0)}\right)\left(\frac{-l}{3^{v_3(dQ)}}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)}.\end{align*}
 If $[L:\bQ]+v_3(dQ)$ is odd, let \[j(a_1)\in\left(-\frac{16}{27}, -\frac{16}{27} + \min_{1\leq i \leq r} \left|j_i + \frac{16}{27}\right|\right)\]
 so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r$.
 If $[L:\bQ]+v_3(dQ)$ is even, by assumption let $j_t$ be the minimal real conjugate of $j_0$ in $( -\frac{16}{27} , 0)$, and \[j(a_1)\in\left(j_t, j_{t+1}\right)\] so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\leq i\leq r, i\neq t$ and $(27j_t + 16) P_{-4l}(j_t) < 0$. In either case we have \[\left(\frac{-4l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-4l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.
 \item Choose a prime $l$ satifying the conditions:
 \begin{itemize}
 \item $l\equiv 19\pmod{24}$,
 \item $\left(\frac{q}{l}\right)=\left(\frac{-l}{q}\right) = 1$ for every prime $q\in \Nm_{L/\bQ}(S) \backslash \{2,3\}$,
 \item $P_{-l}(x)$ has a real root $j(a_1)$ in a subinterval of $(0, \infty)$ to be specified later.
 \end{itemize} Let $D = -l$. By \Cref{lem:red23}, \Cref{lem:pair}, and \cite{MR2414789}*{Lemma 4},
 \begin{align*}
 P_{-l}(x) &\equiv 1 \pmod{2}, \\
 P_{-l}(x) &\equiv \pm x^{h'} \pmod{3}, \\
 P_{-l}(x) &\equiv 3(27x+16)S(x)^2 \pmod{l} \text{ for some }S(x) \in\bZ[x].
 \end{align*}
 Since $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$ and $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, we can choose $d$ such that
 $d\prod_{\sigma\in T} \sigma(j_0)$ is integral for any $T\subset \Hom(L, \overline{\bQ})$ and any prime dividing $d$ lies in $\Nm_{L/\bQ}(S)\backslash\{2\}$, then
 $N, dQ\in \bN$ and $(N, 6) = 1$ .
 Suppose $l\nmid N$, then
 \begin{align*}
 \left(\frac{-l}{N}\right) &= \left(\frac{N}{l}\right) \\
 &= \left(\frac{s(d\Nm_{L/\bQ}(3(27j_0+16)))}{l}\right)\\
 &= \left(\frac{ss'3^{[L:\bQ]}dQ}{l}\right) \\
 &= ss'(-1)^{[L:\bQ]+v_3(dQ)+v_2(Q)}
 \end{align*}
 Let $j_t$ be the minimal real conjugate of $j_0$ in $(0, \infty)$ and $n$ ($0\leq n < r$) be the number of real conjugates of $j_0$ in $(-\frac{16}{27}, 0)$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is odd, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is odd}{(0, j_t)}{$n$ is even}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)>0$. If $[L:\bQ]+v_3(dQ)+v_2(Q)$ is even, let \[\twocase{j(a_1)\in}{(j_t, j_{t+1})}{$n$ is even}{(0, j_t)}{$n$ is odd}\] so that $\Nm_{L/\bQ}(P_{-l}(j_0))\Nm_{L/\bQ}(27j_0+16)<0$. In either case we have \[\left(\frac{-l}{N}\right) = -1\] and there is a prime divisor $p$ of $N$ such that \[\left(\frac{-l}{p}\right) = -1.\] The conditions on $l$ and $(N,6) = 1$ imply that $p\notin \Nm_{L/\bQ}(S)$ and $v_p(\Nm_{L/\bQ}(P_D(j_0))) = v_p(N) > 0$.
 If $l | N$, then since $l\notin \Nm_{L/\bQ}(S)$ by construction, we can choose $p = l$.

\begin{rem} \label{rem:conclude} One can try to further weaken the local conditions by considering Heegner cycles with different forms of discriminant $D$.
For primes $p$ dividing $D$, the unpaired roots of $P_D(x)$ modulo $p$ can be predicted by checking whether a maximal order in a definite quaternion algebra ramified at $2,3,p$ contains two anticommuting CM orders of some given discriminants, as computed in \cite{MR2704678}*{3.4.2}. One can then impose conditions on the number of prime divisors of $D$ and the congruence class modulo $24$ of primes dividing $D$, so that $P_D(x)$ has no unpaired roots or $P_D(x)$ has a single unpaired root from the same elliptic point modulo each $p\nmid D$. In addition, one imposes a congruence condition on $D$ so that $\cP_{D, \cE}$ avoids intersection with one of $\cP_{-3, \cE}$ and $\cP_{-4, \cE}$ by \Cref{lem:avoidintersection}.
The local conditions on $j_0$ are determined so that it avoids intersection with the these $\cP_D$ at $p = 2,3$, and some real conjugate of $j_0$ and some real root of $P_D$ lie in the subinterval corresponding to one geodesic. For $D$ not of the form $-Np$ with $N = 1, 3, 4, 8, 12, 24$, it is possible that $P_D(x)$ has multiple real roots in each subinterval, but we still expect an equidistribution result.
\end{rem}
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More generally, the coarse moduli variety of principally polarized abelian surfaces with potential multiplication by the maximal quaternion order of discriminant $6$ is isomorphic to $\mathbb{P}^1$, and an arithmetic $j$-function (see \eqref{eq:j} in \cref{sec:coord}) is defined in \cite{MR2423455}. In terms of this coordinate, the assumption at primes $\mathfrak{p}$ above $2$ and $3$ in \Cref{thm_specialcase} corresponds to the case $v_{\mathfrak{p}}(j(C)) = 0$ by \cite{MR2414789}*{Proposition 2}, and our main theorem can be stated as follows.

\begin{theorem}\label{thm_main}
Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$,
    and has field of moduli a number field $L$ with at least one real embedding.
    Write $j_0:=j(C)\in L$ and $\Nm_{L/\mathbb{Q}}(j_0) = \frac{n}{d}$ with $n,d\in \mathbb{Z}, (n,d,6) = 1, d>0$.
    Assume at least one of the following conditions:
    \begin{enumerate}
        \item $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $2,3$, and $[L:\mathbb{Q}]+v_3(d\Nm_{L/\mathbb{Q}}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\frac{16}{27},0)$;
        \item $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2$, $v_{\mathfrak{p}}(j_0) \leq 0$ for $\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \infty)$;
        \item $[L:\mathbb{Q}]$ is even, $v_{\mathfrak{p}}(j_0) \geq 0$ for $\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\infty, -\frac{16}{27})\cup (0, \infty)$.
    \end{enumerate}
    Then the Jacobian of $C$ has superspecial reduction at infinitely many primes.
\end{theorem}

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\left(\frac{D}{p}\right) \neq 1$.

As in \cite{MR2414789}, we need to consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$. We use \cite{MR2704678} to generalize \cite{MR2414789} through a more detailed, case-by-case study, where in each case the discriminant $D$ is chosen in a form adapted to the conditions.
In characteristic $2$ and $3$, \cite{MR2414789} shows that the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$, and the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle. The local conditions on $j_0$ at primes above  $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes.
For more on the choice of local conditions and Heegner cycles, and for potential further weakenings of these conditions, see \Cref{rem:conclude}.

\subsection{Notation and conventions}

Assume the following unless specified otherwise.

Let $B = B_{\Delta}$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $\Delta$,  $\Lambda = \Lambda_{\Delta}$ be a maximal order of $B$, and $\Lambda^1 = \Lambda_{\Delta}^1$ be the group of units in $\Lambda$ of norm $1$. Fix an element $\mu \in \Lambda_{\Delta}$ such that $\mu^2 = -\Delta$,\footnote{The element $\mu$ exists because the field $\mathbb{Q}(\sqrt{-\Delta})$ embeds in $B$ by the local-global principle, and any two maximal orders in $B$ are conjugate to each other by strong approximation.} then the involution $\alpha \mapsto \alpha' = \mu^{-1} \bar{\alpha} \mu$ is a positive anti-involution on $B$.

An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\cite{Clark2003}*{p.70} and \cite{MR1097624}*{p.23}), so “it therefore suffices to construct primes of supersingular reduction,” following “Elkies' general strategy.” Given $C$, one finds supersingular reduction of its Jacobian “from its intersection with some Heegner cycle $\cP_D$ of discriminant $D$.” The intersection at a prime $p$ is “captured by $v_p(P_D(j_0))>0$,” where $P_D(x)$ is “the integral minimal polynomial of the $j$-invariants of the points in $\cP_D$,” and when this occurs, supersingular reduction at $p$ is “detected by $\left(\frac{D}{p}\right) \neq 1$.”

As in \cite{MR2414789}, one must “consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$.” Using \cite{MR2704678}, the argument proceeds via “a more detailed, case-by-case study,” where “in each case the discriminant $D$ is chosen in a form adapted to the conditions.” In characteristic $2$ and $3$, \cite{MR2414789} shows “the reduction of any Heegner cycle lies in the superspecial locus $\{j = 0, \infty\}$,” and “the intersection formula of Heegner divisors in \cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle.” Finally, “the local conditions on $j_0$ at primes above $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes,” yielding infinitely many primes of superspecial reduction.

Question: Let $C$ be a genus $2$ curve whose Jacobian $J(C)$ admits an action of the maximal order in the rational quaternion algebra of discriminant $6$, and suppose the field of moduli of $C$ is a number field $L$ having at least one real embedding. Assume moreover that for every prime of $L$ above $2$ or $3$, the curve $C$ has potentially smooth stable reduction there (that is, after a finite extension, its stable reduction becomes smooth). Under these hypotheses, which statement about the reduction of $J(C)$ holds? Here, saying that $J(C)$ has superspecial reduction at a prime means that its reduction modulo that prime is a superspecial abelian surface.

The Jacobian $J(C)$ has superspecial reduction at infinitely many primes.

Choices:
B. The Jacobian $J(C)$ has superspecial reduction at all but finitely many primes.
C. The Jacobian $J(C)$ has superspecial reduction at at least one prime.
D. There exists a finite extension $L'/L$ such that the base change $J(C)_{L'}$ has superspecial reduction at infinitely many primes of $L'$.
E. The Jacobian $J(C)$ has infinitely many primes of supersingular reduction.

The stem does not state the conclusion outright; it only gives hypotheses and asks for the resulting reduction statement. Defining 'superspecial reduction' clarifies terminology but does not reveal which quantified claim is correct.

This is close to a theorem-recall item: the hypotheses are highly specific and the correct option matches the natural theorem conclusion. However, the options introduce meaningful quantifier and property variations, so it is not a pure verbatim restatement.

Some reasoning is needed to distinguish 'infinitely many' from weaker existence, overly strong density-type claims, base-change variants, and supersingular/superspecial confusion. Still, the task is mainly recognition of the exact theorem-level conclusion rather than deep derivation.

The distractors are plausible and mathematically well targeted: one is too strong (all but finitely many), one is a weaker true-looking statement, one alters the field/quantifier dependence, and one exploits the common superspecial vs. supersingular confusion.

A solid theorem-conclusion MCQ with strong distractors and no direct answer leakage, but it remains fairly close to theorem recall rather than testing substantial generative mathematical reasoning.