Approximation theorems (Runge \cite{Run}, Mergelyan \cite{Mer}, Arakelian \cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\subset \mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.

To this end, define the classes
\begin{align*}
H(F)=&\,\lbrace f:U\to\mathbb{C} \text{ holomorphic for some open set $U\supset F$}\rbrace,\\
 A(F)=&\,\lbrace f:F\to\mathbb{C} \text{ holomorphic in }F^\circ\text{ and continuous in F}\rbrace,
\end{align*}
where $F^\circ$ is the interior of $F$.
\begin{dfn}\label{def:hole}
Any bounded connected component $B$ of $\mathbb{C}\setminus F$ is called a hole of $F$.
\end{dfn}
Arakelian's theorem \cite{Ar} gives necessary and sufficient conditions such that every function $f\in A(F)$ can be uniformly approximated on $F$ by entire functions..
\begin{dfn}[Arakelian set]\label{def:Arak}
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\[5pt]
1. $F$ is without holes;\\[5pt]
2. The union of all holes of $F\cup K$ is bounded, for any compact subset $K\subset \mathbb{C}$.
\end{dfn}
\begin{rmk}\label{rem:K}
Point 2. in Definition \ref{def:Arak} is equivalent to the one where $K$ is replaced by $\overline{D}_0(n)=\{z\in\mathbb{C}: |z|\leq n\}$, for any $n\in\mathbb{N}$.
\end{rmk}
\begin{rmk}
In fact, Definition \ref{def:Arak} appears in a later work \cite{RR}, where a new proof of Arakelian's theorem was given (see also \cite{Got}). The original conditions in \cite{Ar} are:\\[5pt]
1. $\mathbb{C}\setminus F$ is connected;\\[5pt]
2. $\mathbb{C}\setminus F$ is locally connected at infinity.\\[5pt]
Nevertheless, they are easily seen to be equivalent.
\end{rmk}

\begin{dfn}[Uniform approximation set]
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\epsilon>0$ and $f\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\epsilon$, for all $z\in F$.
\end{dfn}

\begin{abstract}
Arakelian's classical approximation theorem \cite{Ar} gives necessary and sufficient conditions such that functions can be uniformly approximated in (unbounded) closed sets $F\subset \mathbb{C}$ by entire functions. The conditions are purely topological and concern the connectedness of the complement of $F$. We give a new characterization of Arakelian sets in terms of logarithmic branches of functions $f\in A(F)$, which are continuous in $F$ and holomorphic in its interior $F^\circ$. Our proof is based on a contradiction argument and the counterexample function that we use is furnished by the Weierstrass factorization theorem.
\end{abstract}

Approximation theorems (Runge \cite{Run}, Mergelyan \cite{Mer}, Arakelian \cite{Ar}) play a central role in the field of complex approximation, since they give necessary and sufficient conditions for functions to be uniformly approximated on a given set by model functions, such as polynomials, rational functions or more generally entire/meromorphic functions. Our goal in the present paper is to give a new characterization of closed sets $F\subset \mathbb{C}$ in the complex plane, for which uniform approximation by entire functions is possible.

To this end, define the classes
\begin{align*}
H(F)=&\,\lbrace f:U\to\mathbb{C} \text{ holomorphic for some open set $U\supset F$}\rbrace,\\
 A(F)=&\,\lbrace f:F\to\mathbb{C} \text{ holomorphic in }F^\circ\text{ and continuous in F}\rbrace,
\end{align*}
where $F^\circ$ is the interior of $F$.
\begin{dfn}\label{def:hole}
Any bounded connected component $B$ of $\mathbb{C}\setminus F$ is called a hole of $F$.
\end{dfn}
Arakelian's theorem \cite{Ar} gives necessary and sufficient conditions such that every function $f\in A(F)$ can be uniformly approximated on $F$ by entire functions..
\begin{dfn}[Arakelian set]\label{def:Arak}
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is an Arakelian set, if and only if the following points hold:\\[5pt]
1. $F$ is without holes;\\[5pt]
2. The union of all holes of $F\cup K$ is bounded, for any compact subset $K\subset \mathbb{C}$.
\end{dfn}
\begin{rmk}\label{rem:K}
Point 2. in Definition \ref{def:Arak} is equivalent to the one where $K$ is replaced by $\overline{D}_0(n)=\{z\in\mathbb{C}: |z|\leq n\}$, for any $n\in\mathbb{N}$.
\end{rmk}
\begin{rmk}
In fact, Definition \ref{def:Arak} appears in a later work \cite{RR}, where a new proof of Arakelian's theorem was given (see also \cite{Got}). The original conditions in \cite{Ar} are:\\[5pt]
1. $\mathbb{C}\setminus F$ is connected;\\[5pt]
2. $\mathbb{C}\setminus F$ is locally connected at infinity.\\[5pt]
Nevertheless, they are easily seen to be equivalent.
\end{rmk}

\begin{dfn}[Uniform approximation set]
Let $F\subset \mathbb{C}$ be a closed set. We say that $F$ is a set of uniform approximation, if for every $\epsilon>0$ and $f\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\epsilon$, for all $z\in F$.
\end{dfn}

\subsection{Main result}

We can now state the main result:
\begin{thm}\label{thm:main}
A closed set $F\subset \mathbb{C}$ is an Arakelian set, if and only if
for every $f\in A(F)$ with no zeros in $F$, there exists $g\in A(F)$ such that $e^g=f$ in $F$.
\end{thm}
At first glance, the existence of logarithmic branches of functions $f\in A(F)$ might seem to have little to do with the conditions in Definition \ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\supset F$ whose connected components are simply connected.
\begin{thm}\label{thm:F}
A closed set $F\subset \mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets.
\end{thm}
At least one of the directions in Theorem \ref{thm:main} is clearer now. If a function $f\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.

Theorem \ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\subset \mathbb{C}$ \cite{Ar2} and non-planar Riemann surfaces of finite genus \cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \ref{def:Arak}) are not always sets of uniform approximation, see \cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface.

The one direction of the characterization in Theorem \ref{thm:main} is contained in \cite{GP}. We include the proof here for the sake of completeness.
\begin{prp}[Gauthier-Pouryayevali \cite{GP}]\label{prop:GP}
Suppose that the closed set $F\subset \mathbb{C}$ is an Arakelian set. For every function $f\in A(F)$ with no zeros in $F$, there exists a function $g\in A(F)$ such that $e^g=f$ in $F$.
\end{prp}
\begin{proof}
By Tietze's extension theorem, there exists a continuous extension of $f$ in $\mathbb{C}$, which we
denote by $\widetilde{f}:\mathbb{C}\to\C$. Our assumption implies that the open set $U=\mathbb{C}\setminus \widetilde{f}^{-1}(\{0\})$ contains $F$.
Thus, by Theorem \ref{thm:F} there exists a simply connected open set $V$ with $F\subset V\subset U$.
If we consider the covering map $\exp: \mathbb{C}\to\mathbb{C}\setminus\{0\}$, then the latter implies that
$\widetilde{f}_{\big|V}:V\to\mathbb{C}\setminus\{0\}$ can be lifted
to a continuous function $\widetilde{g}:V\to\mathbb{C}$, such that $\widetilde{f}_{\big|V}=e^{\widetilde{g}}$.
The function $g=\widetilde{g}_{\big|F}$ is obviously continuous and $e^g=f$ in $F$.
Since $f_{\big|F^0}$ is holomorphic, by Lemma \ref{lem:Car}, $g$ is also holomorphic in $F^0$.
Thus, $g\in A(F)$ and the proof of the proposition is complete.
\end{proof}
The reverse direction is contained in the following proposition that, combined with Proposition \ref{prop:GP}, completes the proof of Theorem \ref{thm:main}.
\begin{prp}\label{prop:FNP}
Let $F\subset\mathbb{C}$ be a closed set, such that for every $f\in A(F)$ with no zeros in $F$, there exists a $g\in A(F)$ satisfying $e^g=f$ in $F$. Then $F$ is an Arakelian set.
\end{prp}
\begin{proof}
{\it Step 1. $F$ has no holes.} Arguing by contradiction, suppose that $F$ has a hole $B$. Let $\zeta\in B$ and let $f(z)=z-\zeta$. Since $f$ has no zeros in $F$, there exists $g\in A(F)$, such that $e^g=f$ in $F$.

\begin{thm}[Arakelian \cite{Ar}]\label{thm:Ar}
A closed set $F\subset \mathbb{C}$ is a set of uniform approximation if and only if $F$ is an Arakelian set.
\end{thm}

\begin{thm}\label{thm:main}
A closed set $F\subset \mathbb{C}$ is an Arakelian set, if and only if
for  every $f\in A(F)$ with no zeros in $F$, there exists $g\in A(F)$ such that $e^g=f$ in $F$.
\end{thm}

\subsection{Main result}

We can now state the main result:
\begin{thm}\label{thm:main}
A closed set $F\subset \mathbb{C}$ is an Arakelian set, if and only if
for  every $f\in A(F)$ with no zeros in $F$, there exists $g\in A(F)$ such that $e^g=f$ in $F$.
\end{thm}
At first glance, the existence of logarithmic branches of functions $f\in A(F)$ might seem to have little to do with the conditions in Definition \ref{def:Arak}. Intuitively, it is easier to make the connection by first stating yet another characterization of Arakelian sets, proved in \cite{F}, in terms of simply connected neighborhoods of $F$, that is to say, open sets $V\supset F$ whose connected components are simply connected.
\begin{thm}\label{thm:F}
A closed set $F\subset \mathbb{C}$ is an Arakelian set if and only if it possesses a neighborhood basis of simply connected open sets.
\end{thm}
At least one of the directions in Theorem \ref{thm:main} is clearer now. If a function $f\in A(F)$ could be extended to a neighborhood of $F$, then the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter.

We should give credit here to the influential work of Gauthier-Pouryayevali \cite{GP}, which inspired us to pursue the above characterizations. In fact, the directions in both Theorems \ref{thm:main}, \ref{thm:F} where we assume that $F$ is an Arakelian set, are contained in \cite{GP}.

Our contribution in the present paper is to prove the reverse direction stated in Theorem \ref{thm:main}. We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption. Interestingly, the function $f$ that we use is entire and it is produced by invoking the Weierstrass factorization theorem, unlike the meromorphic functions that are usually employed in such proofs \cite[Chapter IV, \S2.$C_2$]{Gaier}.

\subsection{Outlook}

Theorem \ref{thm:Ar} has been successfully extended to relatively closed subsets of planar domains $G\subset \mathbb{C}$ \cite{Ar2} and non-planar Riemann surfaces of finite genus \cite{GH,Sch}. One of the outstanding open problems in complex approximation is to characterize sets of uniform approximation in Riemann surfaces of infinite genus, where Arakelian sets (Definition \ref{def:Arak}) are not always sets of uniform approximation, see \cite{BG} for a counterexample. Moreover, it was shown by Scheinberg \cite{Sch} that such a characterization, if it exists, cannot be purely topological, but it has to take into account the complex analytic structure of the given Riemann surface.

It would be interesting to seek potential extensions of Theorem \ref{thm:main} to Arakelian sets of planar domains $G\subset \mathbb{C}$, as in \cite{Ar2}. The present logarithmic characterization and our method of proof could be naturally extended to Arakelian sets of simply connected domains $G\subset\mathbb{C}$, see \cite[Corollary 2.15]{F} for one of the directions. However, we do not know how it could be generalized to planar domains $G$ with holes, even when $G$ is an annulus. Finally, coming up with a generalization of our characterization that would take into account the analytic structure of a given domain, could hopefully be of use for characterizing uniform approximation sets in Riemann surfaces.

\subsection{Acknowledgments}

We would like to thank Paul M. Gauthier for his interest in this work and insightful comments.
G.F. and S.P. would like to acknowledge the support of the ERC starting grant 101078061 SINGinGR, under the European Union's Horizon Europe program for research and innovation, and the H.F.R.I. grant 7126, under the 3rd call for H.F.R.I. research projects to support post-doctoral researchers.

For the direction assuming $F$ is Arakelian, the text explains that using Theorem~\ref{thm:F} (Arakelian $\Leftrightarrow$ a neighborhood basis of simply connected open sets), “at least one of the directions in Theorem \ref{thm:main} is clearer now”: if $f\in A(F)$ “could be extended to a neighborhood of $F$,” then “the existence of an intermediate simply connected neighborhood would allow for a logarithmic branch of $f$ to be well-defined in the latter,” yielding $g$ with $e^g=f$ on $F$.

For the reverse direction in Theorem~\ref{thm:main}, the authors state: “We argue by contradiction, assuming $F$ is not an Arakelian set. Then a special function $f\in A(F)$ is constructed, which is finally proven not to satisfy the initial assumption.” They add that this $f$ “is entire and it is produced by invoking the Weierstrass factorization theorem,” in contrast to “the meromorphic functions that are usually employed in such proofs.”

Question:
Let $F\subset \mathbb{C}$ be a closed set, and let
\[
A(F)=\{f:F\to\mathbb{C}: f \text{ is holomorphic on } F^\circ \text{ and continuous on } F\}.
\]
Say that $F$ is a set of uniform approximation if for every $\varepsilon>0$ and every $f\in A(F)$, there exists an entire function $g$ such that $|f(z)-g(z)|<\varepsilon$ for all $z\in F$. Which of the following statements is equivalent to $F$ being a set of uniform approximation?

$F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\subset \mathbb{C}$, the union of all holes of $F\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\mathbb{C}$.

Choices:
B. $F$ is an Arakelian set; that is, $F$ has no holes, and there exists a compact set $K\subset \mathbb{C}$ such that the union of all holes of $F\cup K$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\mathbb{C}$.
C. $F$ has no holes; equivalently, $\mathbb{C}\setminus F$ has no bounded connected components.
D. $F$ is an Arakelian set; that is, $\mathbb{C}\setminus F$ is connected and, for every compact set $K\subset \mathbb{C}$, the complement $\mathbb{C}\setminus (F\cup K)$ is connected.
E. $F$ is an Arakelian set; that is, $F$ has no holes, and for every compact set $K\subset \mathbb{C}$, the union of all connected components of $\mathbb{C}\setminus (F\cup K)$ is bounded, where a hole of a closed set means a bounded connected component of its complement in $\mathbb{C}$.

dropped the condition on holes of $F\cup K$ for all compact $K$

replaced boundedness of holes of $F\cup K$ by connectedness of full complement after adjoining $K$

boundedness applies only to holes, not to all components of the complement

The stem defines the approximation property but does not explicitly name or describe the Arakelian characterization. The correct answer is not leaked directly; the test-taker must know or infer the geometric criterion.

This is essentially a theorem-recall item: one side of the equivalence is given in the stem and the other side is to be selected. It is not a verbatim restatement, but it is still a fairly direct reformulation of Arakelian's theorem.

There is some reasoning pressure because the distractors differ by subtle quantifier changes and related topological conditions. However, the item mainly rewards recognition of the exact theorem statement rather than deeper generative reasoning.

The distractors are strong and mathematically plausible: one weakens a universal quantifier to an existential one, one gives a weaker necessary condition, one confuses bounded-hole conditions with connectedness, and one overstates boundedness to all complement components.

A solid theorem-characterization MCQ with good distractors and no obvious answer leakage, but it leans more toward precise recall/discrimination than genuine generative reasoning.


\begin{remark}\label{r:comb-nondegenerate}
Note that
this triangulation, as well as a triangulation from theorem~\ref{th:lambda-to-0}, is combinatorial
(the star of every vertex has simplicial embedding into $\R^n$, so the stars form a $PL$-atlas of~$\SS^n$),
since the join of combinatorial triangulations of spheres is a combinatorial triangulation of a sphere.

Moreover, these triangulations are isomorphic to the boundary of a convex polytope simplicial in $\R^{n+1}$.
Indeed, if $P\subset\R^{k+1}$ and $Q\subset\R^{m+1}$ are convex polytopes containing the origins in their interiors,
then the join $S=\partial P\star\partial Q$ is canonically embedded into $\R^{k+m+2}=\R^{k+1}\times\R^{m+1}$.
It remains to notice that $S$ is the boundary of the cone over $S$ with the vertex at the origin
(which in fact coincides with $\mathrm{conv}(S)\subset\R^{k+m+2}$).
\end{remark}

\label{s:introduction}

Suppose $f:M\to N$ is a continuous map of closed connected oriented $n$-manifolds, $n\ge1$.
The {\it degree} $\deg f$ is an integer $d$ such that the $n$-th homology homomorphism
$f_*:H_n(M;\mathbb{Z})=\mathbb{Z}\to\mathbb{Z}=H_n(N;\mathbb{Z})$ is the multiplication by $d$.
If the map $f$ is smooth, then $\deg f$ equals the number of preimages of any regular value of $f$,
where the points at which $df$ preserves the orientation are counted as ``$1$'',
and the points at which $df$ reverses the orientation are counted as ``$-1$''.
If $f$ is a simplicial map of triangulated manifolds,
then the degree can be counted in a similar way via preimages of an $n$-simplex.

Namely, we call an $n$-simplex $\sigma\subset M$ {\it positive/negative},
if $f(\sigma)\subset N$ is non-degenerate $n$-simplex and $f|_\sigma$ preserves/reverses orientation.
Then $\deg f$ equals the number of positive perimages minus the number of negative preimages of any $n$-simplex $\tau\subset N$.
This difference does not depend on $\tau$.

Let $\SS_{n+2}^n$ be the standard triangulation of the $n$-sphere with $n+2$ vertices
and suppose $K$ is a simplicial complex such that the geometric realisation $|K|$ is homeomorphic to the sphere~$\mathbb{S}^n$.
Suppose there is a simplicial map $K\to\mathbb{S}^n_{n+2}$ of degree $d$.
One may ask: {\it given $n$ and $d$, how few vertices can $K$ have?}
Denote the minimal number of vertices of such $K$ by $\lambda(n,d)$.

Evidently, if $d=0$ or $d=\pm1$, we have $\lambda(n,d)=n+2$,
since a triangulation of the $n$-sphere cannot have fewer vertices.
It is also easy to see that $\lambda(1,d)=3d$.
For $n=2$ and $d\ge3$ it is known by \cite{sarkaria} that $\lambda(2,d)=2d+2$
(the lower bound for $\lambda(2,d)$ obviously follows from the Euler's formula:
$K$ has at least $4d$ faces, so the number of vertices is at least $2-6d+4d$).

In \cite{musin-deg} K.\,Apolonskaya and O.\,Musin
conjectured that $\lim\sup_{d\to\infty}\frac{\lambda(n,d)}d=\frac{n+2}n$ for all $n>0$.
We disprove this conjecture:

\begin{abstract}
For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere
which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex.
We show that $\lim_{d\to\infty}\frac{\lambda(n,d)}d=0$ for any $n\ge3$,
disproving O.\,Musin's conjecture.
Using similar idea, for any $C$ we construct a triangulation of $\SS^n$, $n\ge3$,
for which $\frac{f_j}{f_i}>C$,
for any $0\le i<j\le n$ such that $i<\lfloor\frac{n-1}2\rfloor$.
All triangulations we obtain are isomorphic to boundaries of convex polytopes in $\R^{n+1}$.
\end{abstract}

In \cite{musin-deg} K.\,Apolonskaya and O.\,Musin
conjectured that $\lim\sup_{d\to\infty}\frac{\lambda(n,d)}d=\frac{n+2}n$ for all $n>0$.
We disprove this conjecture:

The key idea of the proof of theorem~\ref{th:lambda-to-0}
is to construct a triangulation of $\SS^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.
This idea can also be applied to show the following fact.

\begin{theorem}\label{th:f-vectors}
Take any $n\ge3$, and any $C>0$.
Then there is a triangulation of $\SS^n$ for which $\frac{f_j}{f_i}>C$
for any $0\le i<j\le n$ such that $i<\lfloor\frac {n-1}2\rfloor$.
\end{theorem}

\begin{proof}[Proof of theorem~\ref{th:lambda-to-0}]
Let us first prove the case $n>3$.
By corollary~\ref{c:n-est}, for any $k_1,k_2>0$ we have
$$\frac{\lambda(n,k_1k_2)}{k_1k_2}\le\frac{3k_1+3k_2+n-2}{k_1k_2}.$$
In particular, for any $k>0$, the value
$$\frac{\lambda(n,k^2)}{k^2}\le\frac{6k+n-2}{k^2}$$
is arbitrary small.
Namely, take any $\eps>0$.
If $k^2>4\cdot\frac{n-2}\eps$ and $k>\frac{24}\eps$,
then
$$\frac{\lambda(n,k^2)}{k^2}\le\frac{6k+n-2}{k^2}=\frac{n-2}{k^2}+\frac6k<\textstyle\frac24\eps.$$
Further, note that for $l=0,1,\ldots,k-1$ we have
$$
\frac{\lambda(n,k(k+l))}{k(k+l)}
\le
\frac{\lambda(n,k^2)}{k(k+l)}+\frac{3l}{k(k+l)}
\le
\frac{\lambda(n,k^2)}{k^2}+\frac{3}{k}
<\textstyle\frac34\eps.
$$
Finally, for any $s=0,1,\ldots,k-1$
we can apply proposition~\ref{pr:d+1} $s$ times, to obtain
$$
\frac{\lambda(n,k(k+l)+s)}{k(k+l)+s}
\le
\frac{\lambda(n,k(k+l))}{k(k+l)}+\frac{3s}{k(k+1)}
<\eps.
$$
We have shown that $\frac{\lambda(n,d)}d<\eps$ for every $d>\max\{4,\frac{n-2}\eps,\frac{24}\eps\}$.

\begin{remark}
In order to prove theorem~\ref{th:lambda-to-0},
one can use the join of $h$ copies of $\SS^1$, for $h=\lfloor\frac{n+1}2\rfloor$
(as in the proof of theorem~\ref{th:f-vectors}), with the additional join with $\SS^0$ for even $n$.
In this way one can show that $\lim_{d\to\infty}\frac{\lambda(n,k)^{h-1}}d=0$.
\end{remark}

We have estimated the asymptotic of $\lambda(n,d)$ as $d\to\infty$.
However, finding its exact values appears to be a difficult problem.
The estimates for $\lambda(n,d)$ obtained in the proof of theorem~\ref{th:lambda-to-0}
are apparently far from the minimum.

\begin{question}
Is it true that $\lim\sup_{d\to\infty}\frac{\lambda(n,d)^{\lfloor\frac{n+1}2\rfloor}}{d}\ne0$?
\end{question}

\begin{theorem}\label{th:lambda-to-0}
For any $n\ge3$ we have $\lim\limits_{d\to\infty}\frac{\lambda(n,d)}d=0$.
\end{theorem}

The key idea of the proof of theorem~\ref{th:lambda-to-0}
is to construct a triangulation of $\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.
This idea can also be applied to show the following fact.

For a finite simplicial complex $K$,
its {\it $f$-vector} is the sequence $f_0,f_1,\ldots$,
where $f_k$ is the number of $k$-simplices in $K$.

\begin{theorem}\label{th:f-vectors}
Take any $n\ge3$, and any $C>0$.
Then there is a triangulation of $\mathbb{S}^n$ for which $\frac{f_j}{f_i}>C$
for any $0\le i<j\le n$ such that $i<\lfloor\frac {n-1}2\rfloor$.
\end{theorem}

Note that the estimations from theorems~\ref{th:lambda-to-0} and \ref{th:f-vectors}
hold even if we
consider
only {\it combinatorial} triangulations.
Moreover, we can consider only triangulations which are {\it isomorphic to the boundary of a convex simplicial polytope in $\mathbb{R}^{n+1}$}.
See remark~\ref{r:comb-nondegenerate}.

The key idea in establishing the main theorem is to construct a triangulation of $\mathbb{S}^n$ which has quite a lot of $n$-simplices with respect to the number of its vertices.,

Question: For positive integers n and d, let λ(n,d) denote the minimal number of vertices of a triangulation K of the n-sphere S^n such that K admits a degree-d simplicial map to the standard n-sphere S^n_{n+2}, equivalently to the boundary of an (n+1)-simplex. Which statement holds for every integer n \ge 3?

For every n \ge 3, \(\lim_{d\to\infty} \frac{\lambda(n,d)}{d} = 0\).

Choices:
B. For every \(n \ge 3\), there exists a constant \(c_n>0\) such that for all sufficiently large \(d\), \(\frac{\lambda(n,d)}{d} \ge c_n\).
C. For every \(n \ge 3\), there exists a sequence \(d_m\to\infty\) such that \(\lim_{m\to\infty} \frac{\lambda(n,d_m)}{d_m}=0\).
D. There exists a constant \(C>0\) such that for every \(n \ge 3\) and all sufficiently large \(d\), \(\frac{\lambda(n,d)}{d} \le C\).
E. For every \(n \ge 3\), there exists an integer \(d_0=d_0(n)\) such that for every \(d\ge d_0\) one has \(\lambda(n,d)=o\!\left(d\right)\) uniformly in \(d\).

sublinear-vanishing asymptotic replaced by positive linear lower bound

full limit over all d weakened to convergence along a subsequence

vanishing ratio weakened to merely bounded ratio with uniform constant

misstated asymptotic notion by imposing eventual equality with an o(d) condition

The stem does not state or strongly hint at the asymptotic conclusion; it only defines the quantity and asks which limiting statement is true.

The item appears to ask almost exactly for the theorem's main asymptotic conclusion, with choice A giving the direct statement rather than requiring application in a new setting.

Some reasoning is needed to distinguish a full limit from weaker or altered asymptotic claims (e.g. liminf, limsup, positive constant, uniform-in-n bound), but the task is still largely theorem recognition rather than genuine derivation.

The distractors are mathematically plausible and distinct: they test common confusions between limit/liminf/limsup, zero versus positive asymptotic rate, and pointwise-in-n versus uniform-in-n statements.

A solid theorem-recognition MCQ with good distractors and no answer leakage, but it is fairly tautological and only moderately tests generative reasoning.


Question: For positive integers n and d, let λ(n,d) denote the minimal number of vertices of a triangulation K of the n-sphere S^n such that K admits a degree-d simplicial map to the standard n-sphere S^n_{n+2}, equivalently to the boundary of an (n+1)-simplex. Which statement holds for every integer n \ge 3?

Choices:
B. For every \(n \ge 3\), there exists a constant \(c_n>0\) such that for all sufficiently large \(d\), \(\frac{\lambda(n,d)}{d} \ge c_n\).
C. For every \(n \ge 3\), there exists a sequence \(d_m\to\infty\) such that \(\lim_{m\to\infty} \frac{\lambda(n,d_m)}{d_m}=0\).
D. There exists a constant \(C>0\) such that for every \(n \ge 3\) and all sufficiently large \(d\), \(\frac{\lambda(n,d)}{d} \le C\).
E. For every \(n \ge 3\), there exists an integer \(d_0=d_0(n)\) such that for every \(d\ge d_0\) one has \(\lambda(n,d)=o\!\left(d\right)\) uniformly in \(d\).

The stem only defines λ(n,d) and asks which asymptotic statement is true; it does not explicitly or implicitly reveal option A.

The item is not a bare restatement in the stem itself; the student must distinguish among several asymptotic formulations with different quantifiers and strengths.

There is some reasoning pressure because the options differ subtly in limit strength, subsequences, and uniformity. However, this is weakened because option C is a logically weaker consequence of A, so the item does not cleanly force a unique strongest conclusion.

Distractor quality is poor overall: C is also true if A is true, so the question is not genuinely single-correct; E is mathematically malformed/awkward ('λ(n,d)=o(d)' as an eventual equality statement, 'uniformly in d'), which makes it less plausible as a serious distractor.

The question avoids answer leakage and has some asymptotic reasoning content, but it is fundamentally flawed as an MCQ because at least one distractor is also true and another is poorly formulated.


[Dynamical Non-Commutative Bifurcation]\label{thm:main_bifurcation}
Let $P_0(x)$ be a central polynomial over $\mathbb{A} \in \{\mathbb{H}, \mathbb{O}\}$ such that its root variety $V_0 = Z(P_0)$ contains a non-real manifold of dimension $d_{\rm M} > 0$. Let $P_\epsilon(x)$ be a generically non-central analytic deformation. Let $V_\epsilon = Z(P_\epsilon)$. Let $d_{\rm A} = \dim(\mathbb{A})$.
\begin{enumerate}
    \item \textbf{Topological Collapse:} The Hausdorff dimension of the root variety changes discontinuously at $\epsilon=0$.
    \item \textbf{Algebraic Singularity:} The central fiber $V_0$ is characterized by a Jacobian rank deficiency: $\rank(J_{P_0}(x)) = d_{\rm A} - d_{\rm M}$.
    \item \textbf{Dynamical Retraction:} Let $\mathcal{V}_\epsilon(x) = \|P_\epsilon(x)\|^2$. The gradient flow $\dot{x} = -\nabla\mathcal{V}_\epsilon(x)$ realizes the collapse as a deformation retract. The timescale exhibits critical slowing down: $T_{\rm collapse} \propto \epsilon^{-2}$.
    \item \textbf{Thermodynamic Phase Transition:} In a statistical ensemble at temperature $T$, the collapse manifests as a phase transition. An alignment order parameter $m(\epsilon,T)$ exhibits a discontinuity at $(\epsilon, T) = (0, 0)$.
\end{enumerate}

The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\mathbb{H}$) by Niven \cite{niven1941equations} and Eilenberg and Niven \cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\mathbb{R}$), 0-dimensional sets in $\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\mathbb{H}$, $S^6$ in $\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.

This paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.

The motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \cite{baez2002octonions}.

Our main original contributions are:
\begin{enumerate}
    \item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).
    \item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.
    \item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.
\end{enumerate}

\subsection{Main Results}

The central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.

\begin{definition}[Analytic Deformation]\label{def:analytic_deformation}
Let $\mathbb{A}$ be a real division algebra. Let $P_0(x) \in \mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\epsilon(x) \in \mathbb{A}[x]$ parameterized by $\epsilon \in \mathbb{R}$, such that the coefficients depend analytically on $\epsilon$, $P_\epsilon(x) = P_0(x)$ when $\epsilon=0$, and $G_{P_\epsilon}$ is trivial for $\epsilon \neq 0$ in a punctured neighborhood of $\epsilon=0$.
\end{definition}

The study of polynomial roots over non-commutative division algebras has a rich history, dating back to the foundational work on quaternions ($\mathbb{H}$) by Niven \cite{niven1941equations} and Eilenberg and Niven \cite{eilenberg1944fundamental}. A key observation, formalized by Gordon and Motzkin \cite{gordon1965zeros}, is that the algebraic structure of the underlying space profoundly affects the geometry of the solution set (see also \cite{lam2001first}). The failure of commutativity (and associativity in the octonions $\mathbb{O}$) induces a dimensional inflation of the root set. For central polynomials (coefficients in $\mathbb{R}$), 0-dimensional sets in $\mathbb{C}$ inflate into continuous manifolds ($S^2$ in $\mathbb{H}$, $S^6$ in $\mathbb{O}$). The associated symmetry group transitions from a discrete group to a compact Lie group ($SO(3)$ or $G_2$), arising from the action of the automorphism group.

This paper introduces a dynamical framework, Dynamical Non-Commutative Algebraic Geometry (DNCAG), where we treat the polynomial coefficients as parameters. We move beyond the static characterization of root sets to analyze their evolution, stability, and bifurcations. While the underlying algebraic structures are established, our approach utilizes tools from dynamical systems (cf. \cite{guckenheimer1983nonlinear}), algebraic deformation theory (cf. \cite{gerstenhaber1964deformation}), and Morse-Bott theory (cf. \cite{bott1954nondegenerate}) to analyze the stability of these structures under perturbations.

The motivation for developing DNCAG extends beyond pure mathematics into theoretical physics. The geometric structures analyzed here—spherical manifolds arising from algebraic constraints—are ubiquitous. In quaternionic quantum mechanics \cite{adler1995quaternionic}, these manifolds may represent parameter spaces or emergent symmetries. The dynamical collapse of topology offers a novel algebraic model for phenomena involving dimensional reduction and symmetry breaking, such as those hypothesized near black hole singularities or characterizing transitions between topological phases of matter. Furthermore, the extension to octonions is motivated by their potential role in unified theories and their unique algebraic properties \cite{baez2002octonions}.

Our main original contributions are:
\begin{enumerate}
 \item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).
 \item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.
 \item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.
\end{enumerate}

The central contribution of this paper is the characterization of the topological instability induced by non-central perturbations across division algebras.

\begin{definition}[Analytic Deformation]\label{def:analytic_deformation}
Let $\mathbb{A}$ be a real division algebra. Let $P_0(x) \in \mathbb{R}[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\epsilon(x) \in \mathbb{A}[x]$ parameterized by $\epsilon \in \mathbb{R}$, such that the coefficients depend analytically on $\epsilon$, $P_\epsilon(x) = P_0(x)$ when $\epsilon=0$, and $G_{P_\epsilon}$ is trivial for $\epsilon \neq 0$ in a punctured neighborhood of $\epsilon=0$.
\end{definition}

\begin{definition}[Automorphism Stabilizer of the Polynomial]\label{def:aut_stabilizer}
Let $P(x) \in \A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\Aut(\A)$ that fixes all coefficients:
\begin{equation}
G_P = \{g \in \Aut(\A) \mid g(a_k) = a_k \text{ for all } k\}.
\end{equation}
\end{definition}

\abstract{We develop a framework for dynamical non-commutative algebraic geometry (DNCAG) by analyzing the evolution and stability of polynomial root manifolds in real normed division algebras ($\HH$ and $\OO$). We establish a Generalized Inflation Theorem, demonstrating that for central polynomials, the root set forms a homogeneous space $G/H$, where $G$ is the automorphism group of the algebra ($SO(3)$ for $\HH$, $G_2$ for $\OO$). This mechanism generates continuous geometry from non-commutativity. We analyze the dynamics under central modulation (breathing modes), classifying topological bifurcations ($\Delta=0$). We then analyze the topological collapse induced by non-central perturbations, governed by symmetry reduction. We utilize the Localization Theorem (Gordon-Motzkin) to explain the alignment of roots with coefficient subalgebras. We formalize the dynamics of collapse using gradient flow on the potential landscape $\mathcal{V}(x) = \|P(x)\|^2$, characterizing it as a deformation retract and proving that the collapse timescale exhibits critical slowing down with quadratic scaling ($T_{\rm collapse} \propto \epsilon^{-2}$). Finally, we introduce a thermodynamic formalism, proving an Entropy Scaling Law that rigorously characterizes the collapse as a symmetry-breaking phase transition.}

Our main original contributions are:
\begin{enumerate}
 \item The analysis of central dynamics (breathing modes), including spectral characterization of non-linear coupling and the classification of dynamics near central bifurcations (transversal vs. tangential crossings).
 \item The formalization of the topological collapse dynamics using gradient flow, including the proof of critical slowing down and the analysis of basin decomposition.
 \item The introduction of a thermodynamic formalism (Gibbs measure, Entropy Scaling Law, Order Parameter) to characterize the algebraic collapse as a statistical phase transition.
\end{enumerate}

\begin{definition}[Analytic Deformation]\label{def:analytic_deformation}
Let $\A$ be a real division algebra. Let $P_0(x) \in \R[x]$ be a central polynomial. Let $G_P$ be the automorphism stabilizer of $P$ (Definition~\ref{def:aut_stabilizer}). A generically non-central analytic deformation is a family of polynomials $P_\epsilon(x) \in \A[x]$ parameterized by $\epsilon \in \R$, such that the coefficients depend analytically on $\epsilon$, $P_\epsilon(x) = P_0(x)$ when $\epsilon=0$, and $G_{P_\epsilon}$ is trivial for $\epsilon \neq 0$ in a punctured neighborhood of $\epsilon=0$.
\end{definition}

\begin{proof}
We synthesize the results established in the subsequent sections.

\begin{proposition}[Topological Collapse]\label{prop:collapse}
For a generically non-central perturbation $P_\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\epsilon)$ changes discontinuously at $\epsilon=0$. (Part 1 of Main Theorem~\ref{thm:main_bifurcation}).
\end{proposition}

\begin{figure}[htbp]
 \centering
 \includegraphics[width=\textwidth]{topological_collapse.pdf}
 \caption{Visualization of Topological Collapse via Gradient Flow (in $\HH$). The sequence illustrates the transition from $S^2$ to $S^0$ under a non-central perturbation aligned with the $i$-axis. (1) Initial state $V_0$. (2-4) Evolution under the gradient flow $-\nabla\mathcal{V}_\epsilon$. (5) Final state $V_\epsilon$: The manifold has collapsed onto the isolated roots (Alignment Principle). An animation of the collapse is shown in \url{http://youtu.be/yaJQOuftZjE}}
 \label{fig:topological_collapse}
\end{figure}

\begin{theorem}[Symmetry Breaking and Phase Transition]\label{thm:phase_transition}
Consider a perturbation $P_\epsilon(x)$ breaking the symmetry along the $\nu$-axis. We analyze the behavior in the zero temperature limit ($T \to 0$). (Part 4 of Main Theorem~\ref{thm:main_bifurcation}).
\begin{enumerate}
 \item \textbf{Symmetric Phase (Disordered):} For $\epsilon=0$, the measure concentrates uniformly on $S^{d_{\rm A}-2}$. $m(0, 0) = 1/(d_{\rm A}-1)$.
 \item \textbf{Ordered Phase (Aligned):} For $\epsilon \neq 0$, the measure concentrates on the isolated roots aligned with the $\nu$-axis. $m(\epsilon, 0) = 1$.
\end{enumerate}
\end{theorem}

The analysis reveals a unified picture across the real normed division algebras:
\begin{enumerate}
 \item The geometry of central roots is determined by the automorphism group of the algebra (Generalized Inflation Theorem).
 \item Central dynamics are governed by the auxiliary polynomial and its discriminant, independent of associativity (due to power-associativity).
 \item Topological collapse is driven by symmetry reduction (Generalized Symmetry Reduction Theorem) and the Alignment Principle (Localization Theorem), which holds due to alternativity.
 \item The dynamics of collapse follow a gradient flow characterized by Critical Slowing Down ($T \propto \epsilon^{-2}$).
 \item The collapse can be rigorously characterized as a thermodynamic phase transition (Entropy Scaling Law and Order Parameter analysis).
\end{enumerate}

\begin{definition}[Automorphism Stabilizer of the Polynomial]\label{def:aut_stabilizer}
Let $P(x) \in \A[x]$. The automorphism stabilizer of $P$ is the subgroup of $\Aut(\A)$ that fixes all coefficients:
\begin{equation}
G_P = \{g \in \Aut(\A) \mid g(a_k) = a_k \text{ for all } k\}.
\end{equation}
\end{definition}

\begin{proof}
We synthesize the results established in the subsequent sections.

\textbf{Part 1 (Topological Collapse):} For $\epsilon=0$, by the Dimensional Inflation Law (Corollary~\ref{cor:inflation_law}), $d_{\rm M} = d_{\rm A} - 2$. For $\epsilon \neq 0$, $G_{P_\epsilon}$ is trivial. By the Generalized Symmetry Reduction Theorem (Theorem~\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\rm M}=0$). The discontinuity follows (Proposition~\ref{prop:collapse}).

\textbf{Part 2 (Algebraic Singularity):} The tangent space $T_x V_0$ is related to $\Ker(J_{P_0}(x))$. The rank deficiency follows from the positive dimension of the manifold (Theorem~\ref{thm:jacobian_singularity}).

\textbf{Part 3 (Dynamical Retraction):} The potential $\mathcal{V}_\epsilon(x)$ is real-analytic and coercive. The Lojasiewicz gradient inequality \cite{lojasiewicz1963propriete} guarantees convergence of the gradient flow (Theorem~\ref{thm:deformation_retract}). The timescale scaling $T_{\rm collapse} \propto \epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\ref{prop:lojasiewicz_exponent}).

\textbf{Part 4 (Thermodynamic Phase Transition):} We define the Gibbs measure (Definition~\ref{def:gibbs_measure}) and the alignment order parameter $m(\epsilon, T)$ (Definition~\ref{def:order_parameter}). In the limit $T \to 0$, the measure concentrates on $V_\epsilon$. For $\epsilon=0$, symmetry implies $m(0, 0) = 1/(d_{\rm A}-1)$. For $\epsilon \neq 0$, the Localization Theorem (Theorem~\ref{thm:algebraic_alignment}) implies alignment, $m(\epsilon, 0) = 1$. The discontinuity characterizes the phase transition (Theorem~\ref{thm:phase_transition}).
\end{proof}

To prove Theorem~\ref{thm:main_bifurcation} the proof \"synthesizes the results established in the subsequent sections\" and proceeds by parts:

\textbf{Part 1 (Topological Collapse):} For $\epsilon=0$, \"by the Dimensional Inflation Law (Corollary~\ref{cor:inflation_law}), $d_{\rm M} = d_{\rm A} - 2$.\" For $\epsilon\neq 0$, \"$G_{P_\epsilon}$ is trivial\" and \"by the Generalized Symmetry Reduction Theorem (Theorem~\ref{thm:generalized_symmetry_reduction}), the roots are isolated ($d_{\rm M}=0$).\" Hence \"the discontinuity follows (Proposition~\ref{prop:collapse}).\"

\textbf{Part 2 (Algebraic Singularity):} \"The tangent space $T_x V_0$ is related to $\Ker(J_{P_0}(x))$.\" Therefore \"the rank deficiency follows from the positive dimension of the manifold (Theorem~\ref{thm:jacobian_singularity}).\"

\textbf{Part 3 (Dynamical Retraction):} \"The potential $\mathcal{V}_\epsilon(x)$ is real-analytic and coercive.\" Then \"the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow (Theorem~\ref{thm:deformation_retract}).\" The scaling \"$T_{\rm collapse} \propto \epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential (Theorem~\ref{thm:collapse_timescale}) and the generic Lojasiewicz exponent (Proposition~\ref{prop:lojasiewicz_exponent}).\"

\textbf{Part 4 (Thermodynamic Phase Transition):} \"We define the Gibbs measure (Definition~\ref{def:gibbs_measure}) and the alignment order parameter $m(\epsilon, T)$ (Definition~\ref{def:order_parameter}).\" As \"$T\to 0$, the measure concentrates on $V_\epsilon$.\" For $\epsilon=0$, \"symmetry implies $m(0, 0) = 1/(d_{\rm A}-1)$.\" For $\epsilon\neq 0$, \"the Localization Theorem (Theorem~\ref{thm:algebraic_alignment}) implies alignment, $m(\epsilon, 0) = 1$.\" \"The discontinuity characterizes the phase transition (Theorem~\ref{thm:phase_transition}).\"

To prove the main theorem, the proof "synthesizes the results established in the subsequent sections" and proceeds by parts:

\textbf{Part 1 (Topological Collapse):} For $\epsilon=0$, by the following result.

\begin{corollary}[Dimensional Inflation Law]\label{cor:inflation_law}
Let $\A$ be a real normed division algebra ($\C, \HH, \OO$) of dimension $d_{\rm A} \ge 2$. The dimension of the root manifold $M_{x_0}$ is $d_{\rm M} = d_{\rm A} - 2$.
\end{corollary}

For $\epsilon\neq 0$, "$G_{P_\epsilon}$ is trivial" and by the following theorem, the roots are isolated ($d_{\rm M}=0$).

\begin{theorem}[Generalized Symmetry Reduction Theorem]\label{thm:generalized_symmetry_reduction}
The root set $Z(P)$ is invariant under the action of $G_P$. The geometry of the root manifolds is determined by the orbits of $G_P$.
\end{theorem}

Hence the discontinuity follows from the following proposition.

\begin{proposition}[Topological Collapse]\label{prop:collapse}
For a generically non-central perturbation $P_\epsilon(x)$, the root manifold undergoes a topological phase transition. The Hausdorff dimension of the solution set $Z(P_\epsilon)$ changes discontinuously at $\epsilon=0$. (Part 1 of Main Theorem~\ref{thm:main_bifurcation}).
\end{proposition}

\textbf{Part 2 (Algebraic Singularity):} "The tangent space $T_x V_0$ is related to $\Ker(J_{P_0}(x))$." Therefore "the rank deficiency follows from the positive dimension of the manifold" by the following theorem.

\begin{theorem}[Jacobian Singularity of the Central Fiber]\label{thm:jacobian_singularity}
Let $V_0$ be the root manifold of a central polynomial $P_0$. The Jacobian $J_{P_0}(x)$ (viewed as a map $\R^{d_{\rm A}} \to \R^{d_{\rm A}}$) is singular for all $x \in V_0$. (Part 2 of Main Theorem~\ref{thm:main_bifurcation}).
\end{theorem}

\textbf{Part 3 (Dynamical Retraction):} "The potential $\mathcal{V}_\epsilon(x)$ is real-analytic and coercive." Then "the Lojasiewicz gradient inequality ... guarantees convergence of the gradient flow" via the following theorem.

\begin{theorem}[Deformation Retract]\label{thm:deformation_retract}
The gradient flow defines a deformation retract from the initial manifold $V_0$ onto the final set of attractors $V_\epsilon$. (Part 3 of Main Theorem~\ref{thm:main_bifurcation}).
\end{theorem}

The scaling "$T_{\rm collapse} \propto \epsilon^{-2}$ is derived from the quadratic scaling of the restricted potential" using the following theorem and proposition.

\begin{theorem}[Critical Slowing Down (Quadratic Scaling)]\label{thm:collapse_timescale}
The characteristic relaxation time $T_{\rm collapse}$ for the topological collapse under a generic non-central perturbation of magnitude $\epsilon$ scales as $T_{\rm collapse} \propto 1/\epsilon^2$.
\end{theorem}

\begin{proposition}[Lojasiewicz Exponent]\label{prop:lojasiewicz_exponent}
For a generic analytic perturbation (Morse function $f_\epsilon$), the Lojasiewicz exponent characterizing the flatness of the potential near the critical manifold $V_0$ is $\theta = 1/2$. This yields the quadratic timescale $T_{\rm collapse} \propto \epsilon^{-2}$. At non-generic perturbations (where the restricted potential is flatter than quadratic), $\theta$ may be smaller ($0 < \theta < 1/2$), leading to slower collapse timescales $T_{\rm collapse} \propto \epsilon^{-(1+\theta)/\theta}$.
\end{proposition}

\textbf{Part 4 (Thermodynamic Phase Transition):} "We define the Gibbs measure" as follows.

\begin{definition}[Gibbs Measure and Partition Function]\label{def:gibbs_measure}
The Gibbs probability measure $\mu_T$ on $\A$ is defined by
\begin{equation}
d\mu_T(x) = \frac{1}{Z(T)} \exp\left(-\frac{\mathcal{V}(x)}{T}\right) dx,
\end{equation}
where $Z(T) = \int_{\A} \exp(-\mathcal{V}(x)/T) dx$ is the partition function.
\end{definition}

and "the alignment order parameter $m(\epsilon, T)$" as follows.

\begin{definition}[Alignment Order Parameter]\label{def:order_parameter}
Let $\nu$ be a unit imaginary vector defining the perturbation axis. The alignment order parameter $m_\nu$ is the normalized mean square alignment of the coordinate along $\nu$ under the Gibbs measure. For $\HH$ ($d_{\rm A}=4$) aligned along $i$ (coordinate $b$):
\begin{equation}
m(\epsilon, T) = \frac{\langle b^2 \rangle_{\mu_T(\epsilon)}}{\langle b^2+c^2+d^2 \rangle_{\mu_T(\epsilon)}}.
\end{equation}
\end{definition}

As "$T\to 0$, the measure concentrates on $V_\epsilon$." For $\epsilon=0$, "symmetry implies $m(0, 0) = 1/(d_{\rm A}-1)$." For $\epsilon\neq 0$, the following theorem implies alignment, $m(\epsilon, 0) = 1$.

\begin{theorem}[Localization Theorem (Gordon-Motzkin)]\label{thm:algebraic_alignment}
Let $\A$ be a real normed division algebra ($\C, \HH, \OO$). Let $P(x) \in \A[x]$. Any isolated root $x_0$ of $P(x)$ belongs to the coefficient subalgebra $\A(P)$.
\end{theorem}

The discontinuity characterizes the phase transition by the following theorem.

\begin{theorem}[Symmetry Breaking and Phase Transition]\label{thm:phase_transition}
Consider a perturbation $P_\epsilon(x)$ breaking the symmetry along the $\nu$-axis. We analyze the behavior in the zero temperature limit ($T \to 0$). (Part 4 of Main Theorem~\ref{thm:main_bifurcation}).
\begin{enumerate}
    \item \textbf{Symmetric Phase (Disordered):} For $\epsilon=0$, the measure concentrates uniformly on $S^{d_{\rm A}-2}$. $m(0, 0) = 1/(d_{\rm A}-1)$.
    \item \textbf{Ordered Phase (Aligned):} For $\epsilon \neq 0$, the measure concentrates on the isolated roots aligned with the $\nu$-axis. $m(\epsilon, 0) = 1$.
\end{enumerate}
\end{theorem}

Question: Let \(\mathbb{A}\in\{\mathbb{H},\mathbb{O}\}\) and let \(d_{\rm A}=\dim(\mathbb{A})\). Suppose \(P_0(x)\) is a central polynomial over \(\mathbb{A}\), meaning its coefficients lie in \(\mathbb{R}\), and let its root variety be \(V_0=Z(P_0)\). Assume \(V_0\) contains a non-real manifold of dimension \(d_{\rm M}>0\). Let \(P_\epsilon(x)\in \mathbb{A}[x]\) be a generically non-central analytic deformation of \(P_0\), meaning that the coefficients of \(P_\epsilon\) depend analytically on \(\epsilon\in\mathbb{R}\), one has \(P_\epsilon=P_0\) at \(\epsilon=0\), and for \(\epsilon\neq 0\) in some punctured neighborhood of \(0\), the automorphism stabilizer \(G_{P_\epsilon}=\{g\in \operatorname{Aut}(\mathbb{A})\mid g(a_k)=a_k\text{ for all coefficients }a_k\}\) is trivial. Write \(V_\epsilon=Z(P_\epsilon)\), and let \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\). Which statement holds for every such choice of \(\mathbb{A}\), \(P_0\), and deformation \(P_\epsilon\)?

All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).

Choices:
B. All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\) for every analytic deformation; and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).
C. The Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\), the central fiber \(V_0\) has Jacobian rank deficiency, and the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract.
D. All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at every point of the line \(T=0\).
E. All of the following hold: (1) for \(\epsilon\neq 0\) sufficiently small, the root variety \(V_\epsilon\) still contains a non-real manifold of dimension \(d_{\rm A}-2\), so the Hausdorff dimension varies continuously through \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).

The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the resulting asymptotic statement, but the decisive claims in A (dimension jump, exact rank formula, epsilon^{-2} scaling, discontinuous order parameter) are not stated in the prompt itself.

This is essentially a theorem-statement recognition item: the correct choice reproduces a full bundled conclusion under the given hypotheses. The task is mainly to identify the exact theorem-like conclusion rather than reason through a genuinely new situation.

The item does require some discrimination among subtle variants (continuity vs discontinuity, exact vs weaker Jacobian claim, deformation-dependent vs uniform constant), so the answer is not obvious. However, the pressure is mostly on recalling or matching a theorem rather than generating the conclusion from first principles.

The distractors are strong and mathematically meaningful. They reflect common failure modes: reversing continuity behavior, giving a weaker true statement, overclaiming uniformity in constants, or inserting unjustified quantitative/statistical assertions.

Technically sophisticated item with strong distractors and no clear answer leakage, but it functions largely as theorem-recognition rather than a non-tautological test of generative mathematical reasoning.


Question: Let \(\mathbb{A}\in\{\mathbb{H},\mathbb{O}\}\) and let \(d_{\rm A}=\dim(\mathbb{A})\). Suppose \(P_0(x)\) is a central polynomial over \(\mathbb{A}\), meaning its coefficients lie in \(\mathbb{R}\), and let its root variety be \(V_0=Z(P_0)\). Assume \(V_0\) contains a non-real manifold of dimension \(d_{\rm M}>0\). Let \(P_\epsilon(x)\in \mathbb{A}[x]\) be a generically non-central analytic deformation of \(P_0\), meaning that the coefficients of \(P_\epsilon\) depend analytically on \(\epsilon\in\mathbb{R}\), one has \(P_\epsilon=P_0\) at \(\epsilon=0\), and for \(\epsilon\neq 0\) in some punctured neighborhood of \(0\), the automorphism stabilizer \(G_{P_\epsilon}=\{g\in \operatorname{Aut}(\mathbb{A})\mid g(a_k)=a_k\text{ for all coefficients }a_k\}\) is trivial. Write \(V_\epsilon=Z(P_\epsilon)\), and let \(\mathcal{V}_\epsilon(x)=\|P_\epsilon(x)\|^2\). Which statement holds for every such choice of \(\mathbb{A}\), \(P_0\), and deformation \(P_\epsilon\)?

Choices:
B. All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\) for every analytic deformation; and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).
C. The Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\), the central fiber \(V_0\) has Jacobian rank deficiency, and the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract.
D. All of the following hold: (1) the Hausdorff dimension of the root variety changes discontinuously at \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at every point of the line \(T=0\).
E. All of the following hold: (1) for \(\epsilon\neq 0\) sufficiently small, the root variety \(V_\epsilon\) still contains a non-real manifold of dimension \(d_{\rm A}-2\), so the Hausdorff dimension varies continuously through \(\epsilon=0\); (2) the central fiber \(V_0\) is characterized by a Jacobian rank deficiency, namely \(\operatorname{rank}(J_{P_0}(x))=d_{\rm A}-d_{\rm M}\); (3) the gradient flow \(\dot x=-\nabla \mathcal{V}_\epsilon(x)\) realizes the collapse as a deformation retract, and its collapse timescale satisfies \(T_{\rm collapse}\propto \epsilon^{-2}\); and (4) in a statistical ensemble at temperature \(T\), the collapse appears as a phase transition, with an alignment order parameter \(m(\epsilon,T)\) that is discontinuous at \((\epsilon,T)=(0,0)\).

The stem does not explicitly reveal the correct option and does not contain an obvious lexical cue pointing to A. It sets up hypotheses and asks for the valid universal conclusion without giving away the specific strengthened clauses.

The item is very close to a theorem-recall/restatement question: the correct option bundles several theorem-style conclusions almost verbatim, while the distractors are mostly slight perturbations of quantifiers or boundary conditions. It is not purely tautological because there are competing variants, but it is only a mild reformulation.

Some reasoning is needed to compare subtle alternatives such as generic versus uniform claims, exact locus of discontinuity, and continuity versus collapse. However, the task is largely recognition of the strongest theorem statement rather than genuine derivation, and the presence of a weaker true option reduces clear generative pressure.

Several distractors are plausible and mathematically targeted: overgeneralization in B, wrong phase-transition locus in D, and failure of collapse in E. But C appears to be a genuinely true weaker statement, which makes the single-best-answer format ambiguous and lowers distractor quality.

A technically sophisticated but only moderately strong MCQ: it avoids direct answer leakage and uses plausible theorem-adjacent distractors, but it is close to theorem restatement and is weakened by an apparently true weaker alternative, making the item not fully well-posed as a single-answer question.


\label{prop:bound_a}
    If $a!b!=c!$ is a solution of class $k$, i.e., $c-b = k$, then $k < a < k + 2 \lceil \log_2 c \rceil$.

Erd\H{o}s asks the following problem in \cite{ErdosGraham1980}: for which integers $a_1, \ldots, a_t$ with $t \geq 2$ and $a_i \geq 2$ for each $i$ does there exist an integer $c$ so that $\prod_{i=1}^t a_i! = c!$? This question, which is open even for $t=2$, has been the inspiration for dozens of papers in elementary number theory since at least the 1970s. It is widely believed that there are only finitely many ``nontrivial'' solutions, i.e., those for which $c - \max_i a_i > 1$. In 1991, Erd\H{o}s \cite{Erdos91} showed that for sufficiently large $c$, $k \leq 5 \log\log c$; the constant was subsequently lowered, most recently by \cite{BkhatRamachandra10}, to $(1+o(1))/\log 2 \approx 1.44$. In 2007, Luca \cite{Luca07} showed that there are only finitely many nontrivial solutions if the notorious $abc$ Conjecture holds, and Nair-Shorey \cite{NAIR2016307} refined this to a complete list of solutions conditional on an explicit form of the $abc$ Conjecture due to Baker. In the same paper, Luca showed that the set of $c$'s that arise in solutions is of asymptotic density zero. Here, we focus on the $t=2$ case, i.e., $a!b!=c!$, showing that the possible values of $a$ are also rather sparse.

The following definition will play a key role throughout.

\begin{defn}
A solution to $a! b! = c!$ with $a<b$ in the integers is said to be ``class k'' if $c - b = k$.
\end{defn}

Note that class $0$ solutions must have $a=0$ or $a=1$ and $b=c$; it is easy to see that the solution is class $1$ iff $b=a!-1$ and $c=a!$. Thus, a solution to $a!b!=c!$ is said to be ``trivial'' if $k \leq 1$. It is conjectured that the only non-trivial solution is the triple $(6,7,10)$. Already the conjecture that there are no class $2$ solutions is open (see \cite{BerendHarmse06}). In the definition above, and throughout the sequel, we assume that $a < b$. It is straightforward to see that $a \neq b$ for nontrivial solutions. Indeed, if $a=b$, then $c!=a!^2$. However, writing $\nu_p(n!)$ for the $p$-adic order of $n$, this implies that $\nu_p(c!)=2\nu_p(a!)$ for every prime $p$, contradicting that $\nu_p(c!) = 1$ for every prime in $(c/2,c]$, which exists by Bertrand's Postulate.

In Section \ref{sec:primes}, we show that (Corollary \ref{cor:SerreSet}) for any fixed $k>1$, there exists $1/k$ natural density fraction of primes $p$ such that if $a = p-1$, there are no solutions to $a!b!=c!$. The proof makes use of some interesting elementary number theory: old results of Westland-Fl\"uge on irreducible polynomials which are sparse in the falling-factorial basis, Wilson's Theorem, and Dirichlet's Theorem. Then, in Section \ref{sec:equidistribution}, by analyzing the asymptotics of falling factorials, we show that $c$ falls into a small interval defined by $a$ and $k$, implying that if $\sqrt[k]{a!} \pmod 1$ is sufficiently equidistributed as $a,k \rightarrow \infty$ -- a much stronger version of which we conjecture is true -- then the set of $a$’s that appear in a solution is also of asymptotic density zero. Below we write $x^{\underline{k}}$ for the $k$-th falling factorial of $x$, i.e., $x(x-1)\cdots (x-k+1)$.

Here we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.

\begin{cor} \label{cor:SerreSet}
    Fix $k > 1$.  For at least a $1/k$ natural density subset of primes $p$, the equation
    $$
    (p-1)! b! = (b+k)!
    $$
    has no solutions for integers $b$.
\end{cor}

Here we collect a few useful bounds on possible solutions, especially, that $k < a < b < c$.

\section{Solutions Modulo a Prime} \label{sec:primes}

\begin{prop} There are no class $2$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $5 \pmod{6}$.
\end{prop}
\begin{proof}
Note that $c-b=2$ and $a!b!=c!$ implies $a! = c(c-1)$. Suppose $a!=c(c-1)$ where $a+1=p$ is a prime congruent to $5 \pmod{6}$. Then
$$
c^2 - c - a! = 0
$$
so we have $c = \frac{1\pm \sqrt{1-4(-a!)}}{2}$. Thus, $1+4a! = 1 + 4(p-1)!$ is a perfect square. By Wilson's Theorem, $1+4(p-1)! \equiv 1+4(-1) = -3 \pmod{p}$, so $-3$ is a quadratic residue modulo $p$. However, this contradicts Lemma \ref{lem:-3}.
\end{proof}

\begin{prop} \label{thm:pIs4} There are no class $4$ solutions to $a!b!=c!$ in positive integers if $a+1$ is a prime congruent to $2$ or $3 \pmod{5}$.
\end{prop}
\begin{proof}
Suppose $a! = c(c-1)(c-2)(c-3)$, where $a=p-1$ for some prime $p$. Note that $b>0$ implies $c=b+4>4$, so $c!/b! \geq 24 > 3!$, so we may assume $a>3$ and so $p>4$; furthermore, if $b>0$ then $4! = c(c-1)(c-2)(c-3)$ only has the solution $c=4$, but then $b = c-4 = 0$, a contradiction, so $p \neq 5$. Let $r \equiv c/2 \pmod{p}$, and apply Wilson's Theorem to conclude that
$$
2r(2r-1)(2r-2)(2r-3) \equiv -1 \pmod{p}.
$$
Now,
$$
2r(2r-1)(2r-2)(2r-3) = (4r^2-6r+1)^2-1,
$$
so
$$
(4r^2-6r+1)^2 \equiv 0 \pmod{p},
$$
i.e., $p | 4r^2 - 6r + 1$. However, $(2r-3/2)^2 - 5/4 = 4r^2 - 6r + 1$, so $5/4$ is a quadratic residue mod $p$, which happens iff $5$ is a quadratic residue mod $p$. It is known this occurs only for primes $p \equiv 1,4 \pmod{5}$.
\end{proof}

\begin{theorem} \label{thm:PositiveDensityIfIrreducible}Suppose $F_k(x) = x(x-1)\cdots (x-k+1)+1$ is irreducible over $\mathbb{Z}$. Let $P_0$ be the set of primes so that, if $a=p-1$ for some $p \in P_0$, then $a!b!=c!$ has no class $k$ solutions in positive integers. Then the natural density of $P_0$ in the primes is at least $1/k$.
\end{theorem}
\begin{proof}
 Suppose $a!b!=c!$ is class $k$ and $a = p-1$ for some prime $p$. Then $c = b+k$ and $a! = c!/b! = F_k(c)-1$. Then, reducing this equation modulo $p$ yields $c^{\underline{k}} \equiv a! \pmod{p}$, and by Wilson's Theorem, this is equivalent to $F_k(x) \equiv 0 \pmod{p}$. However, by Theorem 1 and 2 of \cite{Serre03}, the set $P_0$ of primes $p$ for which $F_k(x)$ does not have solution modulo $p$ has positive density, and the density is at least $1/k$.
\end{proof}

\begin{prop}\label{thm:WestlundFluge}
 For $k \geq 2$, $F_k(x)$ is reducible over $\mathbb{Z}$ iff $k=4$.
\end{prop}
\begin{proof}
 This is an immediate consequence of a result by Westlund and Fl\"{u}ge, answering a question of Schur, as discussed in \cite{DorwartOre33}.
\end{proof}

\begin{cor} \label{cor:SerreSet}
 Fix $k > 1$. For at least a $1/k$ natural density subset of primes $p$, the equation
 $$
 (p-1)! b! = (b+k)!
 $$
 has no solutions for integers $b$.
\end{cor}
\begin{proof}
 By Theorem \ref{thm:WestlundFluge}, $F_k$ is irreducible if $k \neq 4$, so Theorem \ref{thm:PositiveDensityIfIrreducible} implies that $a!b!=c!$ has no solutions in the integers when $c = b+k$ and $a=p-1$, for an at least $1/k$ density subset of the primes. Proposition \ref{thm:pIs4} yields the remaining case, by Dirichlet's Theorem on primes in arithmetic progressions.
\end{proof}

\begin{prop}
 For $p$ prime, the equation
 $$
 (p-1)!b! = (b+k)!
 $$
 has at most $nk/p + 1$ solutions in the integers with $b \leq n$.
\end{prop}
\begin{proof}
 Suppose $(p-1)! b! = (b+k)!$. Then, $(p-1)! = F_k(b)-1$. Taking this equation modulo $p$ and applying Wilson's Theorem gives
 $$
 -1 \equiv F_k(b) - 1 \pmod{p},
 $$
 i.e., $F_k(b) \equiv 0 \pmod{p}$. However, this equation has at most $k$ solutions $A_k \subseteq \mathbb{Z}/p\mathbb{Z}$. If $b \pmod{p} \not \in A_k$, then $(p-1)!b! \neq (b+k)!$. Since the set of $b \in [1,n]$ so that $b \pmod{p} \in A_k$ has cardinality $\lceil nk/p \rceil \leq nk/p + 1$, the conclusion follows.
\end{proof}

\begin{proof}
    Write $s_q(n)$ for the sum of the base-$q$ digits of $n$. Clearly, $\nu_p(a!)+\nu_p(b!)=\nu_p(c!)$.  Applying Legendre's Formula, we obtain
    $$
    \frac{a-s_p(a)}{p-1} + \frac{b-s_p(b)}{p-1} = \frac{c-s_p(c)}{p-1},
    $$
    from which it follows that $a-(c-b) = s_p(a)+s_p(b)-s_p(c)$ for every prime $p$.
    Since $a < b < c$, in particular
    $$
    | s_2(a)+s_2(b)-s_2(c) | < 2 \lceil \log_2 c \rceil .
    $$
    Thus, $|a-k| < 2 \lfloor \log_2 c \rfloor$.  Now, we argue that $a > k$.  First,
    \begin{align*}
    a! & = \frac{c!}{b!} = c(c-1)\cdots(c-k+1) \\
    & > (c-k+1)^k = (b+1)^k \geq (a+2)^k.
    \end{align*}
    Then taking the log of both sides above and applying Stirling's approximation (that $\log a! < a \log a$) gives
    $$
    a \log a > \log a! > k \log (a+2)
    $$
    so that $k < a \log a / \log(a+2) < a$.
\end{proof}

Using Legendre's formula with digit sums $s_p(n)$, from $\nu_p(a!)+\nu_p(b!)=\nu_p(c!)$ one gets
\[
\frac{a-s_p(a)}{p-1}+\frac{b-s_p(b)}{p-1}=\frac{c-s_p(c)}{p-1},
\]
so for every prime $p$,
\[
a-(c-b)=s_p(a)+s_p(b)-s_p(c).
\]
With $k=c-b$ and $a<b<c$, this yields the base-$2$ bound
\[
|s_2(a)+s_2(b)-s_2(c)|<2\lceil\log_2 c\rceil,
\]
hence $|a-k|<2\lfloor\log_2 c\rfloor$, giving the upper bound $a<k+2\lceil\log_2 c\rceil$.

To show $a>k$, rewrite
\[
a!=\frac{c!}{b!}=c(c-1)\cdots(c-k+1)>(c-k+1)^k=(b+1)^k\ge (a+2)^k.
\]
Taking logs and using Stirling's approximation (namely $\log a!<a\log a$) gives
\[
a\log a>\log a!>k\log(a+2),
\]
so $k<a\log a/\log(a+2)<a$, i.e. $k<a$.

Question: Let $a,b,c$ be nonnegative integers with $a<b$ such that $a!\,b!=c!$. Such a solution is said to be of class $k$ if $c-b=k$. Under these assumptions, which quantitative estimate for $a$ holds?

If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+2\left\lceil \log_2 c\right\rceil.\]

Choices:
B. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k\le a<k+2\left\lceil \log_2 c\right\rceil.\]
C. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[a<k+2\left\lceil \log_2 c\right\rceil.\]
D. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+2\left\lceil \log_2 a\right\rceil.\]
E. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+\left\lceil \log_2 c\right\rceil.\]

The stem does not explicitly reveal the correct choice. It introduces necessary notation and hypotheses, but does not single out the exact bound in A beyond what is needed to read the options.

The item appears to ask for essentially the exact theorem conclusion, with the correct option being a near-verbatim statement rather than a conclusion derived from a fresh scenario.

There is some reasoning pressure because the choices differ in subtle but meaningful ways (strict vs. weak inequality, dependence on c vs. a, special role of base 2, weaker true statement). However, the task is still mainly recognition of the precise theorem statement rather than substantial derivation.

The distractors are well designed: B tests boundary sharpness, C is a weaker true statement, D swaps the parameter inside the logarithm, and E overgeneralizes the base-2 feature to all primes. These are plausible mathematical failure modes.

A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it is weak on tautology avoidance because it largely restates the target result instead of requiring deeper generative reasoning.


Question: Let $a,b,c$ be nonnegative integers with $a<b$ such that $a!\,b!=c!$. Such a solution is said to be of class $k$ if $c-b=k$. Under these assumptions, which quantitative estimate for $a$ holds?

Choices:
B. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k\le a<k+2\left\lceil \log_2 c\right\rceil.\]
C. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[a<k+2\left\lceil \log_2 c\right\rceil.\]
D. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+2\left\lceil \log_2 a\right\rceil.\]
E. If $a!\,b!=c!$ is a class $k$ solution (that is, $c-b=k$), then \[k<a<k+\left\lceil \log_2 c\right\rceil.\]

The stem does not explicitly state the target estimate or give away the correct option. It only defines the setup and asks which bound on a holds.

The item is essentially asking the test-taker to identify the correct formal estimate among nearby variants of a theorem-like statement. It is not a verbatim restatement in the stem, but it mainly tests recognition/recall of the exact bound rather than deriving a new conclusion from substantive premises.

There is some reasoning pressure because the options differ in subtle but meaningful ways: strict versus weak inequality, dependence on c versus a, and the factor 2. However, without intermediate data or a proof context, success is driven more by memory of the result than by strong generative reasoning.

The distractors are plausible and mathematically well targeted. They reflect common failure modes: weakening a strict bound, dropping the lower bound, replacing c by a in the logarithm, and losing the factor 2. They are distinct and close enough to the correct statement to discriminate understanding.

A solid theorem-recognition MCQ with no answer leakage and strong distractors, but only moderate generative depth because it mainly tests recall of the exact estimate rather than substantial reasoning.


A collection $H$ of $k$-element subsets of $[n]$  is {\itshape shifted} if for every $i<j$, $j\in S\in H$, $i\notin S$, also $(S\setminus\{j\})\cup \{i\}\in H$. After fixing a field, a canonical operation of {\itshape algebraic shifting}, which associates with every simplicial complex $K$ a shifted simplicial complex $\Delta(K)$,
was introduced by Kalai \cite{K ch, K Alg Sh}. This operation preserves some important properties of $K$, such as its $f$-vector and $\beta$-vector, while loses others, such as the homotopy type of $K$.
Various properties of $K$ are described in terms of its shifting, such as its homology and Cohen-Macaulyness over the fixed field.

Two different variations of algebraic shifting were introduced
by Kalai: {\itshape exterior
shifting} and {\itshape symmetric
shifting}. Given a simplicial complex $K$ we denote by $\Delta^e(K)$ and $\Delta^s(K)$ its exterior and symmetric shifting respectively, and sometimes omit the superscript for  statements that hold for both types of algebraic shifting. Throughout the paper all results on exterior shifting are satisfied over any field, and all results on symmetric shifting are satisfied over any field of characteristic zero.

We start our exposition with a simple known example. Let $n$ be a positive integer. The {\itshape graphic matroid} of the complete graph on $n$ vertices is a collection of all spanning trees on the vertex set $[n]$. Let $G$ be such a spanning tree. Note that since algebraic shifting preserves Betti numbers, for every edge $e\in\Delta(G)$, we have $1\in e$. Note also that since algebraic shifting preserves $f$-vector, it then follows that $\Delta(G)=\{12,13,\ldots,1n\}$. Conversely, one can similarly verify that every $G\in\Delta^{-1}(\{12,13,\ldots,1n\})$ is a spanning tree on $[n]$. Thus, we conclude that $\Delta^{-1}(\{12,13,\ldots,1n\})$ coincides with the graphic matroid of the complete graph on $n$ vertices. The main goal of this paper is to study this phenomenon in a general setting.

Let $H$ be a shifted $k$-uniform hypergraph on $[n]$ (equivalently, the collection of facets of a pure simplicial complex).
Denote by
$$M(H)=\left\{G\subseteq \binom{[n]}k|\ \Delta^e(G)=H\right\}$$
all $k$-uniform hypergraphs whose exterior shifting is $H$.
We say that
$H$ is {\itshape matroidal} if $M(H)$ is the set of bases of a matroid; we abuse notation and also denote that matroid by $M(H)$.

For example, if $H$ is the collection of $k$-subsets of $[n]$ containing $1$, then $M(H)$ is the Crapo-Rota simplicial matroid \cite{CR1, CR2},
whose independent sets are the $(k-1)$-acyclic collections of $k$-subsets of $[n]$, namely those that do not support any $(k-1)$-homology cycle over a fixed
field; see e.g.
~\cite[Chapter 6]{White} and \cite[Section 4, Remark (2)]{K Symm} for further background. If $H=\{ij:\ 1\le i<j\le n, \ i\le k\}$ then $M(H)$ is Kalai's $k$-hyperconnectivity matroid~\cite{K Hyper}; see e.g. \cite[Section 4, Remark (2)]{K Symm} and \cite[Section 2.7]{K Alg Sh} for further background. If $H=\{1ij:\ 2\le i<j\le n, \ i\le 3\}$ then $M(H)$ is the area-rigidity matroid, as recently proved in~\cite[Theorem 1.2]{BNP}.

We conjecture a combinatorial characterization of the algebraic property of being \emph{matroidal}:
\begin{conjecture}\label{conj:any_k}
For every natural numbers $1\le k\le n$, a shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order
(initial lex-segment for short).
\end{conjecture}

The "if" part follows easily from a result of Kalai~\cite[Section 4]{K Symm}.
We prove the "only if" part for $k=2$ (i.e. for graphs) and for $k=3$:

A collection $H$ of $k$-element subsets of $[n]$ is {\itshape shifted} if for every $i<j$, $j\in S\in H$, $i\notin S$, also $(S\setminus\{j\})\cup \{i\}\in H$. After fixing a field, a canonical operation of {\itshape algebraic shifting}, which associates with every simplicial complex $K$ a shifted simplicial complex $\Delta(K)$,
was introduced by Kalai \cite{K ch, K Alg Sh}. This operation preserves some important properties of $K$, such as its $f$-vector and $\beta$-vector, while loses others, such as the homotopy type of $K$.
Various properties of $K$ are described in terms of its shifting, such as its homology and Cohen-Macaulyness over the fixed field.

Two different variations of algebraic shifting were introduced
by Kalai: {\itshape exterior
shifting} and {\itshape symmetric
shifting}. Given a simplicial complex $K$ we denote by $\Delta^e(K)$ and $\Delta^s(K)$ its exterior and symmetric shifting respectively, and sometimes omit the superscript for statements that hold for both types of algebraic shifting. Throughout the paper all results on exterior shifting are satisfied over any field, and all results on symmetric shifting are satisfied over any field of characteristic zero.

Let $H$ be a shifted $k$-uniform hypergraph on $[n]$ (equivalently, the collection of facets of a pure simplicial complex).
Denote by
$$M(H)=\left\{G\subseteq \binom{[n]}k|\ \Delta^e(G)=H\right\}$$
all $k$-uniform hypergraphs whose exterior shifting is $H$.
We say that
$H$ is {\itshape matroidal} if $M(H)$ is the set of bases of a matroid; we abuse notation and also denote that matroid by $M(H)$.

For example, if $H$ is the collection of $k$-subsets of $[n]$ containing $1$, then $M(H)$ is the Crapo-Rota simplicial matroid \cite{CR1, CR2},
whose independent sets are the $(k-1)$-acyclic collections of $k$-subsets of $[n]$, namely those that do not support any $(k-1)$-homology cycle over a fixed
field; see e.g.
~\cite[Chapter 6]{White} and \cite[Section 4, Remark (2)]{K Symm} for further background. If $H=\{ij:\ 1\le i<j\le n, \ i\le k\}$ then $M(H)$ is Kalai's $k$-hyperconnectivity matroid~\cite{K Hyper}; see e.g. \cite[Section 4, Remark (2)]{K Symm} and \cite[Section 2.7]{K Alg Sh} for further background. If $H=\{1ij:\ 2\le i<j\le n, \ i\le 3\}$ then $M(H)$ is the area-rigidity matroid, as recently proved in~\cite[Theorem 1.2]{BNP}.

We conjecture a combinatorial characterization of the algebraic property of being \emph{matroidal}:
\begin{conjecture}\label{conj:any_k}
For every natural numbers $1\le k\le n$, a shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order
(initial lex-segment for short).
\end{conjecture}

The "if" part follows easily from a result of Kalai~\cite[Section 4]{K Symm}.
We prove the "only if" part for $k=2$ (i.e. for graphs) and for $k=3$:

We conjecture a combinatorial characterization of the algebraic property of being \emph{matroidal}:
\begin{conjecture}\label{conj:any_k}
For every natural numbers $1\le k\le n$, a shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the hyperedges of $H$ form an initial segment w.r.t. the lexicogrpaphic order
(initial lex-segment for short).
\end{conjecture}

For $k\ge 4$ we prove the "only if" part for $k$-uniform hypergraphs satisfying some technical condition on their structure.

\begin{theorem} \label{main theorem}
Let $k\ge 4$ and
let $H\subseteq\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $s_H\notin H$, where $s_H$ is defined in terms of the lex-maximum element of $H$ by \eqref{S for case 1}-\eqref{S for case 2}. Then $H$ is matroidal if and only if the hyperedges of $H$ form an initial lex-segment.
\end{theorem}

Let $X=(x_{ij})_{1\le i,j\le n}$ be an $n\times n$-matrix over $\mathbb L$ which is generic
over $\mathbb F$. For a $k$-uniform hypergraph $H\subseteq\binom{[n]}k$ define a matroid $M'(H)$ on $\binom{[n]}k$ as follows: $B\subseteq\binom{[n]}k$ is independent in $M'(H)$ iff the rows of $X(B,H)$ are linearly independent.
\begin{remark} \cite[Section 4, Remark (2)]{K Symm} \label{Remark: Kalai's construction for hypperconnectivity}
 For $H=\{S\subseteq\binom{[n]}2:S\cap[k]\ne\emptyset\},\ M'(H)$ is the $k$-hyperconnectivity matroid.
\end{remark}
In fact, a more general statement is straightforward from the construction above. We call a subset $H\subseteq\binom{[n]}{k}$ an {\itshape initial lex-segment}
if $H=\left\{T'\in\binom{[n]}{k}|\ T'\le_{\text{lex}}T\right\}$ for some $T\in\binom{[n]}{k}$.
\begin{corollary}
\label{Initial segment is a matroid}
 Let $H\subseteq\binom{[n]}k$ be a $k$-uniform hypergraph whose hyperedges form an initial lex-segment. Then $H$ is matroidal.
\end{corollary}
\begin{proof}
We will verify that $M(H)$ is the set of bases of the matroid $M(H')$, or $M(H)=M'(H)$ for short.
Let $B\in M(H)$, then $H$ is a column basis chosen in the greedy way for the matrix $X(B,\binom{[n]}{k})$. Thus, the columns of the matrix $X(B,H)$ are linearly independent, hence $B\in M'(H)$.

\begin{lemma}[Isolated vertex] \label{lemma: partial star with an additional edge 2,3 is not a matroid}
 Let $H\subseteq\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $n$ is an isolated vertex and $\{2,3,\ldots,k+1\}\in H$.
 Then $H$ is not matroidal.
\end{lemma}
\begin{proof}
 Consider two $k$-uniform hypergraphs $H_1=H$ and $H_2$ obtained from $H$ by flipping the vertex $1$ with the vertex $n$ (see Fig. \ref{graphs which do not satisfy the exchange exiom} for illustration). By its structure $H_1$ contains the facets of the boundary complex $\partial \sigma$ of the simplex $\sigma=\{1,2,\ldots,k+1\}$. In order to show that the pair $H_1, H_2$ violates the exchange axiom we have to find $e_1\in H_1\setminus H_2$ such that for every $e_2\in H_2\setminus H_1$ we have that
 $(H_1\setminus e_1)\cup e_2\notin M(H)$. Let us take as $e_1$ an arbitrary facet of
 $\partial \sigma$
 which contains $1$. Such $e_1$ indeed lies in $H_1\setminus H_2$ since $1$ is an isolated vertex in $H_2$. Note that $\beta_{k-1}(H_1\setminus e_1)<\beta_{k-1}(H_1)$. As $e_2$ we can take only hyperedges containing the vertex $n$, which is an isolated vertex in $H_1$. Thus, $e_2$
 does not belong to any $(k-1)$-homology cycle in $(H_1\setminus e_1)\cup e_2$. Thus, $\beta_{k-1}((H_1\setminus e_1)\cup e_2)=\beta_{k-1}(H_1\setminus e_1)<\beta_{k-1}(H_1)$. However, since algebraic shifting preserves Betti numbers, see Theorem \ref{Basic properties of algebraic shifting}(4), we have $\beta_{k-1}(\Delta((H_1\setminus e_1)\cup e_2))=\beta_{k-1}((H_1\setminus e_1)\cup e_2)=\beta_{k-1}(H_1\setminus e_1)<\beta_{k-1}(H_1)=\beta_{k-1}(H)$. Thus, $((H_1\setminus e_1)\cup e_2)\notin M(H)$.
\end{proof}

\begin{example} \label{example: of not matroidal graph}
Consider the $2$-uniform hypergraph $H=\{12,13,23\}$ on $[5]$. Note that the (hyper)edges of $H$ do not form an initial lex-segment. Thus, by Theorem \ref{main theorem} $H$ is not matroidal. In this case one can verify this directly. Since algebraic shifting preserves $f$-vector and $\beta$-vector (see Theorem \ref{Basic properties of algebraic shifting}(3,4)) and the only shifted $2$-uniform hypergraph on $[5]$ with $3$ (hyper)edges and an one cycle is $H$, we conclude that $M(H)$ consists of all $2$-uniform hypergraph on $[5]$ with
$3$ edges forming a triangle boundary.
However, the pair of elements
$\{12,13,23\},\ \{14,15,45\}\in M(H)$ does not satisfy the exchange axiom.
\end{example}

\begin{definition}
Let $H$ be a shifted $k$-uniform hypergraph on $[n]$. $H$ is {\itshape s-matroidal} if $M^s(H)$ is the set of bases of a matroid on $\binom{[n]}{k}$.
\end{definition}
\begin{proposition} \label{thm: symmetric case for graphs}
 A simple graph $H\subseteq\binom{[n]}{2}$ is s-matroidal if and only if the edges of $H$ form an initial lex-segment.
\end{proposition}

\begin{proposition} \label{thm: lex-matroidal}
 A shifted $k$-uniform hypergraph $H\subseteq\binom{[n]}{k}$ is lex-matroidal iff the hyperedges of $H$ form an initial lex-segment.
\end{proposition}
\begin{proof}
 By Corollary \ref{Initial segment is a matroid} if the hyperedges of $H$ form an initial lex-segment, then $H$ is matroidal. For every hyperedge $S$ of $H$ the set $H(S)$ is an initial lex-segment, and thus it is matroidal.

\begin{equation} \label{S for case 1}
s_H=\{a,\ldots,b,c-1,n-(l+1),n-l,\ldots,n-1\}.
\end{equation}

\begin{equation} \label{S for case 2}
s_H=\{a,\ldots,b,c-1,n-m-l,n-m-l+1,\ldots,n-m-1,n-m,\ldots,n\}.
\end{equation}

For $k\ge 4$ we prove the "only if" part for $k$-uniform hypergraphs satisfying some technical condition on their structure.

\begin{theorem} \label{main theorem}
Let $k\ge 4$ and
let $H\subseteq\binom{[n]}{k}$ be a shifted $k$-uniform hypergraph such that $s_H\notin H$, where $s_H$ is defined in terms of the lex-maximum element of $H$ by \eqref{S for case 1}-\eqref{S for case 2}. Then $H$ is matroidal if and only if the hyperedges of $H$ form an initial lex-segment.
\end{theorem}

For the "only if" part of Theorem \ref{thm: main thm 2 and 3} we first make a reduction of the general problem to a certain restricted subset of hypergraphs; and for each of them we find a suitable permutation $\pi:[n]\rightarrow [n]$ such that the pair $(H, \pi(H))$, where $\pi(H)=\{\pi(T):\ T\in H\}$, violates the exchange axiom among the elements of $M(H)$. For the "only if" part of Theorem \ref{main theorem} we find a suitable such permutation $\pi=\pi(s_M)$ for each $H$.

The analog of Theorem~\ref{thm: main thm 2 and 3}
for symmetric algebraic shifting is proved
for $k=2$ in Proposition~\ref{thm: symmetric case for graphs}.
For $H=\{ij:\ 1\le i\le j \le n, i\le k\}$ the corresponding matroid is the (generic) $k$-rigidity matroid, see e.g. \cite{Lee} and  \cite[Section 2.7]{K Alg Sh}, \cite[Section 3.2]{Nevo phd} for further background, references and for a discussion of this correspondence.
\

This paper is organized as follows. In Section \ref{sec: preliminaries} we give a background on algebraic shifting and matroids. In Section \ref{sec: Preparatory lemmas} we prove lemmas needed for proofs of Theorem \ref{thm: main thm 2 and 3} and Theorem \ref{main theorem}, and in Section \ref{sec: Proofs of main results} we prove these theorems. In Section \ref{sec: Matroids arising from symmetric shifting} we consider matroids arising from symmetric shifting, and in Proposition \ref{thm: symmetric case for graphs} classify all matroidal
simple graphs
w.r.t. symmetric shifting. In Section \ref{sec: Some partial results} we replace the matroidal property by a stronger algebraic property in Conjecture~\ref{conj:any_k} that allows us to prove the resulted "iff" combinatorial characterization.
In Section \ref{sec: Concluding remarks} we conclude with related open problems.

For the ``only if'' part of Theorem~\ref{thm: main thm 2 and 3} the approach is: (i) ``first make a reduction of the general problem to a certain restricted subset of hypergraphs''; (ii) for each hypergraph in this restricted subset, ``find a suitable permutation $\pi:[n]\rightarrow [n]$ such that the pair $(H, \pi(H))$, where $\pi(H)=\{\pi(T):\ T\in H\}$, violates the exchange axiom among the elements of $M(H)$.''

For the ``only if'' part of Theorem~\ref{main theorem} (the $k\ge 4$ theorem under the condition $s_H\notin H$), the method is similarly to ``find a suitable such permutation $\pi=\pi(s_M)$ for each $H$.''

For the ``only if'' part of \begin{theorem} \label{thm: main thm 2 and 3}
For $k=2,3$,
a shifted
$k$-uniform
hypergraph $H\subseteq\binom{[n]}{k}$ is matroidal if and only if the (hyper)edges of $H$ form an
initial lex-segment.
\end{theorem}
the approach is: (i) ``first make a reduction of the general problem to a certain restricted subset of hypergraphs''; (ii) for each hypergraph in this restricted subset, ``find a suitable permutation $\pi:[n]\rightarrow [n]$ such that the pair $(H, \pi(H))$, where $\pi(H)=\{\pi(T):\ T\in H\}$, violates the exchange axiom among the elements of $M(H)$.''

For the ``only if'' part of the main theorem (the $k\ge 4$ theorem under the condition $s_H\notin H$), the method is similarly to ``find a suitable such permutation $\pi=\pi(s_M)$ for each $H$.''

Question:
Let k\in\{2,3\}, let [n]=\{1,\dots,n\}, and let H\subseteq \binom{[n]}{k} be a shifted k-uniform hypergraph, meaning that for every i<j, whenever j\in S\in H and i\notin S, one also has (S\setminus\{j\})\cup\{i\}\in H. Define
\[
M(H)=\{G\subseteq \tbinom{[n]}{k}: \Delta^e(G)=H\},
\]
where \(\Delta^e\) denotes exterior algebraic shifting. Say that H is matroidal if \(M(H)\) is the set of bases of a matroid on \(\binom{[n]}{k}\). Also, call H an initial lex-segment if there exists \(T\in \binom{[n]}{k}\) such that
\[
H=\{T'\in \tbinom{[n]}{k}: T'\le_{\mathrm{lex}} T\},
\]
with \(\le_{\mathrm{lex}}\) the usual lexicographic order on k-subsets. Which statement holds for every such H?

H is matroidal if and only if the (hyper)edges of H form an initial lex-segment.

Choices:
B. H is matroidal if and only if the (hyper)edges of H form a lex-segment after relabeling the vertices, i.e. there exists a permutation \(\pi\in S_n\) such that \(\pi(H)=\{T'\in \tbinom{[n]}{k}: T'\le_{\mathrm{lex}} T\}\) for some \(T\in \tbinom{[n]}{k}\).
C. If the (hyper)edges of H form an initial lex-segment, then H is matroidal.
D. H is matroidal if and only if the (hyper)edges of H form an initial colex-segment.
E. If H is matroidal, then there exists a permutation \(\pi:[n]\to[n]\) such that the pair \((H,\pi(H))\) satisfies the exchange axiom inside \(M(H)\), and hence H is matroidal exactly when this happens.

fixed-label initial lex vs up-to-permutation lex

lexicographic order replaced by colexicographic order

permutation in proof used to witness failure of exchange axiom, reversed into existence criterion for matroidality

The stem does not state or trivially imply the correct equivalence. It defines the relevant notions, but the correct answer is not leaked beyond introducing the concepts under comparison.

This is essentially a direct theorem-identification item: the correct option is the theorem statement itself, with nearby variants as alternatives. It tests recall of the precise characterization more than a nontrivial derived conclusion.

There is some reasoning pressure because the options differ in subtle ways (iff vs one-way, fixed lex vs relabeling, lex vs colex). However, the item mostly rewards recognizing the exact known theorem rather than generating a conclusion from premises.

The distractors are mathematically plausible and target common failure modes: weakening an equivalence, allowing relabeling, confusing lex with colex, and misusing an exchange-style condition. They are distinct and well aligned with likely misconceptions.

A solid theorem-recall MCQ with strong distractors and no major answer leakage, but it is fairly tautological and only moderately tests generative reasoning.