\label{thm:main}
    Let $Q=\Spec(A)$ be an affine algebraic rack with unit $e\in Q$, and let $\mathfrak{q}\coloneq \Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\cdot,\cdot]\colon\mathfrak{q}\times\mathfrak{q}\to\mathfrak{q}$ by convolution with respect to the corack operation $\nabla$ of $A$ (see Section \ref{sec:nabla}):
    \[
    [D,E]\coloneq (D\otimes E)\circ\nabla.
    \]
    Then the assignment $Q\mapsto(\mathfrak{q},[\cdot,\cdot])$ defines a covariant functor $\Rack(\Aff)\to\leib$ sending morphisms of affine algebraic racks $\varphi\colon Q\to R$ to their derivatives $(d\varphi)_e\colon \mathfrak{q}\to\mathfrak{r}$ at $e$.

    Furthermore, if $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\mathfrak{q},[\cdot,\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\mathbb{R}$ or $\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\mathfrak{q},[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.

\begin{rmk}\label{rmk:recover}
    If $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\q$ at $e$. By the definition of $\Conj(G)$, the adjoint representations $\Ad$ of $Q$ and $\ad$ of $\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, \cite{milne}*{Rem.\ 10.24}). Hence, the claim follows from Proposition \ref{prop:adj}.\footnote{Alternatively, one can use Example \ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\otimes E)\circ\nabla$ agrees with the usual Lie bracket $(D\otimes E-E\otimes D)\circ\Delta$ on $\q=\operatorname{Lie}(G)$ (see \cite{waterhouse}*{Sec.\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\eps\circ S=\eps$ from the axiom $\mu\circ (S\otimes\id)\circ\Delta=\eta\circ\eps$ by post-composing with $\eps$, and then deduce from these two equalities that $D\circ S=-D$.}

    Similarly, if $k=\R$ or $\mathbb{C}$ and $Q(k)$ is an algebraic Lie rack, then our definitions of $\Ad$ and $\ad$ coincide with the ones for Lie racks in \cite{kinyon}, so $(\q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
\end{rmk}

\begin{thm}\label{thm:main}
    Let $Q=\Spec(A)$ be an affine algebraic rack with unit $e\in Q$, and let $\q\coloneq \Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\cdot,\cdot]\colon\q\times\q\to\q$ by convolution with respect to the corack operation $\nabla$ of $A$ (see Section \ref{sec:nabla}):
    \[
    [D,E]\coloneq (D\otimes E)\circ\nabla.
    \]
    Then the assignment $Q\mapsto(\q,[\cdot,\cdot])$ defines a covariant functor $\Rack(\Aff)\to\leib$ sending morphisms of affine algebraic racks $\phi\colon Q\to R$ to their derivatives $(d\phi)_e\colon \q\to\mathfrak{r}$ at $e$.

    Furthermore, if $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\R$ or $\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
\end{thm}

This paper introduces \emph{algebraic racks}, which are pointed racks internal to the category of schemes of finite type over a field $k$. We show that affine algebraic racks provide a solution to a version of the well-known \emph{coquecigrue} problem for Leibniz algebras over arbitrary fields $k$.

\subsubsection{}
Loday's \cite{loday} original version of the \emph{coquecigrue} problem from 1993 seeks algebraic structures on smooth manifolds that differentiate to Leibniz algebras in a way that recovers the Lie group--Lie algebra correspondence. \emph{Leibniz algebras} are vector spaces $\mathfrak{q}$ equipped with a bilinear mapping $[\cdot,\cdot]\colon \mathfrak{q}\times\mathfrak{q}\to\mathfrak{q}$ called the \emph{Leibniz bracket} that satisfies the derivation form of the Jacobi identity
 \[
 [X,[Y,Z]]=[[X,Y],Z]+[Y,[X,Z]].
 \]
Bloh \cite{bloh} introduced Leibniz algebras in 1965 to generalize Lie algebras, which are precisely those Leibniz algebras whose Leibniz bracket is alternating. See \cite{leibniz} for an introduction to Leibniz algebras.

As a solution to the \emph{coquecigrue} problem over $\mathbb{R}$ and $\mathbb{C}$, Kinyon \cite{kinyon} in 2007 introduced \emph{Lie racks}, which are pointed rack objects (see Section \ref{sec:pro}) in the category of smooth manifolds; see the survey \cite{ongay} for further discussion. \emph{Racks} \cite{fenn} and similar structures called \emph{quandles} \citelist{\cite{joyce}\cite{matveev}} were originally introduced as invariants of knots and 3-manifolds; see \citelist{\cite{book}\cite{quandlebook}} for introductions to the theory.

\subsection{Main theorem}
Kinyon also posed an analogue of the \emph{coquecigrue} problem over arbitrary fields $k$ that replaces Lie groups with affine algebraic groups. Save for unpublished work of Ba\v{s}i\'{c} and \v{S}koda \cite{skoda} in 2008 in the case that $\operatorname{char}(k)=0$, the author is unaware of any literature in this direction.
To address this, we prove the following. Let $\Rack(\Aff)$ and $\leib$ denote the categories of affine algebraic racks over $k$ and Leibniz algebras over $k$, respectively.

\begin{proof}
 We prove the theorem piecemeal in Propositions \ref{prop:adj}, \ref{prop:jacobi}, and \ref{prop:functor} and Remark \ref{rmk:recover}.
\end{proof}

\subsubsection{}
To make Definition \ref{def:corack} more concrete, let $A$ be a $k$-algebra with (commutative and associative) multiplication $\mu\colon A\otimes A\to A$, and let $\tau\colon A\otimes A\to A\otimes A$ denote the transposition $a\otimes b\mapsto b\otimes a$. A corack algebra structure on $A$ consists of two $k$-algebra homomorphisms $\nabla,\nabla\inv\colon A\to A\otimes A$ called the \emph{corack operations} and a $k$-algebra homomorphism $\eps\colon A\to k$ called the \emph{counit}, \emph{coidentity}, or \emph{augmentation} satisfying the following five identities:
\begin{enumerate}[(C1)]
 \item (Co-invertability) $(\mathord{\id}\otimes \mu)\circ(\tau\otimes\mu)\circ(\mathrm{\id}\otimes\nabla\inv\otimes\id)\circ(\nabla\circ\id)=(\mathord{\id}\otimes \id)=(\mu\otimes\id)\circ(\mu\otimes\tau)\circ(\mathord{\id}\otimes\nabla\otimes\id)\circ(\mathord{\id}\otimes\nabla\inv)$.
 \item\label{ax:cold} (Co-left distributivity) $(\mathord{\id}\otimes\nabla)\circ\nabla=(\mu\otimes\mathord{\id}\otimes\id)\circ(\mathord{\id}\otimes\tau\otimes\id)\circ(\nabla\otimes\nabla)\circ\nabla$.
 \item (Co-right distributivity) $(\nabla\inv\otimes\id)\circ\nabla\inv=(\mathord{\id}\otimes\mathord{\id}\otimes\mu)\circ(\mathord{\id}\otimes\tau\otimes\id)\circ(\nabla\inv\otimes\nabla\inv)\circ\nabla\inv$.
 \item\label{ax:co-fixing} (Co-fixing) $(\eps\otimes \id)\circ\nabla=\id$.
 \item\label{ax:co-fixed} (Co-fixedness) $(\mathord{\id}\otimes \eps)\circ\nabla=\eps$.
\end{enumerate}
We identify $k\otimes A\cong A\cong A\otimes k$ in identities \ref{ax:co-fixing} and \ref{ax:co-fixed}. Actually, this paper will only have occasion to use identities \ref{ax:cold} and \ref{ax:co-fixing}; the other three identities are less important for our purposes than the corresponding axioms of algebraic racks.

For example, let $Q=\Spec(A)$ be an affine algebraic rack. Then for all $k$-algebras $R$ and $R$-points $x\in Q(R)$ (thought of as $k$-algebra homomorphisms $x^\sharp\colon A\to R$), the automorphism $L_x\colon Q(R)\bij Q(R)$ from \eqref{eq:lx} induces an automorphism
\begin{align}
L_x^\sharp\colon R\otimes_k A &\bij R\otimes_k A\nonumber\\
 r\otimes f&\mapsto(r\otimes 1)\cdot ((x^\sharp\otimes\id_{A})\circ \nabla)(f).\label{eq:lx-sharp-full}
\end{align}

\begin{definition}
 The \emph{Leibniz algebra of $Q$} is the vector space
 \[
 \q\coloneq \Der_k(A,k)
 \]
 equipped with the Leibniz bracket $[\cdot,\cdot]\colon \q\times\q\to \q$ defined by
 \[
 [D,E]\coloneq (D\otimes E)\circ\nabla,
 \]
 the \emph{convolution} operation with respect to $\nabla$.
\end{definition}
Writing $\nabla f=\sum \fo\otimes\ft$, we thus have
\[
[D,E](f)=\sum D(\fo)E(\ft)
\]
for all $f\in A$. Note that $[\cdot,\cdot]$ is $k$-bilinear.

\subsubsection{}
Similarly, since $\partial D$ is a $k[\delta]$-point of $Q$ and $\Ad\colon Q\to\GL(\q)$ is a morphism of schemes, we obtain a $k[\delta]$-point of $\GL(\q)$ \[\Ad_{\partial D}\in\GL(\q)(k[\delta])=\GL(k[\delta]\otimes_k \q).\] In particular, under the identification $k[\delta]\otimes_kA \bij A[\delta]$, the definition of the derivative shows that
\begin{equation}\label{eq:adj-partial}
 \Ad_{\partial D}( 1\otimes E)= E\circ L_{\partial D}^\sharp\colon A[\delta]\to k[\delta]
\end{equation}
for all $k$-derivations $E\in\q$. We use this to show the following. \begin{prop}\label{prop:adj}
 When considered as a map $\ad\colon\q\to\mathfrak{gl}(\q)$, the adjoint representation of $\q$ is given by the left argument of the Leibniz bracket $D\mapsto [D,\cdot]$. That is, $\ad_D(E)=[D,E]$ for all $k$-derivations $D,E\in\q$. In particular, $\q$ is closed under $[\cdot,\cdot]$.
\end{prop}

\begin{rmk}\label{rmk:recover}
 If $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\q$ at $e$. By the definition of $\Conj(G)$, the adjoint representations $\Ad$ of $Q$ and $\ad$ of $\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, \cite{milne}*{Rem.\ 10.24}). Hence, the claim follows from Proposition \ref{prop:adj}.\footnote{Alternatively, one can use Example \ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\otimes E)\circ\nabla$ agrees with the usual Lie bracket $(D\otimes E-E\otimes D)\circ\Delta$ on $\q=\operatorname{Lie}(G)$ (see \cite{waterhouse}*{Sec.\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\eps\circ S=\eps$ from the axiom $\mu\circ (S\otimes\id)\circ\Delta=\eta\circ\eps$ by post-composing with $\eps$, and then deduce from these two equalities that $D\circ S=-D$.}

\begin{enumerate}
 \item Does the Leibniz algebra of a (not necessarily affine) algebraic rack have a natural Leibniz algebra structure? That is, does the functor $\Rack(\Aff)\to\leib$ from Theorem \ref{thm:main} extend to a functor $\Rack(\Sch)\to\leib$?
 \item If $G$ is an algebraic group, then its Lie algebra $\mathrm{Lie}(G)$ is isomorphic to the space of left-invariant $k$-derivations $D\colon \O(G)\to\O(G)$, that is, $k$-derivations that satisfy the identity $\Delta\circ D=(\mathord{\id}\otimes D)\circ\Delta$. (See, for example, \cite{milne}*{Sec.\ 10e}.) Do Leibniz algebras of algebraic racks have a similar interpretation?
 \item Mirroring the representation theory of algebraic groups and comodules of commutative Hopf algebras (see, for example, \cite{milne}*{Sec.\ 4e}), develop a theory of affine representations of affine algebraic racks $Q$ and $\O(Q)$-comodules. Also, study their relationships with the induced omni-representations (see \cite{omni}) of Leibniz algebras. See Comment \ref{com:rep} for further discussion.
 \item Building upon the previous question, introduce a cohomology theory for algebraic racks and their representations. Study its relationships with Leibniz algebra cohomology \cite{lp}, traditional rack and quandle cohomology \cite{covez2}, and (in the case of conjugation quandle schemes) the cohomology of algebraic groups and their representations (see \cite{bendel}).
 \item More broadly, which theorems about affine algebraic groups and their Lie algebras extend to affine algebraic racks and their Leibniz algebras? (Cf.\ Remark \ref{rmk:triv}.)
 \item Determine which Leibniz algebras are \emph{algebraic} or ``integrate to an algebraic rack.'' That is, given $\q$, determine whether there exists an algebraic rack whose Leibniz algebra is $\q$.
 \item La Rosa and Mancini \cite{larosa}*{Thm.\ 4.6} proved that if $Q$ is a Lie rack with Leibniz algebra $\q$, then $\q$ is a Lie algebra if and only if $Q$ is a quandle. Does the same hold for algebraic racks?
 \item Study non-pointed algebraic racks (that is, \emph{rack objects} in $\Sch$; cf.\ \cite{rack-roll}). In particular, generalize the results of Takahashi \citelist{\cite{takahashi}\cite{takahashi2}} from quandle varieties to rack schemes.
 \item Develop the general theory of corack algebras, perhaps in analogy to the theory of Hopf algebras. What relationships do they have with rack bialgebras (see \citelist{\cite{bardakov}\cite{alexandre}})?
 \item Racks \cite{fenn} and quandles \citelist{\cite{joyce}\cite{matveev}} were originally introduced to construct invariants of knots and 3-manifolds; see \citelist{\cite{book}\cite{quandlebook}}. In the past decade, racks (for example, \citelist{\cite{randal}\cite{shusterman}}) and quandles (for example, \citelist{\cite{fan}\cite{bianchi}\cite{bianchi2}\cite{takahashi1}}) have also enjoyed applications as invariants of various algebro-geometric objects. Use algebraic racks to enhance these invariants, and compare them to other algebro-geometric invariants.
\end{enumerate}

\begin{proof}
    We prove the theorem piecemeal in Propositions \ref{prop:adj}, \ref{prop:jacobi}, and \ref{prop:functor} and Remark \ref{rmk:recover}.
\end{proof}

\subsubsection{}
Theorem \ref{thm:main} is an algebraic analogue of a result of Kinyon's result for Lie racks in \cite{kinyon}*{Thm.\ 3.4}.
Compared to that result, two advantages of Theorem \ref{thm:main} are that it holds for all fields $k$ and gives an explicit formula for the Leibniz bracket $[\cdot,\cdot]$ that does not require computing any derivatives. That said, the Leibniz bracket agrees with the derivative of the \emph{adjoint representation} of $Q$ at the identity; see Proposition \ref{prop:adj}.

The last part of Theorem \ref{thm:main} states that our construction recovers the usual functors sending affine algebraic groups to their Lie algebras (see, for example, \cite{waterhouse}*{Sec.\ 12.2}) and affine algebraic Lie racks over $\mathbb{R}$ or $\mathbb{C}$ to their Leibniz algebras (see \cite{kinyon}). As racks and quandles have enjoyed increasing interest in both quantum algebra and algebraic geometry in the past decade, this paper lays the foundations for what we expect will be a very rich area of study; cf.\ Section \ref{sec:open}.

\subsection{Literature review}
While there does not appear to be any published work on the algebraic \emph{coquecigrue} problem over arbitrary fields $k$---see \cite{skoda} for a preliminary report of unpublished work of Ba\v{s}i\'{c} and \v{S}koda from 2008 in the case that $\operatorname{char}(k)=0$---there is a rich literature on the \emph{coquecigrue} problem for smooth manifolds over $\mathbb{R}$ and $\mathbb{C}$. See \cite{ongay} for a discussion of the literature up to 2014.

Although the first work on the \emph{coquecigrue} problem is due to Datuashvili \cite{Datuashvili} in 2004, most modern approaches to the problem employ Lie racks, which Kinyon \cite{kinyon} introduced in 2007 in response to the problem.
For example, Covez \cite{covez} in 2013 introduced \emph{local augmented Lie racks}, which address the \emph{coquecigrue} problem with the caveat that an algebraic structure is only defined in a neighborhood of the identity. By contrast, Bordemann and Wagemann \cite{bordemann} in 2016 introduced a global rack-theoretic solution with the caveat of not being functorial.

Within the last decade, various authors have classified special types of Lie racks \citelist{\cite{abchir2}\cite{benayadi}\cite{dibartolo}}. These include Lie racks corresponding to certain kinds of nilpotent Leibniz algebras \citelist{\cite{larosa}\cite{larosa2}} and symmetric Leibniz algebras \cite{abchir}, as well as all Lie racks corresponding to Lie algebras \cite{larosa}*{Thm.\ 4.6}. For further work on the \emph{coquecigrue} problem, see \citelist{\cite{alexandre}\cite{dherin}}. For non-rack-theoretic approaches, see \citelist{\cite{smith}\cite{rodriguez}\cite{uslu}}.

\subsubsection{}
Although there does not appear to be existing literature on rack schemes, we note that quandle varieties and ordinary set-theoretic racks have enjoyed increasing attention in algebraic geometry in the past decade. In particular, quandle varieties provide algebro-geometric analogues of Riemannian symmetric spaces \citelist{\cite{takahashi}\cite{takahashi2}} and applications to geometric invariant theory \cite{fan}.

Ordinary racks and quandles provide invariants of not only knots and 3-manifolds but also various objects of study in algebraic geometry, particularly arithmetic algebraic geometry. For example, quandles provide invariants of arithmetic schemes \cite{takahashi1} and categorical constructions of Hurwitz spaces \citelist{\cite{bianchi}\cite{bianchi2}}. Racks have similar applications in the theory of Hurwitz spaces and algebraic function fields \citelist{\cite{randal}\cite{shusterman}}. The increasing attention racks and quandles have received in algebraic geometry provides another motivation for developing a modern scheme-theoretic treatment of algebraic racks.

\begin{notationC}
    We work over a field $k$ of arbitrary characteristic.
    By a \emph{scheme}, we mean a scheme of finite type over $k$. Unless we are specifically discussing Leibniz algebras or Lie algebras, by a \emph{$k$-algebra} we mean a finitely generated commutative associative $k$-algebra.
    Let $\Sch$, $\Aff$, $\Alg$, $\leib$, $\mathsf{Grp}$, $\mathsf{Grp}(\Sch)$, and $\mathsf{Set}$ denote the categories of schemes, affine schemes, $k$-algebras, Leibniz algebras over $k$, groups, algebraic groups (that is, group objects in $\Sch$), and sets, respectively. Denote the $k$-algebra of \emph{dual numbers} by $k[\delta]\coloneq k[\delta]/(\delta^2)$. Given a morphism of affine schemes $\varphi\colon\Spec(B)\to\Spec(A)$, let $\varphi^\sharp$ denote the induced homomorphism $A\to B$.
\end{notationC}

\subsection{Organization of the paper}

In Section \ref{sec:ptd-racks}, we recall the definition of the category of pointed racks $\Rack$ and introduce pointed rack objects internal to cartesian monoidal categories.

In Section \ref{sec:alg-racks}, we introduce the categories of algebraic racks $\Rack(\Sch)$ and corack algebras $\Corack(\Alg)$. We also discuss the functor of points approach for algebraic racks.

In Section \ref{sec:corack}, we specialize our attention to the category of affine algebraic racks $\Rack(\Aff)$, which is equivalent to the opposite category $(\Corack(\Alg))\op$. As examples, we compute the corack algebras corresponding to three classes of affine algebraic racks. We also show that corack algebras are almost never coassociative or cocommutative, unlike Hopf algebras of algebraic groups.

In Section \ref{sec:pre-pf}, we discuss our approach to proving Theorem \ref{thm:main}, and we recall various facts about $k$-derivations and tangent spaces.

In Section \ref{sec:pf}, we prove Theorem \ref{thm:main}. A major step in the proof is to show that the Leibniz bracket $[\cdot,\cdot]$ in Theorem \ref{thm:main} agrees with the scheme-theoretic derivative $\ad\coloneq (d\Ad)_e$ of the \emph{adjoint representation} of $Q$ at the identity $e$.

In Section \ref{sec:ol}, we illustrate Theorem \ref{thm:main} with a class of affine algebraic racks $\OL_n$ whose Leibniz algebras $\ol_n(k)$ are not generally Lie algebras.
Then we provide several results relating the structure of an affine algebraic rack to that of its Leibniz algebra, including sufficient conditions for the latter to be abelian. These results recover the analogous results for Lie algebras of affine algebraic groups.

In Section \ref{sec:open}, we propose directions for future research.

\begin{ack}
    An early draft of this paper was submitted as the final project for a course on affine algebraic groups taught by Prof.\ Bogdan Ion at the University of Pittsburgh in fall 2025.
\end{ack}

We "prove the theorem piecemeal in Propositions \ref{prop:adj}, \ref{prop:jacobi}, and \ref{prop:functor} and Remark \ref{rmk:recover}." A "major step in the proof" is "to show that the Leibniz bracket $[\cdot,\cdot]$ in Theorem \ref{thm:main} agrees with the scheme-theoretic derivative $\ad\coloneq (d\Ad)_e$ of the \emph{adjoint representation} of $Q$ at the identity $e$."

We prove the theorem piecemeal in the following results. \begin{prop}\label{prop:adj}
    When considered as a map $\ad\colon\q\to\mathfrak{gl}(\q)$, the adjoint representation of $\q$ is given by the left argument of the Leibniz bracket $D\mapsto [D,\cdot]$. That is, $\ad_D(E)=[D,E]$ for all $k$-derivations $D,E\in\q$. In particular, $\q$ is closed under $[\cdot,\cdot]$.
\end{prop} \begin{prop}\label{prop:jacobi}
    The convolution operation $[\cdot,\cdot]$ satisfies the Jacobi identity.
\end{prop} \begin{prop}\label{prop:functor}
    The assignments $Q\mapsto\q$ and $\phi\mapsto \phi'\coloneq (d\phi)_e$ define a covariant functor $\Rack(\Aff)\to\leib$.
\end{prop} We also use the following remark. \begin{rmk}\label{rmk:recover}
    If $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\q,[\cdot,\cdot])$ is the Lie algebra of $G$. To see this, note that $Q$ and $G$ have the same underlying scheme and identity $e$, so they also have the same tangent space $\q$ at $e$. By the definition of $\Conj(G)$, the adjoint representations $\Ad$ of $Q$ and $\ad$ of $\q$ are therefore the same as those of $G$ and its Lie algebra (see, for example, J. S. Milne, \emph{Algebraic Groups, Lie Groups, and their Arithmetic Subgroups}*{Rem.\ 10.24}). Hence, the claim follows from the preceding proposition.\footnote{Alternatively, one can use Example \ref{ex:corack-conj} and the Hopf algebra axioms to directly (if tediously) verify that the Leibniz bracket $[D,E]=(D\otimes E)\circ\nabla$ agrees with the usual Lie bracket $(D\otimes E-E\otimes D)\circ\Delta$ on $\q=\operatorname{Lie}(G)$ (see W. C. Waterhouse, \emph{Introduction to Affine Group Schemes}*{Sec.\ 12.2}); we leave the details to the reader. As a hint, first deduce the identity $\eps\circ S=\eps$ from the axiom $\mu\circ (S\otimes\id)\circ\Delta=\eta\circ\eps$ by post-composing with $\eps$, and then deduce from these two equalities that $D\circ S=-D$.}

    Similarly, if $k=\R$ or $\mathbb{C}$ and $Q(k)$ is an algebraic Lie rack, then our definitions of $\Ad$ and $\ad$ coincide with the ones for Lie racks in \cite{kinyon}, so $(\q,[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.
\end{rmk} A major step in the proof is to show that the Leibniz bracket $[\cdot,\cdot]$ in the main theorem agrees with the scheme-theoretic derivative $\ad\coloneq (d\Ad)_e$ of the \emph{adjoint representation} of $Q$ at the identity $e$.

\label{thm:main}
    Let $Q=\Spec(A)$ be an affine algebraic rack with unit $e\in Q$, and let $\mathfrak{q}\coloneq \Der_k(A,k)$ be the vector space of $k$-derivations of $A$. Define a Leibniz bracket $[\cdot,\cdot]\colon\mathfrak{q}\times\mathfrak{q}\to\mathfrak{q}$ by convolution with respect to the corack operation $\nabla$ of $A$ (see below):
    \[
    [D,E]\coloneq (D\otimes E)\circ\nabla.
    \]
    Then the assignment $Q\mapsto(\mathfrak{q},[\cdot,\cdot])$ defines a covariant functor $\Rack(\Aff)\to\leib$ sending morphisms of affine algebraic racks $\varphi\colon Q\to R$ to their derivatives $(d\varphi)_e\colon \mathfrak{q}\to\mathfrak{r}$ at $e$.

    Furthermore, if $G$ is an affine algebraic group and $Q=\Conj(G)$ is its conjugation quandle scheme, then $(\mathfrak{q},[\cdot,\cdot])$ is the Lie algebra of $G$. On the other hand, if $k=\mathbb{R}$ or $\mathbb{C}$ and $Q(k)$ is a Lie rack, then $(\mathfrak{q},[\cdot,\cdot])$ is the Leibniz algebra of $Q(k)$ as a Lie rack.,

Question:
Let $k$ be a field, and let $Q=\operatorname{Spec}(A)$ be an affine algebraic rack over $k$ with unit $e\in Q$. Write $\nabla\colon A\to A\otimes_k A$ for the corack operation of $A$, and define
\[
\mathfrak q:=\operatorname{Der}_k(A,k),\qquad [D,E]:=(D\otimes E)\circ \nabla\quad(D,E\in\mathfrak q).
\]
Here $\Rack(\Aff)$ is the category of affine algebraic racks over $k$, and $\leib$ is the category of Leibniz algebras over $k$. Also, for an affine algebraic group $G$, let $\operatorname{Conj}(G)$ denote $G$ equipped with its conjugation quandle structure. Which statement holds for every such $Q$ (and, when applicable, in the special cases just described)?

The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q=\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ extends to a covariant functor $\Rack(\Aff)\to\leib$ that sends each morphism of affine algebraic racks $\varphi\colon Q\to R$ to its derivative at the unit, $(d\varphi)_e\colon \mathfrak q\to\mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ is the usual Lie algebra of $G$; and if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$.

Choices:
B. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q=\operatorname{Der}_k(A,k)$ into a Lie algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ extends to a contravariant functor $\Rack(\Aff)\to\leib$ that sends each morphism of affine algebraic racks $\varphi\colon Q\to R$ to the pullback on derivations induced by $\varphi^\sharp$. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ is the usual Lie algebra of $G$; and if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$.
C. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q=\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and if $G$ is an affine algebraic group with $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ is the usual Lie algebra of $G$. Moreover, if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$.
D. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q=\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ extends to a covariant functor $\Rack(\Aff)\to\leib$ that sends each morphism of affine algebraic racks $\varphi\colon Q\to R$ to its derivative at the unit, $(d\varphi)_e\colon \mathfrak q\to\mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ is the usual Lie algebra of $G$ if and only if $\operatorname{char}(k)=0$; and if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$.
E. The bracket $[D,E]=(D\otimes E)\circ\nabla$ makes $\mathfrak q=\operatorname{Der}_k(A,k)$ into a Leibniz algebra, and the assignment $Q\mapsto (\mathfrak q,[\cdot,\cdot])$ extends to a covariant functor $\Rack(\Aff)\to\leib$ that sends each morphism of affine algebraic racks $\varphi\colon Q\to R$ to its derivative at the unit, $(d\varphi)_e\colon \mathfrak q\to\mathfrak r$. Moreover, if $G$ is an affine algebraic group and $Q=\operatorname{Conj}(G)$, then $(\mathfrak q,[\cdot,\cdot])$ identifies with the Lie algebra of $G$ only after replacing the convolution bracket by its antisymmetrization $(D\otimes E-E\otimes D)\circ\nabla$; and if $k=\mathbb R$ or $\mathbb C$ and $Q(k)$ is a Lie rack, then $(\mathfrak q,[\cdot,\cdot])$ is the Leibniz algebra of the Lie rack $Q(k)$.

covariant derivative-at-unit functoriality and non-antisymmetric Leibniz nature

drops the functorial-extension assertion

adds a spurious characteristic-zero restriction in the conjugation-case recovery

misstates the recovery step by requiring antisymmetrization instead of equality of the convolution bracket with the Lie bracket in the conjugation case

The stem gives the setup and definitions but does not reveal the correct conclusion. It does not explicitly state the functoriality, covariance, or the special-case identifications that distinguish the correct option.

This is essentially a theorem-statement recognition item: the correct choice is a near-complete restatement of the full result, with distractors formed by small perturbations. It tests recall of the theorem more than selection among independently motivated conclusions.

There is some reasoning pressure because the options differ in mathematically meaningful ways (Leibniz vs. Lie, covariant vs. contravariant, exact vs. weaker statement, characteristic restriction, antisymmetrization). However, the task is still mainly to recognize the exact theorem statement rather than generate or derive it.

The distractors are strong and plausible: they target common confusions about duality/functoriality, antisymmetry, characteristic assumptions, and omission of a stronger functorial claim. The weaker-true option is especially effective.

A technically strong but theorem-recall-heavy MCQ. It avoids answer leakage and uses high-quality distractors, but it is largely a direct restatement of a result rather than a genuinely generative reasoning question.


\begin{equation}\label{eq:bound.logm}
    \log m \le \big(M+o_m(1)\big) \sqrt{st}.
\end{equation}

Notice that any graph $H$ on $m$ vertices provides a red/blue edge-coloring of $K_m$ implying that $m < R(\omega(H)+1, \alpha(H)+1)$. Thus, by \eqref{eq:bound.logm}, for every subgraph $G'$ of $G$ on $m$ vertices, we have
\begin{equation*}
     \omega(G') \cdot \alpha(G') \ge
\Big( \frac{1}{M^2} + o_m(1) \Big) (\log m)^2.
\end{equation*}
Since $\omega(G') \le \omega(G)$, we obtain that \begin{equation*}
    \alpha(G') \ge
\Big( \frac{1}{M^2} + o_m(1) \Big) \frac{(\log m)^2}{\omega(G)}.
\end{equation*}
Hence, together with the fact that $\log m = \big(1+o_n(1)\big) \log n$, \Cref{lem:erdos} yields
\begin{equation*}
    \chi(G) \le \big(M^2 +o_n(1)\big)\frac{n\cdot\omega(G)}{(\log n)^2} + \frac{n}{(\log n)^3}.
\end{equation*}
Finally, we conclude that
\begin{equation*}
    \frac{\chi(G)}{\omega(G)} \le \big(M^2 +o_n(1)\big)\frac{n}{(\log n)^2}.\qedhere
\end{equation*}
\end{proof}

\section{Proof of Theorem~\ref{thm:g_upper}} \label{subsec:numerics}
In this section, we prove Theorem~\ref{thm:g_upper}. The currently best known upper bound for $R(k,k)$ is
\begin{equation}\label{eq:diagonal_upper_bound}
    R(k,k) \le \left( 4e^{-0.14e^{-1}} \right)^{k+o(k)}.
\end{equation}

\label{sec:intro}

Ramsey Theory, often described as the investigation of when order arises from chaos, is one of the most influential areas in Graph Theory. The \emph{Ramsey number} $R(s,t)$ is the minimum $n$ such that every red/blue edge-coloring of the complete graph on $n$ vertices contains either a clique on $s$ vertices colored red or a clique on $t$ vertices colored blue.

The first explicit upper bound for Ramsey numbers was obtained by Erd\H{o}s and Szekeres~\cite{E-Sz} in 1935, who showed that $R(s,t) \leqslant \binom{s+t-2}{s-1}$. Note that in the \emph{diagonal} case, this shows that $R(k,k)\leqslant 4^{k+o(k)}$.
On the other hand, Erd\H{o}s~\cite{Ramsey.lower} established the lower bound $R(k,k)\geqslant 2^{k/2+o(k)}$ in 1947 in one of the first uses of the probabilistic method. Subsequent improvements on the lower bound have only refined it by a constant multiplicative factor (see, e.g., Spencer’s bound obtained via the Lov\'asz Local Lemma~\cite{Spencer.LLL}). Since then, one of the major problems in modern combinatorics has been to determine the correct asymptotic behavior of the diagonal Ramsey number $R(k,k)$. More specifically, one could ask whether the limit
\begin{equation}\label{limit1}
    \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}
 exists and, if so, what is its value. This question appears as Problem 1 in~\cite{chung}, as Problem 77 in~\url{https://www.erdosproblems.com/77}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}.

In 1967, Erd\H{o}s~\cite{erdos} asked a seemingly unrelated question. Define $f(n)$ to be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{\chi(G)}{\omega(G)}$, where $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ the clique number of $G$. Formally, we have
\begin{equation*}
    f(n)\coloneq \max \Big\{ \ \frac{\chi(G)}{\omega(G)} \ :
    \ G \text{ is a graph on } n \text{ vertices } \Big\}.
\end{equation*}

Erd\H{o}s showed that $f(n) = \Theta(n/(\log n)^2)$ and asked whether the following limit exists:
\begin{equation}\label{limit2}
    \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}
This question also appears in~\cite{erdos2}, as Problem 627 in~\url{https://www.erdosproblems.com/627}, as Problem 53 in~\cite{chung}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.

\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \leqslant k^2$ we have that
    \begin{equation*}
        R(s,t) \leqslant R(k, k).
    \end{equation*}
\end{conjecture}

We remark that Conjecture~\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \eqref{limit1} and \eqref{limit2}, should they exist, under the assumption of Conjecture~\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.

\label{sec:intro}

The first explicit upper bound for Ramsey numbers was obtained by Erd\H{o}s and Szekeres~\cite{E-Sz} in 1935, who showed that $R(s,t) \leqslant \binom{s+t-2}{s-1}$. Note that in the \emph{diagonal} case, this shows that $R(k,k)\leqslant 4^{k+o(k)}$.
On the other hand, Erd\H{o}s~\cite{Ramsey.lower} established the lower bound $R(k,k)\geqslant 2^{k/2+o(k)}$ in 1947 in one of the first uses of the probabilistic method. Subsequent improvements on the lower bound have only refined it by a constant multiplicative factor (see, e.g., Spencer’s bound obtained via the Lov\'asz Local Lemma~\cite{Spencer.LLL}). Since then, one of the major problems in modern combinatorics has been to determine the correct asymptotic behavior of the diagonal Ramsey number $R(k,k)$. More specifically, one could ask whether the limit
\begin{equation}\label{limit1}
 \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}
 exists and, if so, what is its value. This question appears as Problem 1 in~\cite{chung}, as Problem 77 in~\url{https://www.erdosproblems.com/77}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}.

In 1967, Erd\H{o}s~\cite{erdos} asked a seemingly unrelated question. Define $f(n)$ to be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{\chi(G)}{\omega(G)}$, where $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ the clique number of $G$. Formally, we have
\begin{equation*}
 f(n)\coloneq \max \Big\{ \ \frac{\chi(G)}{\omega(G)} \ :
 \ G \text{ is a graph on } n \text{ vertices } \Big\}.
\end{equation*}

Erd\H{o}s showed that $f(n) = \Theta(n/(\log n)^2)$ and asked whether the following limit exists:
\begin{equation}\label{limit2}
 \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}
This question also appears in~\cite{erdos2}, as Problem 627 in~\url{https://www.erdosproblems.com/627}, as Problem 53 in~\cite{chung}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.

\begin{conjecture}\label{conj:weak.mult.RDC}
 For every $s, t, k \in \N$ such that $st \leqslant k^2$ we have that
 \begin{equation*}
 R(s,t) \leqslant R(k, k).
 \end{equation*}
\end{conjecture}

We remark that Conjecture~\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \eqref{limit1} and \eqref{limit2}, should they exist, under the assumption of Conjecture~\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.

\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \le k^2$ we have that
    \begin{equation*}
        R(s,t) \le R(k, k).
    \end{equation*}
\end{conjecture}

\begin{equation}\label{limit1}
    \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}

\begin{equation}\label{limit2}
    \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}

Erd\H{o}s showed that $f(n) = \Theta(n/(\log n)^2)$ and asked whether the following limit exists:
\begin{equation}\label{limit2}
 \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}
This question also appears in~\cite{erdos2}, as Problem 627 in~\url{https://www.erdosproblems.com/627}, as Problem 53 in~\cite{chung}, and on the webpage~\url{https://mathweb.ucsd.edu/~erdosproblems/}. In this note, we establish a connection between these two problems. In particular, we show that if the following conjecture is true, a solution to the former problem results in a solution to the latter problem.

We remark that Conjecture~\ref{conj:weak.mult.RDC} does not seem to appear in any existing work, but it is closely related to the Ramsey Diagonal Conjecture. In Appendix~\ref{sec:DiagonalConjecture}, we briefly discuss these conjectures. The following theorem establishes a connection between the limits in \eqref{limit1} and \eqref{limit2}, should they exist, under the assumption of Conjecture~\ref{conj:weak.mult.RDC}. Throughout this note, all logarithms are in base 2.

In~\cite{erdos}, Erd\H{o}s mentioned that by their method, it would be easy to prove that
\begin{equation*}
 \big(c+o(1)\big) \frac{n}{(\log n)^2} \le f(n)
 \le \big(C+o(1) \big) \frac{n}{(\log n)^2},
\end{equation*}
for constants $c=\frac{1}{4}$ and $C=1$. However, this appears to be a typographical error, as a careful application of their method yields the constants $c=\frac{1}{4}$ and $C=4$. Aided by recent breakthrough developments in Ramsey theory, we can prove the following.

\begin{theorem}\label{thm:g_upper}
 The function $f(n)$ satisfies
\begin{equation}\label{eq:thm.bound.f.1}
 f(n) \le \big( 3.71943+o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
Moreover, if Conjecture~\ref{conj:weak.mult.RDC} holds, then
 \begin{equation}\label{eq:thm.bound.f.2}
 f(n) \le \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
\end{theorem}

\section{Proof of Theorem~\ref{thm:diagonal_to_ratio}}
\label{sec:proof_1.2}
In this section, we establish upper and lower bounds sufficient to prove Theorem~\ref{thm:diagonal_to_ratio}. We write $h = o_n(t)$ if the functions $t,h\colon \N \to \RR$ satisfy $\lim\limits_{n \to \infty} \frac{h(n)}{t(n)} = 0$. When the variable $n$ is clear from the context, we simply write $h=o(t)$. Let us define
\begin{equation*}
g(n) \coloneqq \frac{(\log n)^2}{n} f(n),\quad
L \coloneqq \liminf\limits_{k \to \infty} \; \frac{\log (R(k,k))}{k}
\quad \text{ and } \quad
M \coloneq \limsup_{t \to \infty } \; \max_{s\le t} \ \frac{\log(R(s,t))}{\sqrt{st \ }}
\end{equation*}

\begin{fact}\label{fact} Under the assumption of Conjecture~\ref{conj:weak.mult.RDC}, equality holds in \eqref{ineq:limsups}.
\end{fact}
\begin{proof}
For $0 < s\le t$, let $k$ be the positive integer such that $(k-1)^2 < st \le k^2$. Let
\[D\coloneq\limsup_{k \to \infty} \;\frac{\log (R(k,k))}{k}.\]
Conjecture~\ref{conj:weak.mult.RDC} implies that
\begin{equation*}
 R(s,t) \le R(k,k) \le 2^{(D+o_k(1))k}
\end{equation*}
 and, therefore, we have
\begin{equation*}
 \frac{\log(R(s,t))}{\sqrt{st \ }} \le \frac{\log(R(k,k))}{k-1} \le D +o_k(1) = D +o_t(1). \qedhere
\end{equation*}
\end{proof}

Let $G$ be a graph on $n$ vertices with no clique or independent set of size $k_n+1$. In other words, $\alpha(G) \le k_n$ and $\omega(G) \le k_n$. As $\chi(G) \ge n/\alpha(G)$, we conclude that
\begin{equation}\label{eq:lower.bound.inf.g}
 g(n)=\frac{f(n)}{n/(\log n)^2} \ge \frac{\chi(G)}{\omega(G)}\cdot\frac{(\log n)^2}{n} \ge
\frac{(\log n)^2}{\alpha(G)\omega(G)}\ge
\frac{(\log n)^2}{k_n^2} \ge
L^2 + o_n(1),
\end{equation}
where the last inequality is true by \eqref{eq:bound.logn}. Hence, $\liminf\limits_{n \to \infty} g(n) \ge L^2$.

\begin{proof}[Proof of Theorem~\ref{thm:g_upper}]
If Conjecture~\ref{conj:weak.mult.RDC} holds, then Theorem~\ref{thm:upper}, Fact~\ref{fact}, and \eqref{eq:diagonal_upper_bound} imply~\eqref{eq:thm.bound.f.2}. Indeed,
\begin{equation*}
 f(n) \le \big( ( \log 4e^{-0.14e^{-1}} )^2 + o(1) \big) \frac{n}{(\log n)^2} < \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation*}

\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \le k^2$ we have that
    \begin{equation*}
        R(s,t) \le R(k, k).
    \end{equation*}
\end{conjecture}

\begin{equation}\label{limit1}
    \lim_{k \to \infty} \frac{\log(R(k,k))}{k}
\end{equation}

\begin{equation}\label{limit2}
    \lim_{n \to \infty} \frac{f(n)}{n/(\log n)^2}.
\end{equation}

\begin{theorem}\label{thm:diagonal_to_ratio}
    If Conjecture~\ref{conj:weak.mult.RDC} holds, and $\lim\limits_{k \to \infty} \frac{\log(R(k,k))}{k}$ exists and is equal to $\mathcal{L}$, then
    \begin{equation*}
        f(n)=\big(\mathcal{L}^2+o(1)\big)\frac{n}{(\log n)^2} .
    \end{equation*}
\end{theorem}

\begin{theorem}\label{thm:g_upper}
    The function $f(n)$ satisfies
\begin{equation}\label{eq:thm.bound.f.1}
 f(n) \le \big( 3.71943+o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
Moreover, if Conjecture~\ref{conj:weak.mult.RDC} holds, then
 \begin{equation}\label{eq:thm.bound.f.2}
    f(n) \le \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
\end{theorem}

\begin{theorem}\label{thm:upper}
    The function $g(n)$ satisfies
    \begin{equation*}
         \limsup_{n \to \infty} g(n) \le M^2 .
    \end{equation*}
\end{theorem}

In~\cite{erdos}, Erd\H{o}s mentioned that by their method, it would be easy to prove that
\begin{equation*}
    \big(c+o(1)\big) \frac{n}{(\log n)^2} \leqslant f(n)
    \leqslant \big(C+o(1) \big)  \frac{n}{(\log n)^2},
\end{equation*}
for constants $c=\frac{1}{4}$ and $C=1$. However, this appears to be a typographical error, as a careful application of their method yields the constants $c=\frac{1}{4}$ and $C=4$. Aided by recent breakthrough developments in Ramsey theory, we can prove the following.

\begin{theorem}\label{thm:g_upper}
    The function $f(n)$ satisfies
\begin{equation}\label{eq:thm.bound.f.1}
 f(n) \leqslant \big( 3.71943+o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
Moreover, if Conjecture~\ref{conj:weak.mult.RDC} holds, then
 \begin{equation}\label{eq:thm.bound.f.2}
    f(n) \leqslant \big( 3.70831 + o(1) \big) \frac{n}{(\log n)^2}.
\end{equation}
\end{theorem}

This upper bound on $f(n)$ is the first improvement in the asymptotics of $f(n)$ since 1967.

\subsection*{Organization of the paper} In Section~\ref{sec:proof_1.2}, we introduce some notation used throughout the paper and state bounds on $f(n)$, namely Theorems~\ref{thm:lower} and~\ref{thm:upper}, establishing a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$. Theorem~\ref{thm:diagonal_to_ratio} is an immediate consequence of these results, whose proofs appear at the end of Section~\ref{sec:proof_1.2}. In Section~\ref{subsec:numerics}, we establish the numerical bounds of Theorem~\ref{thm:g_upper}. In Appendix~\ref{sec:DiagonalConjecture}, we discuss versions of the Ramsey Diagonal Conjecture and how they relate to Conjecture~\ref{conj:weak.mult.RDC}.

The post-theorem introduction says that in Section~\ref{sec:proof_1.2} the authors "state bounds on $f(n)$, namely Theorems~\ref{thm:lower} and~\ref{thm:upper}, establishing a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$." It then states that "Theorem~\ref{thm:diagonal_to_ratio} is an immediate consequence of these results, whose proofs appear at the end of Section~\ref{sec:proof_1.2}."

The post-theorem introduction says that next the authors state bounds on $f(n)$, namely the following results.

\begin{theorem}\label{thm:lower}
    The function $g(n)$ satisfies
    \begin{equation*}
         \liminf_{n \to \infty} g(n) \ge L^2
    \quad \text{ and } \quad \limsup_{n \to \infty} g(n) \ge M^2.
    \end{equation*}
\end{theorem}

\begin{theorem}\label{thm:upper}
    The function $g(n)$ satisfies
    \begin{equation*}
         \limsup_{n \to \infty} g(n) \le M^2 .
    \end{equation*}
\end{theorem}

These establish a connection between the asymptotic behavior of $f(n)$ and the Ramsey numbers $R(s,t)$. It then states that, in establishing the main theorem, it is an immediate consequence of these results, whose proofs appear later.

\label{thm:diagonal_to_ratio}
If the following conjecture holds:
\begin{conjecture}\label{conj:weak.mult.RDC}
    For every $s, t, k \in \N$ such that $st \le k^2$ we have that
    \begin{equation*}
        R(s,t) \le R(k, k).
    \end{equation*}
\end{conjecture}
and $\lim\limits_{k \to \infty} \frac{\log(R(k,k))}{k}$ exists and is equal to $\mathcal{L}$, then
\begin{equation*}
    f(n)=\big(\mathcal{L}^2+o(1)\big)\frac{n}{(\log n)^2} .
\end{equation*}

Question:
Let $R(s,t)$ denote the Ramsey number, i.e. the least integer $N$ such that every graph on $N$ vertices contains either a clique of size $s$ or an independent set of size $t$. For each $n$, define
\[
f(n)\coloneq \max\left\{\frac{\chi(G)}{\omega(G)}: G \text{ is a graph on } n \text{ vertices}\right\},
\]
where $\chi(G)$ is the chromatic number of $G$ and $\omega(G)$ is the clique number of $G$. Assume that for every $s,t,k\in\mathbb N$ with $st\le k^2$, one has
\[
R(s,t)\le R(k,k),
\]
and also that the limit
\[
\lim_{k\to\infty}\frac{\log(R(k,k))}{k}
\]
exists and is equal to $\mathcal L$ (with logarithms in base $2$). Under these assumptions, which statement about $f(n)$ holds as $n\to\infty$?

One has
\[
f(n)=\big(\mathcal L^2+o(1)\big)\frac{n}{(\log n)^2},
\]
as $n\to\infty$.

Choices:
B.
One has
\[
f(n)=\big(\mathcal L+o(1)\big)\frac{n}{(\log n)^2},
\]
as $n\to\infty$.
C.
One has
\[
f(n)=O\!\left(\frac{n}{(\log n)^2}\right),
\]
as $n\to\infty$.
D.
One has
\[
\limsup_{n\to\infty}\frac{f(n)}{n/(\log n)^2}=\mathcal L^2,
\]
as $n\to\infty$.
E.
One has
\[
f(n)=\big(\mathcal L^2+o(1)\big)\frac{n}{\log n},
\]
as $n\to\infty$.

square-on-the-constant from combining lower/upper bounds

dropped the precise asymptotic constant \mathcal L^2 and retained only the growth order

full limit/equality conclusion replaced by only a limsup statement

normalization by (\log n)^2 in g(n)=f(n)/(n/(\log n)^2)

The stem does not explicitly state the conclusion or uniquely signal choice A. It gives definitions and hypotheses, then asks for the asymptotic consequence for f(n).

This is essentially a direct theorem-recall item: under the stated assumptions, the student is asked to identify the theorem's exact conclusion. It does not substantially reframe the result into a new setting or inference task.

Some reasoning is needed to distinguish the exact asymptotic from nearby variants such as a weaker O-bound, a limsup statement, or an incorrect constant/normalization. However, the main task is still recognition of the precise theorem statement rather than generating a conclusion from first principles.

The distractors are plausible and mathematically meaningful: they vary the constant, weaken the mode of convergence, or alter the logarithmic normalization. These reflect realistic failure modes in asymptotic reasoning.

A mathematically well-constructed but theorem-recall-heavy MCQ. It avoids answer leakage and has strong distractors, but it is close to a direct restatement of a result rather than a genuinely generative reasoning question.


\label{T:1}
Suppose $n\geq 3$ is an odd integer and $p\geq 5$ is a prime with
$\gcd(p-1,n)=1$.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\mathbb{Z}[x]$ of degree $n$
that are Eisenstein at the prime $p$.
Define $\delta_{p,n}(X)$ to be the
proportion of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which Heilbronn's criterion applies.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\varepsilon(p)\right\}
\,.
$$
Here $\varepsilon(p)\to 0$ and is effectively computable.

\label{S:reallyFinal}

\begin{proof}[Proof of Theorem~\ref{T:2}]

Let $Y>0$ be a parameter to be chosen later.
Let $\mathcal{F}^*_{p,n}$ denote the subset of
$\mathcal{F}_{p,n}$ consisting of polynom

\begin{proof}
First note that
    $f(k) = {p\choose k}/p^k$
    is a decreasing function. Indeed, for $k\le p-1$
    $$\frac{f(k)}{f(k+1)} = \frac{(k+1)p}{p-k} > 0,$$
    so $f(k) > f(k+1)$ for all $k\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\ge 2$,
    $$\frac{{p\choose 2}}{p^2} - \frac{{p\choose 3}}{p^3} \le \frac{A_p(n)}{p^n} \le \frac{{p\choose 2}}{p^2}.$$
    \end{proof}

\section{Proof of the main result}\label{S:final}

\begin{proof}[Proof of Theorem~\ref{T:1}]
Adopt the notation from the statement of the theorem.
Further, write $N_{p,n}(X)$ to denote the number of
$f\in\mathcal{F}_{p,n}$ with $\Ht(f)\leq X$.
We have $N_{p,n}(X)\sim E(p)(2X)^n$ where $E(p)$
is given in (\ref{E:eisenstein}).
This follows from~\cite{MR1995564} or alternatively,
from Proposition~\ref{P:count} with $t=0$.

First we will establish the lower bound
$\liminf_{X\to\infty}\delta_{p,n}(X)\geq 2/27$.
Assume $p\neq 7,11,19$.
By Lemma~\ref{lem: Froby}, we can write $p=2u+3v$ with
$(2,u)=1$ and $(3,v)=1$ for $p\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.
Therefore, provided $f(x)$ has no roots in $\F_2$
or $\F_3$, Heilbronn's criterion applies.
(Recall Lemma~\ref{L:heilbronn} and the discussion in~\S\ref{S:EH}.)
Write $N^*_{p,n}(X)$ to denote the counting function for this collection
of polynomials.
Using Proposition~\ref{P:count}
with $q_1=2$, $q_2=3$, we have
$$
  N^*_{p,n}(X)\sim E_p(n)C_2(n)C_3(n)(2X)^n
  \,,
$$
and therefore,
via Lemma~\ref{lem: A_p formula} with $n\geq 3$,
we find
$$
\liminf_{X\to\infty}
\delta_{n,p}(X)\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27
\,.
$$

In the case where $p\in\{7,11,19\}$,
we can write $p=2u+5v$ with
$(2,u)=1$ and $(5,v)=1$.
The same approach proves
$$
\liminf_{X\to\infty}\delta_{n,p}(X)\geq C_2(n)C_5(n)\geq(1/4)(8/25)>2/27
\,.
$$
Indeed, by Lemma~\ref{L:Cbounds},
as $n\geq 3$ is odd,
one has $C_5(n)\geq 8/25$.

It remains to establish a lower bound $\delta_{p,n}(X)\geq 1-\eps(p)$
where $\eps(p)\to 0$.
We may assume that $p$ is large.
Choose $Y$ maximal such that \mbox{$Y^4<p$}.
Let $q_1<\dots<q_t\leq Y$ be all the primes less than $Y$.
In particular, Lemma~\ref{lem: Froby} applies
to any pair of primes in $Q\colonequals\{q_1,\dots,q_t\}$
and we have $t=\pi(Y)\leq \pi(p^{1/4})$.
Before proceeding with the final estimate,
observe that by Lemma~\ref{L:Cbounds}, one has
$$
  D\colonequals\max_{j=1,\dots,t}\left\{\frac{C(q_j)}{1-C(q_j)}\right\} \le 1  \,.
$$
We wish to bound from above the
number of $p$-Eisenstein polynomials with $\Ht(f)\leq X$
such that Heilbronn's criterion does not apply for any
pair of primes in $Q$.
This means that $f$ has no root modulo $q$ for at most one $q\in Q$.
Using Proposition~\ref{P:count}
in the situation where
$\{q\in Q\mid f \text{ has a root in }\F_q\}$
equals $Q$, $Q\setminus\{q_1\}$, $Q\setminus\{q_2\}$, and so on,
we obtain
\begin{align}\label{E:liminf}
  &1-\liminf_{X\to\infty}\delta_{n,p}(X)
  \notag\\
  &\quad\leq
  \prod_{i=1}^t(1-C(q_i))+
  \sum_{j=1}^tC(q_j)\left((1-C(q_j))^{-1}\prod_{i=1}^t(1-C(q_i)\right)
  \notag\\
  &\quad\leq
  (1+tD)\prod_{i=1}^t(1-C(q_i))
  \notag\\
  &\quad\leq
  \left(1+\pi\left(p^{1/4}\right)\right)\left(\frac{3}{4}\right)^{\pi(p^{1/4})}=:\eps(p)\,.
\end{align}
In the last step, we have used
the fact that $C_p(n)\ge 1/4$ for $p\geq 2$
which follows from Lemma~\ref{L:Cbounds}.
\end{proof}

We are interested in proving statistical results concerning
polynomials with integer coefficents.  Let $\mathcal{F}\subseteq\mathbb{Z}[x]$.
Before proceeding, we must have an ordering on our set,
which is typically endowed by a height function.
Let $\mathcal{F}\subseteq\mathbb{Z}[x]$ be a collection of polynomials.
Most importantly, a height function $H:\mathcal{F}\to\R_{\geq 0}$ should satisfy the following property:
For any $X>0$, the set
$$
\{f\in\mathcal{F}\mid H(f)\leq X\}
$$
is finite.
Given $$f(x)=a_n x^n+\dots+a_1 x+a_0\in\mathbb{Z}[x]\,,$$
we define the standard height function
$$
\Ht(f)=\max\{|a_0|,|a_1|,\dots,|a_n|\}
\,.
$$
One might consider other height functions, but here we will always
use $\Ht(f)$ as defined above.
Given a subset $\mathcal{F}\subseteq\mathbb{Z}[x]$, we define
the associated counting function
$$
  N(X)=\#\{f\in\mathcal{F}\mid \Ht(f)\leq X\}
  \,.
$$
Given a distinguished subset $\mathcal{F}^*\subseteq\mathcal{F}$,
we let $\delta(X)$ denote the proportion of elements of $\mathcal{F}$ that belong to $\mathcal{F}^*$; that is
$$
  \delta(X)=\frac{N^*(X)}{N(X)}
  \,,
$$
where we have written $N^*(X)$ to denote the counting function associated to $\mathcal{F}^*$.
One then defines the density of $\mathcal{F}^*$ in $\mathcal{F}$ to be
$$
\delta\colonequals \lim_{X\to\infty}\delta(X)
\,,
$$
provided the limit exists.
In cases where it is difficult to prove this limit exists,
one might seek bounds on the lower and upper densities:
$$
\underline{\delta}\colonequals
\liminf_{X\to\infty}\delta(X)
\,,\;\;
\overline{\delta}\colonequals
\limsup_{X\to\infty}\delta(X)
\,.$$

We begin with an important example.
Fix a positive integer $n$.
Denote by $\mathcal{F}_n$ the collection
of all monic polynomials in $\mathbb{Z}[x]$ of degree $n$.
It is easy to show that
$$
   N(X)\sim (2X)^n
   \,.
$$
Let $\mathcal{F}_n^*$ be the
subset of $\mathcal{F}_n$ consisting of Eisenstein polynomials.
Dubickas considers the probability that a randomly chosen
polynomial of degree $n$ is Eisenstein (see~\cite{MR1995564}).  Indeed, he
proves that the associated density exists and that it equals
\begin{equation}\label{E:dub}
  \delta=1-\prod_p\left(1-\frac{1}{p^n}+\frac{1}{p^{n+1}}\right)
  \,.
\end{equation}
This appears to be a topic of interest, as this paper led to
the papers~\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.

On the other hand, one of the central objects of study in algebraic number theory
are number fields.  As an Eisenstein polynomial
$f\in\mathcal{F}^*_n$ is always irreducible,
it generates a number field $K$ of degree $n$.
That is, if we define $K=\mathbb{Q}[x]/(f)$ then one has $[K:\mathbb{Q}]=n$.
We can ask various questions about properties of
this number field as we vary the polynomial.
As is standard, we let $\O_K$ denote the ring of integers in $K$,
and let $N:K\to\mathbb{Q}$ denote the norm map.

One classical question is regarding when the Euclidean algorithm holds in $K$.
We call a number field $K$ norm-Euclidean if for every $\alpha,\beta\in\O_K$, $\beta\neq 0$,
there exists $\gamma,\rho\in\O_K$ such that $\alpha=\gamma\beta+\rho$
and $|N(\rho)|<|N(\gamma)|$.  This is the generalization of the
division property of $\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps.  The statement that $K$ is norm-Euclidean is equivalent to $\O_K$ being a Euclidean domain with respect to the function $\partial(\alpha)=|N(\alpha)|$.
The quadratic fields that are norm-Euclidean have been completely determined
(see~\cite{MR41883, MR41885, MR53162, MR1191702}),
but this has not been accomplished in full for any degree $n>2$.
The reader may be interested to consult the survey article~\cite{MR1362867}.

One knows that
all norm-Euclidean fields have class number one, but the converse is not
true in general. Recall that $K$ having class number one is equivalent to $\O_K$ being a unique factorization domain.
We expect the norm-Euclidean property to be rather rare in general, but proving this is quite difficult.
On the other hand, if $K$ is generated by an Eisenstein polynomial, we actually have a tool available to us, due to the existence of a totally ramified prime.  More specifically, if $f$ is Eisenstein at the prime $p$, then $p$ is totally ramified in $K$; that is, $(p)=\mathfrak{p}^n$ for some prime ideal $\mathfrak{p}$.  The tool we refer to is as follows
(see~\cite{MR1574525, MR35313, MR43134, MR752120}):

\begin{lemma}[Heilbronn's criterion]\label{L:heilbronn}
Let $K$ be a number field of degree $n$.
Let $p$ be a prime that is totally ramified in $K$.
If we can write $p=a+b$ where $\{a,-b\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.
\end{lemma}
For clarity, recall that we refer to $a\in \mathbb{Z}$ as a norm (from $\O_K$) when
there exists $\alpha\in \O_K$ such that $N(\alpha)=a$.
Note that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence
one can simply write $\{a,b\}$ where we have written $\{a,-b\}$
in the lemma above.  Additionally, note that the criterion never applies
when $p=2$ or $p=3$, so we should always be considering $p\geq 5$ in our applications
of this result.

A natural question to ask is:  How often does Heilbronn's criterion apply to a number field $K$?
One could ask this question in the case of all number fields of fixed degree,
ordered by their discriminant.
There are some results in this direction when
$n=3$ and $n=5$ (see~\cite{MR4613609, MR4649640}).
However, such questions are intractable
at present when $n>5$ (see, for example, the introduction to~\cite{MR2745272}).
On the other hand, if we consider the corresponding
statistical questions for polynomials, then something can be said.  For convenience, we will say Heilbronn's criterion applies to a polynomial $f\in\mathbb{Z}[x]$ if the criterion applies to the number field
$K=\mathbb{Q}[x]/(f)$.  We establish the following result:

We are interested in proving statistical results concerning
polynomials with integer coefficents. Let $\mathcal{F}\subseteq\mathbb{Z}[x]$.
Before proceeding, we must have an ordering on our set,
which is typically endowed by a height function.
Let $\mathcal{F}\subseteq\mathbb{Z}[x]$ be a collection of polynomials.
Most importantly, a height function $H:\mathcal{F}\to\R_{\geq 0}$ should satisfy the following property:
For any $X>0$, the set
$$
\{f\in\mathcal{F}\mid H(f)\leq X\}
$$
is finite.
Given $$f(x)=a_n x^n+\dots+a_1 x+a_0\in\mathbb{Z}[x]\,,$$
we define the standard height function
$$
\Ht(f)=\max\{|a_0|,|a_1|,\dots,|a_n|\}
\,.
$$
One might consider other height functions, but here we will always
use $\Ht(f)$ as defined above.
Given a subset $\mathcal{F}\subseteq\mathbb{Z}[x]$, we define
the associated counting function
$$
 N(X)=\#\{f\in\mathcal{F}\mid \Ht(f)\leq X\}
 \,.
$$
Given a distinguished subset $\mathcal{F}^*\subseteq\mathcal{F}$,
we let $\delta(X)$ denote the proportion of elements of $\mathcal{F}$ that belong to $\mathcal{F}^*$; that is
$$
 \delta(X)=\frac{N^*(X)}{N(X)}
 \,,
$$
where we have written $N^*(X)$ to denote the counting function associated to $\mathcal{F}^*$.
One then defines the density of $\mathcal{F}^*$ in $\mathcal{F}$ to be
$$
\delta\colonequals \lim_{X\to\infty}\delta(X)
\,,
$$
provided the limit exists.
In cases where it is difficult to prove this limit exists,
one might seek bounds on the lower and upper densities:
$$
\underline{\delta}\colonequals
\liminf_{X\to\infty}\delta(X)
\,,\;\;
\overline{\delta}\colonequals
\limsup_{X\to\infty}\delta(X)
\,.$$

We begin with an important example.
Fix a positive integer $n$.
Denote by $\mathcal{F}_n$ the collection
of all monic polynomials in $\mathbb{Z}[x]$ of degree $n$.
It is easy to show that
$$
 N(X)\sim (2X)^n
 \,.
$$
Let $\mathcal{F}_n^*$ be the
subset of $\mathcal{F}_n$ consisting of Eisenstein polynomials.
Dubickas considers the probability that a randomly chosen
polynomial of degree $n$ is Eisenstein (see~\cite{MR1995564}). Indeed, he
proves that the associated density exists and that it equals
\begin{equation}\label{E:dub}
 \delta=1-\prod_p\left(1-\frac{1}{p^n}+\frac{1}{p^{n+1}}\right)
 \,.
\end{equation}
This appears to be a topic of interest, as this paper led to
the papers~\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.

On the other hand, one of the central objects of study in algebraic number theory
are number fields. As an Eisenstein polynomial
$f\in\mathcal{F}^*_n$ is always irreducible,
it generates a number field $K$ of degree $n$.
That is, if we define $K=\mathbb{Q}[x]/(f)$ then one has $[K:\mathbb{Q}]=n$.
We can ask various questions about properties of
this number field as we vary the polynomial.
As is standard, we let $\O_K$ denote the ring of integers in $K$,
and let $N:K\to\mathbb{Q}$ denote the norm map.

One classical question is regarding when the Euclidean algorithm holds in $K$.
We call a number field $K$ norm-Euclidean if for every $\alpha,\beta\in\O_K$, $\beta\neq 0$,
there exists $\gamma,\rho\in\O_K$ such that $\alpha=\gamma\beta+\rho$
and $|N(\rho)|<|N(\gamma)|$. This is the generalization of the
division property of $\mathbb{Z}$, which ensures that the Euclidean algorithm terminates after a finite number of steps. The statement that $K$ is norm-Euclidean is equivalent to $\O_K$ being a Euclidean domain with respect to the function $\partial(\alpha)=|N(\alpha)|$.
The quadratic fields that are norm-Euclidean have been completely determined
(see~\cite{MR41883, MR41885, MR53162, MR1191702}),
but this has not been accomplished in full for any degree $n>2$.
The reader may be interested to consult the survey article~\cite{MR1362867}.

\begin{lemma}[Heilbronn's criterion]\label{L:heilbronn}
Let $K$ be a number field of degree $n$.
Let $p$ be a prime that is totally ramified in $K$.
If we can write $p=a+b$ where $\{a,-b\}$ are not norms and $a$ is an $n$-th power residue modulo $p$, then $K$ is not norm-Euclidean.
\end{lemma}
For clarity, recall that we refer to $a\in \mathbb{Z}$ as a norm (from $\O_K$) when
there exists $\alpha\in \O_K$ such that $N(\alpha)=a$.
Note that if $n$ is odd, then $b$ is a norm if and only if $-b$ is norm and hence
one can simply write $\{a,b\}$ where we have written $\{a,-b\}$
in the lemma above. Additionally, note that the criterion never applies
when $p=2$ or $p=3$, so we should always be considering $p\geq 5$ in our applications
of this result.

A natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?
One could ask this question in the case of all number fields of fixed degree,
ordered by their discriminant.
There are some results in this direction when
$n=3$ and $n=5$ (see~\cite{MR4613609, MR4649640}).
However, such questions are intractable
at present when $n>5$ (see, for example, the introduction to~\cite{MR2745272}).
On the other hand, if we consider the corresponding
statistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\in\mathbb{Z}[x]$ if the criterion applies to the number field
$K=\mathbb{Q}[x]/(f)$. We establish the following result:

Suppose $n\geq 3$ is odd and $p\geq 5$ is prime with
$\gcd(p-1,n)=1$.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\Z[x]$ of degree $n$
that are Eisenstein at the prime $p$.
We order our polynomials by the natural height $\Ht(f)$.
Define $\delta_{p,n}(X)$ to be the
proportion of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which Heilbronn's criterion applies.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\eps(p)\right\}
\,,
$$
where $\eps(p)\to 0$ and is effectively computable.
In particular, the lower density tends to $1$ as $p\to\infty$
uniformly in $n$.
We also give a version of this result where we weaken the
condition on $\gcd(p-1,n)$.

We are interested in proving statistical results concerning
polynomials with integer coefficents. Let $\mathcal{F}\subseteq\Z[x]$.
Before proceeding, we must have an ordering on our set,
which is typically endowed by a height function.
Let $\mathcal{F}\subseteq\Z[x]$ be a collection of polynomials.
Most importantly, a height function $H:\mathcal{F}\to\R_{\geq 0}$ should satisfy the following property:
For any $X>0$, the set
$$
\{f\in\mathcal{F}\mid H(f)\leq X\}
$$
is finite.
Given $$f(x)=a_n x^n+\dots+a_1 x+a_0\in\Z[x]\,,$$
we define the standard height function
$$
\Ht(f)=\max\{|a_0|,|a_1|,\dots,|a_n|\}
\,.
$$
One might consider other height functions, but here we will always
use $\Ht(f)$ as defined above.
Given a subset $\mathcal{F}\subseteq\mathbb{Z}[x]$, we define
the associated counting function
$$
 N(X)=\#\{f\in\mathcal{F}\mid \Ht(f)\leq X\}
 \,.
$$
Given a distinguished subset $\mathcal{F}^*\subseteq\mathcal{F}$,
we let $\delta(X)$ denote the proportion of elements of $\mathcal{F}$ that belong to $\mathcal{F}^*$; that is
$$
 \delta(X)=\frac{N^*(X)}{N(X)}
 \,,
$$
where we have written $N^*(X)$ to denote the counting function associated to $\mathcal{F}^*$.
One then defines the density of $\mathcal{F}^*$ in $\mathcal{F}$ to be
$$
\delta\colonequals \lim_{X\to\infty}\delta(X)
\,,
$$
provided the limit exists.
In cases where it is difficult to prove this limit exists,
one might seek bounds on the lower and upper densities:
$$
\underline{\delta}\colonequals
\liminf_{X\to\infty}\delta(X)
\,,\;\;
\overline{\delta}\colonequals
\limsup_{X\to\infty}\delta(X)
\,.$$

We begin with an important example.
Fix a positive integer $n$.
Denote by $\mathcal{F}_n$ the collection
of all monic polynomials in $\Z[x]$ of degree $n$.
It is easy to show that
$$
 N(X)\sim (2X)^n
 \,.
$$
Let $\mathcal{F}_n^*$ be the
subset of $\mathcal{F}_n$ consisting of Eisenstein polynomials.
Dubickas considers the probability that a randomly chosen
polynomial of degree $n$ is Eisenstein (see~\cite{MR1995564}). Indeed, he
proves that the associated density exists and that it equals
\begin{equation}\label{E:dub}
 \delta=1-\prod_p\left(1-\frac{1}{p^n}+\frac{1}{p^{n+1}}\right)
 \,.
\end{equation}
This appears to be a topic of interest, as this paper led to
the papers~\cite{MR3063896, MR3278954, MR3544517, MR3893668}, among others.

A natural question to ask is: How often does Heilbronn's criterion apply to a number field $K$?
One could ask this question in the case of all number fields of fixed degree,
ordered by their discriminant.
There are some results in this direction when
$n=3$ and $n=5$ (see~\cite{MR4613609, MR4649640}).
However, such questions are intractable
at present when $n>5$ (see, for example, the introduction to~\cite{MR2745272}).
On the other hand, if we consider the corresponding
statistical questions for polynomials, then something can be said. For convenience, we will say Heilbronn's criterion applies to a polynomial $f\in\Z[x]$ if the criterion applies to the number field
$K=\Q[x]/(f)$. We establish the following result:

When $p$ is large, there are many ways to write $p=a+b$ and hence
determining the exact density of polynomials that satisfy Heilbronn's
criterion is likely very difficult. However, the previous
theorem says that the lower density tends to $1$ as $p\to\infty$,
uniformly in the variable $n$.

\begin{theorem}\label{T:2}
Suppose $n\geq 2$ and $p\geq 5$ is a prime with
$$
\gcd(p-1,n)<\frac{p^{1/2}}{(\log p)^2}
\,.
$$
Define $\delta_{p,n}(X)$ as in Theorem~\ref{T:1}.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq 1-\hat{\eps}(p)
\,,
$$
where $\hat{\eps}(p)\to 0$ and is effectively computable.
\end{theorem}

Before proceeding, we require a couple definitions.
First we define
\begin{equation}\label{E:eisenstein}
E_p(n)\colonequals
\frac{1}{p^n}-\frac{1}{p^{n+1}}
\,,
\end{equation}
which is the $p$-Eisenstein density that contributes
to (\ref{E:dub}).
Secondly, we write $C_q(n)$ to denote --- the number of
$(a_0,\dots,a_{n-1})\in(\Z/q\Z)^n$ such that the corresponding polynomial
in $\F_q[x]$ has no roots in $\F_q$ ---
divided by $q^n$.
We are now ready to prove the following result, which
will be used in the proof of Theorem~\ref{T:1}.
\begin{prop}\label{P:count}
Let $n\in\Z^+$.
Suppose $q_1<\dots<q_t<p$ are primes.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\Z[x]$ of degree $n$
that are Eisenstein at the prime $p$.
The number of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which
$f$ has no roots in $\F_q$ for all $q\in\{q_1,\dots,q_t\}$
equals
$$
 E_p(n)\left(\prod_{i=1}^t C_{q_i}(n)\right)(2X)^n+O\left(n 2^n q_1\dots q_t p^2 (2X)^{n-1}\right)
 \,.
$$
\end{prop}

It remains to establish a lower bound $\delta_{p,n}(X)\geq 1-\eps(p)$
where $\eps(p)\to 0$.
We may assume that $p$ is large.
Choose $Y$ maximal such that \mbox{$Y^4<p$}.
Let $q_1<\dots<q_t\leq Y$ be all the primes less than $Y$.
In particular, Lemma~\ref{lem: Froby} applies
to any pair of primes in $Q\colonequals\{q_1,\dots,q_t\}$
and we have $t=\pi(Y)\leq \pi(p^{1/4})$.
Before proceeding with the final estimate,
observe that by Lemma~\ref{L:Cbounds}, one has
$$
 D\colonequals\max_{j=1,\dots,t}\left\{\frac{C(q_j)}{1-C(q_j)}\right\} \le 1 \,.
$$
We wish to bound from above the
number of $p$-Eisenstein polynomials with $\Ht(f)\leq X$
such that Heilbronn's criterion does not apply for any
pair of primes in $Q$.
This means that $f$ has no root modulo $q$ for at most one $q\in Q$.
Using Proposition~\ref{P:count}
in the situation where
$\{q\in Q\mid f \text{ has a root in }\F_q\}$
equals $Q$, $Q\setminus\{q_1\}$, $Q\setminus\{q_2\}$, and so on,
we obtain
\begin{align}\label{E:liminf}
 &1-\liminf_{X\to\infty}\delta_{n,p}(X)
 \notag\\
 &\quad\leq
 \prod_{i=1}^t(1-C(q_i))+
 \sum_{j=1}^tC(q_j)\left((1-C(q_j))^{-1}\prod_{i=1}^t(1-C(q_i)\right)
 \notag\\
 &\quad\leq
 (1+tD)\prod_{i=1}^t(1-C(q_i))
 \notag\\
 &\quad\leq
 \left(1+\pi\left(p^{1/4}\right)\right)\left(\frac{3}{4}\right)^{\pi(p^{1/4})}=:\eps(p)\,.
\end{align}
In the last step, we have used
the fact that $C_p(n)\ge 1/4$ for $p\geq 2$
which follows from Lemma~\ref{L:Cbounds}.
\end{proof}

\begin{theorem}\label{T:1}
Suppose $n\geq 3$ is an odd integer and $p\geq 5$ is a prime with
$\gcd(p-1,n)=1$.
Let $\mathcal{F}_{p,n}$ denote the collection
of monic polynomials $f\in\Z[x]$ of degree $n$
that are Eisenstein at the prime $p$.
Define $\delta_{p,n}(X)$ to be the
proportion of polynomials $f\in\mathcal{F}_{p,n}$
with $\Ht(f)\leq X$
for which Heilbronn's criterion applies.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq \max\left\{\frac{2}{27}\,,\;1-\eps(p)\right\}
\,.
$$
Here $\eps(p)\to 0$ and is effectively computable.
\end{theorem}

When $p$ is large, there are many ways to write $p=a+b$ and hence
determining the exact density of polynomials that satisfy Heilbronn's
criterion is likely very difficult.  However, the previous
theorem says that the lower density tends to $1$ as $p\to\infty$,
uniformly in the variable $n$.

As a result, we immediately obtain the following consequence:
When $n\geq 3$ is odd,
a positive proportion of Eisenstein polynomials of degree $n$
fail to generate norm-Euclidean fields.

The condition from Theorem~\ref{T:1} that $\mathrm{gcd}(p-1,n) = 1$
is not a severe restriction as there are infinitely
many primes $p$ that satisfy this condition for each fixed $n$.
Nonetheless, we show that we can
relax this condition at the expense of considering larger primes.
This will allow us to deal with the case where $n$ is even.

\begin{theorem}\label{T:2}
Suppose $n\geq 2$ and $p\geq 5$ is a prime with
$$
\gcd(p-1,n)<\frac{p^{1/2}}{(\log p)^2}
\,.
$$
Define $\delta_{p,n}(X)$ as in Theorem~\ref{T:1}.
One has
$$
\liminf_{X\to\infty}
\delta_{p,n}(X)\geq 1-\hat{\varepsilon}(p)
\,,
$$
where $\hat{\varepsilon}(p)\to 0$ and is effectively computable.
\end{theorem}

\begin{corollary}\label{C:1}
Fix an integer $n\geq 2$.
A positive proportion of Eisenstein polynomials of degree $n$
fail to generate norm-Euclidean fields.
\end{corollary}

In \S\ref{S:EH} we discuss Eisenstein polynomials and Heilbronn's criterion,
and outline the strategy of our proof.  In addition, we establish some preliminary
lemmas.  In the case where the proofs are very short, we err on the side of giving
proofs of known results for the sake of completeness.
In \S\ref{S:gotime} we prove our main counting result for Eisenstein polynomials
satisfying certain other local conditions.  In \S\ref{S:local} we prove a formula
for the relevant local densities, which is then used to establish the necessary bounds.
In \S\ref{S:final}
we put everything together and prove Theorem~\ref{T:1}. Finally, in \S\ref{S:reallyFinal}, we prove Theorem \ref{T:2}.

In \,\S\ref{S:EH} the authors “discuss Eisenstein polynomials and Heilbronn's criterion, and outline the strategy of our proof,” and “establish some preliminary lemmas.” Then “in \,\S\ref{S:gotime} we prove our main counting result for Eisenstein polynomials satisfying certain other local conditions.” Next, “in \,\S\ref{S:local} we prove a formula for the relevant local densities, which is then used to establish the necessary bounds.” Finally, “in \,\S\ref{S:final} we put everything together and prove Theorem~\ref{T:1}.”

\label{S:EH}

Let $f\in\Z[x]$ and write
$$
  f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0
  \,.
$$
We say that $f$ is Eisenstein at the prime $p$
(or $p$-Eisenstein for short) if
$p \mid a_i$
for $i=0,\,

\begin{proof}[Proof of Lemma~\ref{L:heilbronn}]
Suppose $p$ is totally ramified and we can write $p=a+b$ as in
the statement of the result.
By way of contradiction, suppose $K$ is norm-Euclidean.
By hypothesis, $(p)=\mathfrak{p}^n$, where $N(\mathfrak{p})=p$.
It follows that $p=u\pi^n$ in $K$,
for some $u\in\mathcal{O}_K^\times$ and some
$\pi\in\mathcal{O}_K$ irreducible with $N(\pi)=p$.
Since $a$ is an $n$-th power residue modulo $n$, there exists $x\in\Z$ such that $x^n\equiv a\pmod{p}$.
Using the norm-Euclidean condition, we find
$x=\gamma\pi+\rho$ for some $\gamma,\rho\in\mathcal{O}_K$
with $|N(\rho)|<|N(\pi)|=p$.  We have $x\equiv \rho\pmod{\pi}$
and hence $x\equiv \rho^\sigma\pmod{\pi}$ for any
embedding $\sigma:K\to\C$.  This leads to
$a\equiv x^n\equiv N(\rho)\pmod{p}$.
We have $0<a<p$ and $-p<N(\rho)<p$.
Thus either $N(\rho)=a$ or $N(\rho)+p=a$.
This implies one of $\{a,-b\}$ is a norm,
a contradiction.
\end{proof}

\section{Counting $p$-Eisenstein polynomials with local specifications}\label{S:gotime}

Given $a=(a_0,\dots,a_{n-1})\in\Z^n$, we define
$$
\Ht(a)=\max\{|a_0|,\dots,|a_{n-1}|\}
$$
so that it agrees  with our polynomial height.
The next result is what allows us to count polynomials
of bounded height with local specifications.
This type of result is certainly well-known,
but the proof is short and so we include it for the
sake of completeness.
\begin{lemma}\label{L:sieve}
Let $S$ be a subset of $(\Z/m\Z)^n$.  We have
$$
\#\{a\in\Z^n\mid a\hspace{-1ex}\mod{m}\in S\,,\;\Ht(a)\leq X\}
=
\frac{|S|}{m^n}(2X)^n+O\left(n2^n m(2X)^{n-1}\right)
\,.
$$
\end{lemma}

\begin{proof}
Let $\mathcal{A}\colonequals [-X,X]^n\cap\Z^n$.
Choose the maximal $k\in\Z^+$ such that
$$
  \mathcal{B}\colonequals (-km,km]^n\cap\Z^n\subseteq \mathcal{A}
  \,.
$$
The set $\mathcal{B}$ is chosen so that
$$
\#\{a\in\mathcal{B}\mid a\hspace{-1ex}\mod{m}\in S\}
=
|S|(2k)^n
$$
Using the Binomial Theorem,
one has
$$
\left(\frac{X}{m}\right)^n \ge k^n \ge \left(\frac{X}{m}-1\right)^n
=
\left(\frac{X}{m}\right)^n+O\left(n2^n\left(\frac{X}{m}\right)^{n-1}\right)
\,,
$$
and therefore
$$
\#\{a\in\mathcal{B}\mid a\hspace{-1ex}\mod{m}\in S\}=
\frac{|S|}{m^n}(2X)^n+O\left(n 2^n m(2X)^{n-1}\right)
\,.
$$
Next one bounds
$$
  \#(\mathcal{A}\setminus\mathcal{B})
  \leq
  (2X+1)^n-(2km)^n
  \ll
  n2^n m(2X)^{n-1}
  \,.
$$
The result follows.
\end{proof}

\label{S:local}

The following lemma gives a formula for our local density
$C_p(n)=A_p(n)/p^n$.  Here $C_p(n)$ is the quantity
defined immediately before Proposition~\ref{P:count}
and $A_p(n)$ is defi

\begin{proof}
First note that
    $f(k) = {p\choose k}/p^k$
    is a decreasing function. Indeed, for $k\le p-1$
    $$\frac{f(k)}{f(k+1)} = \frac{(k+1)p}{p-k} > 0,$$
    so $f(k) > f(k+1)$ for all $k\le p-1$. Since $f(k) = 0$ for $k > p$, and $f(k) > 0$ for $k\le p$, it follows that $f(k)$ is decreasing. Since $f(k)$ is decreasing and $A_p(n)$ is an alternating sum of positive decreasing terms, we get for $n\ge 2$,
    $$\frac{{p\choose 2}}{p^2} - \frac{{p\choose 3}}{p^3} \le \frac{A_p(n)}{p^n} \le \frac{{p\choose 2}}{p^2}.$$
    \end{proof}

\section{Proof of the main result}\label{S:final}

\begin{proof}[Proof of Theorem~\ref{T:1}]
Adopt the notation from the statement of the theorem.
Further, write $N_{p,n}(X)$ to denote the number of
$f\in\mathcal{F}_{p,n}$ with $\Ht(f)\leq X$.
We have $N_{p,n}(X)\sim E(p)(2X)^n$ where $E(p)$
is given in (\ref{E:eisenstein}).
This follows from~\cite{MR1995564} or alternatively,
from Proposition~\ref{P:count} with $t=0$.

First we will establish the lower bound
$\liminf_{X\to\infty}\delta_{p,n}(X)\geq 2/27$.
Assume $p\neq 7,11,19$.
By Lemma~\ref{lem: Froby}, we can write $p=2u+3v$ with
$(2,u)=1$ and $(3,v)=1$ for $p\ge 36$. One can then manually check that the only examples of primes up to 36 that cannot be expressed in such a way are $p = 7, 11, 19$.
Therefore, provided $f(x)$ has no roots in $\F_2$
or $\F_3$, Heilbronn's criterion applies.
(Recall Lemma~\ref{L:heilbronn} and the discussion above.)
Write $N^*_{p,n}(X)$ to denote the counting function for this collection
of polynomials.
Using Proposition~\ref{P:count}
with $q_1=2$, $q_2=3$, we have
$$
  N^*_{p,n}(X)\sim E_p(n)C_2(n)C_3(n)(2X)^n
  \,,
$$
and therefore,
via Lemma~\ref{lem: A_p formula} with $n\geq 3$,
we find
$$
\liminf_{X\to\infty}
\delta_{n,p}(X)\geq C_2(n)C_3(n)= (1/4)(8/27)=2/27
\,.
$$

In the case where $p\in\{7,11,19\}$,
we can write $p=2u+5v$ with
$(2,u)=1$ and $(5,v)=1$.
The same approach proves
$$
\liminf_{X\to\infty}\delta_{n,p}(X)\geq C_2(n)C_5(n)\geq(1/4)(8/25)>2/27
\,.
$$
Indeed, by Lemma~\ref{L:Cbounds},
as $n\geq 3$ is odd,
one has $C_5(n)\geq 8/25$.

It remains to establish a lower bound $\delta_{p,n}(X)\geq 1-\eps(p)$
where $\eps(p)\to 0$.
We may assume that $p$ is large.
Choose $Y$ maximal such that \mbox{$Y^4<p$}.
Let $q_1<\dots<q_t\leq Y$ be all the primes less than $Y$.
In particular, Lemma~\ref{lem: Froby} applies
to any pair of primes in $Q\colonequals\{q_1,\dots,q_t\}$
and we have $t=\pi(Y)\leq \pi(p^{1/4})$.
Before proceeding with the final estimate,
observe that by Lemma~\ref{L:Cbounds}, one has
$$
  D\colonequals\max_{j=1,\dots,t}\left\{\frac{C(q_j)}{1-C(q_j)}\right\} \le 1  \,.
$$
We wish to bound from above the
number of $p$-Eisenstein polynomials with $\Ht(f)\leq X$
such that Heilbronn's criterion does not apply for any
pair of primes in $Q$.
This means that $f$ has no root modulo $q$ for at most one $q\in Q$.
Using Proposition~\ref{P:count}
in the situation where
$\{q\in Q\mid f \text{ has a root in }\F_q\}$
equals $Q$, $Q\setminus\{q_1\}$, $Q\setminus\{q_2\}$, and so on,
we obtain
\begin{align}\label{E:liminf}
  &1-\liminf_{X\to\infty}\delta_{n,p}(X)
  \notag\\
  &\quad\leq
  \prod_{i=1}^t(1-C(q_i))+
  \sum_{j=1}^tC(q_j)\left((1-C(q_j))^{-1}\prod_{i=1}^t(1-C(q_i)\right)
  \notag\\
  &\quad\leq
  (1+tD)\prod_{i=1}^t(1-C(q_i))
  \notag\\
  &\quad\leq
  \left(1+\pi\left(p^{1/4}\right)\right)\left(\frac{3}{4}\right)^{\pi(p^{1/4})}=:\eps(p)\,.
\end{align}
In the last step, we have used
the fact that $C_p(n)\ge 1/4$ for $p\geq 2$
which follows from Lemma~\ref{L:Cbounds}.
\end{proof}

This completes the proof of the main theorem.

Question:
Let \(n\ge 3\) be an odd integer and \(p\ge 5\) a prime such that \(\gcd(p-1,n)=1\). Let \(\mathcal F_{p,n}\) be the set of monic polynomials
\[
f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0\in \mathbb Z[x]
\]
of degree \(n\) that are Eisenstein at \(p\), i.e. \(p\mid a_i\) for every \(0\le i\le n-1\) and \(p^2\nmid a_0\). For such a polynomial, define its height by
\[
\operatorname{Ht}(f)=\max\{1,|a_{n-1}|,\dots,|a_0|\}.
\]
For each \(X>0\), let \(\delta_{p,n}(X)\) be the proportion of polynomials \(f\in \mathcal F_{p,n}\) with \(\operatorname{Ht}(f)\le X\) for which Heilbronn's criterion applies to the number field \(K=\mathbb Q[x]/(f)\), meaning that there exist integers \(a,b\) with \(p=a+b\), such that \(a\) is an \(n\)-th power residue modulo \(p\) and neither \(a\) nor \(-b\) is a norm from \(\mathcal O_K\). Under these assumptions, which quantitative estimate holds?

One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(p)\right\},
\]
where \(\varepsilon(p)\) is effectively computable and satisfies \(\varepsilon(p)\to 0\) as \(p\to\infty\).

Choices:
B.
One has
\[
\lim_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(p)\right\},
\]
where \(\varepsilon(p)\) is effectively computable and satisfies \(\varepsilon(p)\to 0\) as \(p\to\infty\).
C.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \frac{2}{27}.
\]
D.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge 1-\varepsilon(p),
\]
where \(\varepsilon(p)\to 0\) as \(p\to\infty\).
E.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(n)\right\},
\]
where \(\varepsilon(n)\) is effectively computable and satisfies \(\varepsilon(n)\to 0\) as \(n\to\infty\).

The stem does not disclose the conclusion; it only gives the setup and asks which asymptotic statement is valid. There is no explicit answer leakage, though the phrasing strongly signals that one option is the exact theorem statement.

This is essentially a direct theorem-recall item: the question asks for the precise limiting statement that holds under the stated hypotheses. It does not substantially transform the result into an application or independent inference task.

There is some reasoning pressure because the options differ by subtle logical strength (liminf vs limit vs limsup, retaining or dropping the max term, weaker true consequence). However, the main task is still recognizing the exact theorem statement rather than generating a conclusion from first principles.

The distractors are plausible and mathematically sharp: they test common errors involving quantifier strength, asymptotic notions, and weakening/strengthening of bounds. They are distinct and aligned with realistic failure modes.

Well-constructed distractors and no real answer leakage, but the item is primarily a theorem-restatement/recall question rather than a genuine generative reasoning task.


Question:
Let \(n\ge 3\) be an odd integer and \(p\ge 5\) a prime such that \(\gcd(p-1,n)=1\). Let \(\mathcal F_{p,n}\) be the set of monic polynomials
\[
f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0\in \mathbb Z[x]
\]
of degree \(n\) that are Eisenstein at \(p\), i.e. \(p\mid a_i\) for every \(0\le i\le n-1\) and \(p^2\nmid a_0\). For such a polynomial, define its height by
\[
\operatorname{Ht}(f)=\max\{1,|a_{n-1}|,\dots,|a_0|\}.
\]
For each \(X>0\), let \(\delta_{p,n}(X)\) be the proportion of polynomials \(f\in \mathcal F_{p,n}\) with \(\operatorname{Ht}(f)\le X\) for which Heilbronn's criterion applies to the number field \(K=\mathbb Q[x]/(f)\), meaning that there exist integers \(a,b\) with \(p=a+b\), such that \(a\) is an \(n\)-th power residue modulo \(p\) and neither \(a\) nor \(-b\) is a norm from \(\mathcal O_K\). Under these assumptions, which quantitative estimate holds?

Choices:
B.
One has
\[
\lim_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(p)\right\},
\]
where \(\varepsilon(p)\) is effectively computable and satisfies \(\varepsilon(p)\to 0\) as \(p\to\infty\).
C.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \frac{2}{27}.
\]
D.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge 1-\varepsilon(p),
\]
where \(\varepsilon(p)\to 0\) as \(p\to\infty\).
E.
One has
\[
\liminf_{X\to\infty}\delta_{p,n}(X)\ge \max\left\{\frac{2}{27},\,1-\varepsilon(n)\right\},
\]
where \(\varepsilon(n)\) is effectively computable and satisfies \(\varepsilon(n)\to 0\) as \(n\to\infty\).

The stem does not explicitly reveal the correct option. It sets up hypotheses and asks for the valid estimate, without directly embedding the exact conclusion.

This is essentially a theorem-recall item: the hypotheses are stated in full and the choices are near-verbatim variants of the theorem’s conclusion. It tests recognition of the exact statement more than mathematical inference.

There is some reasoning pressure in distinguishing liminf vs. limit, the max bound vs. weaker sub-bounds, and the dependence on p rather than n. However, the item mainly rewards recall of the precise theorem statement rather than generative reasoning from the setup.

The distractors are strong: they are plausible theorem variants involving common errors such as strengthening liminf to limit, dropping a fallback term, or changing parameter dependence. They are distinct and mathematically meaningful.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than recall.