\input{preamble.tex}
\unittitle{Convergence of solutions of $p$-adic differential equations}

In this unit, we consider the radius of convergence of
a local horizontal section of such a differential module. In particular,
we define a fundamental invariant, the \emph{generic radius of convergence},
and some related invariants, the \emph{subsidiary radii}.

\section{Radius of convergence on a disc}

View $K \langle \alpha/t, t/\beta \rangle$ 
as a differential ring with derivation $d = \frac{d}{dt}$,
the formal differentiation in the variable $t$. Note that this
does indeed carry $K \langle \alpha/t,t/\beta \rangle$ into itself.
\begin{prop} \label{P:free}
Any finite differential module over $K \langle
\alpha/t, t/\beta \rangle$ is torsion-free, and hence free. 
\end{prop}
Thus I could omit saying ``free'' in many statements, though I will continue
to do so for emphasis.
\begin{proof}
Exercise.
\end{proof}

Let $M$ be a finite free differential module over $K \langle t/\beta 
\rangle$.
The fundamental theorem of $p$-adic ordinary differential
equations, if it were true,
 would say that $M \otimes K \langle t/\rho \rangle$ 
admits a basis of horizontal elements (elements in the kernel of $D$)
for any $\rho \in [0, \beta)$.
Unfortunately, this is simply false, as we saw in the introduction;
it already fails for the module $M = K \langle t/\beta \rangle$
with $D(x) = x$,
when $\beta > p^{-1/(p-1)}$.

We define the \emph{radius of convergence} for $M$ around $0$, denoted
$R(M)$, as the
supremum of the set of
$\rho \in (0, \beta)$ such that $M \otimes K \langle t/\rho \rangle$
has a basis of horizontal elements. We will see a bit later 
(Corollary~\ref{C:rad pos}) that the set is nonempty, so this supremum
makes sense. 

Writing everything in terms of such a basis, we see that
the only elements of any $M \otimes K \langle t /\rho \rangle$
that can be horizontal are the $K$-linear combinations of the basis
elements; we call such linear combinations \emph{local horizontal sections}
of $M$. (As I may have done before, I use the word ``section'' because I
am thinking geometrically, in terms of a vector bundle equipped with
a connection.)

Here are some easy consequences of the definition; note the parallels
with properties of the truncated spectral norm. In particular,
the proof of (c) below may clarify the proof of the corresponding property
of the truncated spectral norm.
\begin{lemma} \label{L:radius}
Let $M, M_1, M_2$ be finite free differential modules over
$K \langle t/\beta \rangle$.
\begin{enumerate}
\item[(a)]
If $0 \to M_1 \to M \to M_2 \to 0$ is exact, then
\[
R(M) = \min\{R(M_1), R(M_2)\}.
\]
\item[(b)]
We have
\[
R(M^\dual) = R(M).
\]
\item[(c)]
We have
\[
R(M_1 \otimes M_2) \geq \min\{R(M_1), R(M_2)\},
\]
with equality when $R(M_1) \neq R(M_2)$.
\end{enumerate}
\end{lemma}
\begin{proof}
For (a), it is clear that
$R(M) \leq \min\{R(M_1), R(M_2)\}$; we must check that equality
holds. Choose $\lambda < \min\{R(M_1), R(M_2)\}$,
so that $M_1 \otimes K \langle t/\lambda \rangle$ and
$M_2 \otimes K \langle t /\lambda \rangle$ are both trivial.
If we choose a basis of $M$ compatible with the sequence, then
the action of $D$ will be block upper triangular nilpotent,
and trivializing $M$ amounts to antidifferentiating the entries in the
nonzero block. We may not be able to perform this
antidifferentiation in
$K \langle t /\lambda \rangle$, but we can do it in
$K \langle t /\lambda' \rangle$ for any $\lambda' < \lambda$.
Since we can make $\lambda$ and $\lambda'$ as close to $\min\{R(M_1),R(M_2)\}$
as we like, we find $R(M) \geq \min\{R(M_1), R(M_2)\}$.

For (b), we obtain $R(M^\dual) \geq R(M)$ from the fact that if
$M \otimes K \langle t / \lambda \rangle$ is trivial, then so is
its dual $M^\dual \otimes K \langle t / \lambda \rangle$. Since
$M$ and $M^\dual$ enter symmetrically, we get $R(M^\dual)=R(M)$.

For (c), the inequality is clear from the fact that the tensor product
of two trivial modules over $K \langle t /\lambda \rangle$ is also
trivial. If $R(M_1) < R(M_2)$, then we have
\begin{align*}
R(M_1) &= \min\{R(M_1), R(M_2)\} \\
&\leq \min\{R(M_1 \otimes M_2), R(M_2^\dual) \} \\
&\leq R(M_1 \otimes M_2 \otimes M_2^\dual).
\end{align*}
Moreover, $M_2 \otimes M_2^\dual$ contains a trivial submodule (the trace),
so $M_1 \otimes M_2 \otimes M_2^\dual$ contains a copy of $M$;
hence by (a), $R(M_1 \otimes M_2 \otimes M_2^\dual) \leq R(M_1)$.
We thus obtain a chain of inequalities leading to $R(M_1) \leq R(M_1)$;
this forces the intermediate equality 
$R(M_1) = \min\{R(M_1 \otimes M_2), R(M_2^\dual)\}$. Since $R(M_1) \neq R(M_2)
= R(M_2^\dual)$,
we can only have $R(M_1) = R(M_1 \otimes M_2)$.
\end{proof}

Here is a simple example.
\begin{prop} \label{P:example1}
Let $M$ be the differential module of rank $1$ over $K \langle t/\beta \rangle$
defined by $D(v) = \lambda v$ with $\lambda \in K$. Then
\[
R(M) = \min\{\beta, |p|^{-1/(p-1)} |\lambda|^{-1} \}.
\]
\end{prop}
\begin{proof}
Exercise.
\end{proof}

In general, the radius of convergence is difficult to compute. To get
a better handle on it, we introduce another invariant which looks more
complicated to define but has much simpler behavior; this is the generic
radius of convergence introduced below.

\section{Generic radius of convergence}

For $\rho > 0$, let $F_\rho$ be the completion of $K(t)$ under the
$\rho$-Gauss norm $|\cdot|_\rho$.
Put $d = \frac{d}{dt}$ on $K(t)$; then $d$ extends
by continuity to $F_\rho$, and
\[
|d|_{F_\rho} = \rho^{-1}, \qquad |d|_{\spect,F_\rho} =
\lim_{n \to \infty} |n!|^{1/n} \rho^{-1} =  p^{-1/(p-1)} \rho^{-1}.
\]
Let $(V,D)$ be a finite differential module over $F_\rho$. We define the \emph{generic
radius of convergence} 
(or for short, the \emph{generic radius}) of $V$ to be
\[
R(V) = p^{-1/(p-1)} |D|_{\tspect,V}^{-1};
\]
note that $R(V) > 0$.
We will see later (Proposition~\ref{P:generic radius})
 that this does indeed compute the
radius of convergence of horizontal sections of $V$ on a ``generic disc''.

In the interim, we note the following
relationship with the usual radius of convergence. In the language of
Dwork, this is a \emph{transfer theorem}, because it transfers convergence
information from one disc to another.

\begin{theorem} \label{T:transfer}
For any finite free differential module $M$ over $K \langle t/\rho \rangle$,
$R(M) \geq R(M \otimes F_\rho)$. That is, the radius of convergence
is at least the generic radius.
\end{theorem}
\begin{proof}
Suppose $\lambda < \rho$ and $\lambda < p^{-1/(p-1)}
|D|_{\tspect,V}^{-1}$. We claim that for any $x \in M$, 
the Taylor series
\begin{equation} \label{eq:taylor}
y = \sum_{i=0}^\infty \frac{(-t)^i}{i!} D^i(x)
\end{equation}
converges 
under $|\cdot|_\lambda$.
To see this, pick $\epsilon>0$ such that
$\lambda p^{1/(p-1)} (|D|_{\spect,V} + \epsilon) < 1$;
then there exists $c>0$ such that 
$|D^i(x)| \leq c (|D|_{\spect,V} + \epsilon)^i$ for all $i$. The $i$-th
term of the sum defining $y$ thus has norm at most
$\lambda^i p^{i/(p-1)} c (|D|_{\spect,V} + \epsilon)^i$, which tends to 0
as $i \to \infty$.

By differentiating the series expression, we find that
\begin{align*}
Dy &= \sum_{i=0}^\infty \frac{(-t)^i}{i!} D^{i+1}(x) +
\sum_{i=1}^\infty \frac{-(-t)^{i-1}}{(i-1)!} D^i(x) \\
&= \sum_{i=0}^\infty \frac{(-t)^i}{i!} D^{i+1}(x) -
\sum_{i=0}^\infty \frac{(-t)^i}{i!} D^{i+1}(x) = 0.
\end{align*}
That is, $y$ is a
horizontal section of $V \otimes K \langle t/\lambda \rangle$.

If we run this construction over a basis of $M$, we obtain 
horizontal sections of $V \otimes K \langle t/\lambda \rangle$ 
whose reductions modulo $t$ form a basis; they thus form a basis themselves
by Nakayama's lemma (and the fact that finite differential modules
over a PID are free). This proves the claim.
\end{proof}
\begin{cor} \label{C:rad pos}
For any finite free differential module $M$ over $K \langle t/\rho \rangle$,
$R(M) > 0$.
\end{cor}

We can translate some basic properties of the truncated spectral radius
into properties of generic radii, leading to the following analogue of
Lemma~\ref{L:radius}. Alternatively, one can first check
Proposition~\ref{P:generic radius} and then simply invoke
Lemma~\ref{L:radius} itself around a generic point.
\begin{lemma} \label{L:radius2}
Let $V, V_1, V_2$ be finite differential modules over $F_\rho$.
\begin{enumerate}
\item[(a)]
For $0 \to V_1 \to V \to V_2 \to 0$ exact,
\[
R(V) = \min\{R(V_1), R(V_2)\}.
\]
\item[(b)]
We have
\[
R(V^\dual) = R(V).
\]
\item[(c)]
We have
\[
R(V_1 \otimes V_2) \geq \min\{R(V_1), R(V_2)\},
\]
with equality when $R(V_1) \neq R(V_2)$.
\end{enumerate}
\end{lemma}

In some situations, it is more natural to consider the
\emph{intrinsic generic radius of convergence}, or for short
the \emph{intrinsic radius}, defined as
\[
IR(V) = \rho^{-1} R(V) = \frac{|d|_{\spect,F_\rho}}{|D|_{\tspect,V}}
\in (0,1].
\]
For emphasis, I will sometimes refer to the usual generic radius of
convergence as the \emph{extrinsic} generic radius of convergence.
The term ``intrinsic'' is used to connote a certain independence from
scale; see Proposition~\ref{P:tame}. (It also is more natural
from the point of view of the definition of truncated spectral norm given
by Baldassarri and di Vizio, as described in a previous unit.)

\section{Some examples in rank $1$}

An important class of examples is given as follows.
For $\lambda \in K$, let $V_\lambda$ be the differential module of rank
1 over $F_\rho$ defined by $D(v) = \lambda t^{-1} v$.
\begin{prop} \label{P:example2}
We have $IR(V_\lambda) = 1$ if and only if $\lambda \in \ZZ_p$.
\end{prop}
\begin{proof}
Exercise.
\end{proof}

\begin{prop}
We have $V_\lambda \cong V_{\lambda'}$ if and only if $\lambda - \lambda' \in 
\ZZ$.
\end{prop}
\begin{proof}
Note that $V_\lambda \cong V_{\lambda'}$
if and only if $V_{\lambda-\lambda'}$ is trivial, so we may reduce to the
case $\lambda' = 0$. By 
Proposition~\ref{P:example2}, $V_\lambda$ is nontrivial whenever $\lambda
\notin \ZZ_p$; by direct inspection, $V_\lambda$ is trivial whenever
$\lambda \in \ZZ$.

It remains to deduce a contradiction assuming that
$V_\lambda$ is trivial, $\lambda \in \ZZ_p$, and $\lambda \notin \ZZ$.
There is no harm in enlarging $K$ now, so we may assume that $K$ contains
a scalar of norm $\rho$; by rescaling, we may reduce to the case $\rho=1$.
We now have $f \in F_1^\times$ such that $t\frac{df}{dt} = \lambda f$; by multiplying
by an element of $K^\times$, we can force $|f|_1 = 1$.

Let $\lambda_1$ be an integer such that
$\lambda \equiv \lambda_1 \pmod{p}$. Then
\[
\left| \frac{d(f t^{-\lambda_1})}{dt} \right|_1 = |(\lambda - \lambda_1) ft^{-\lambda_1-1}|_1 \leq p^{-1}.
\]
On the other hand, the map $d$ induces the derivation 
$\frac{d}{dt}$ on the residue field
$\kappa_K(t)$ of $F_1$, 
and likewise on $\kappa_K((t))$. In the latter, it is clear that the
kernel of $\frac{d}{dt}$ is precisely the series with only exponents
divisible by $p$. 
In particular, if we expand the residue of $f$ as a power series
around $t=0$, it can only have
 exponents congruent to $\lambda$ modulo $p$.

By considering the reduction of $f$ modulo $p^n$ and arguing similarly,
we find that the image of
$f$ in $\kappa_K((t))$ has only exponents congruent to $\lambda$
modulo $p^2, p^3, \dots$. But since $\lambda \notin \ZZ$, this means
that the image of $f$ in $\kappa_K((t))$ cannot have any terms at all,
contradiction.
\end{proof}

\section{Open discs and annuli}

Although we have been talking about closed discs so far, it is clearly
quite natural to consider open discs. After all, if
$M$ is a finite free differential module over $K \langle t/\beta \rangle$
with radius of convergence $M$, that only guarantees that the local 
horizontal sections exist on the union of the closed discs of radii
strictly less than 1.

We will stick to the following convention, which I'll explain just in the
half-open case (since I can consider a fully open disc or annulus as the
union of two half-opens pasted together along a common boundary).
By a finite differential module $M$ on the
half-open annulus $\alpha \leq |t| < \beta$
(which becomes a disc if $\alpha = 0$), let us mean a sequence
of finite free differential modules $M_i$ over 
$K \langle \alpha/t,t/\beta_i \rangle$
with $\beta_1, \beta_2, \dots$ an increasing sequence with limit $\beta$,
together with isomorphisms $M_{i+1} \otimes K \langle \alpha/t,t/\beta_i
\rangle \cong M_i$. 
In geometric language,
this corresponds to a locally free coherent sheaf on the corresponding rigid  or
Berkovich analytic space, equipped with a connection.

It is unambiguous to refer to the generic radius of convergence
$R(M \otimes F_\rho)$ for $\rho \in[\alpha, \beta)$; simply restrict to
some interval $[\alpha, \beta_i]$ containing $\rho$ and make the definition
there. 

\section{A cautionary note}

Let $M$ be a differential module over $K \langle t/\beta \rangle$
for which $IR(M \otimes F_\beta) = 1$. Then by 
Theorem~\ref{T:transfer}, 
for any $\rho \in [0, \beta)$, $M \otimes K \langle t/\rho \rangle$
is trivial, so we have an isomorphism
\[
M \otimes K \langle t/\rho \rangle
\cong K \langle t/\rho \rangle^{\oplus n}
\]
of differential modules.

Now let $M$ be a differential module over $K \langle \alpha/t, t/\beta \rangle$
for which $IR(M \otimes F_\rho) = 1$ for all $\rho \in [\alpha, \beta]$.
This most favorable situation was originally thought to be analogous
to the situation of regular singularities in the complex setting. In 
particular, it was believed that for any $\alpha < \gamma \leq \delta
< \beta$, it would be
possible to write
\[
M \otimes K \langle \gamma/t, t/\delta\rangle \cong
M_{\lambda_1} \oplus \dots \oplus M_{\lambda_n}
\]
for some $\lambda_1, \dots, \lambda_n \in \ZZ_p$,
where $M_{\lambda}$ is the differential module of rank
defined by $D(v) = \lambda t^{-1} v$ (as in the previous section).

This hope was dashed when a counterexample was exhibited by Monsky;
it is the rank 2 differential module associated to the 
differential polynomial $p(1-x)T^2 - xT - a$, where $a \in \ZZ_p$
is constructed so that
\begin{equation} \label{eq:monsky}
\liminf_{m \to +\infty} |a+m|^{1/m} < 1,
\qquad
\liminf_{m \to +\infty} |a-m|^{1/m} = 1.
\end{equation}
(The existence of such $a$ is left as an exercise, or see
[DR77, \S 7.20].) 
I plan to expand on this in a further unit; in the
meantime, see [DR77, \S 7] for further discussion.

What this suggests is the hypothesis on the intrinsic radius needs to
be supplemented with some extra hypotheses in order to get the
decomposition we want. We will see one such hypothesis later (the existence
of a Frobenius structure); in the interim, see the notes for further
discussion.

\section{Geometric interpretation}

You might be wondering why we call $R(V)$ the generic radius of convergence;
here is the construction that explains the name. 
First, the geometric motivation. Say we have $f \in F_\rho$
given by a limit of a Cauchy sequence $f_1, f_2, \dots$ in $K(t)$.
For any complete extension $L$ of $K$
and any $t_\rho \in L$ of norm $\rho$, we can evaluate the $f_i(t_\rho)$
to get a convergent sequence
provided that $f_i$ does not have a pole in the disc $|t-t_\rho| < \rho$.

However, the poles of elements of $K(t)$ belong to $F^{\alg}$. 
So suppose we take $t_\rho$ to be a \emph{generic point of norm $\rho$ relative to $F$},
i.e., an element of some larger complete nonarchimedean field 
such that the disc $|t-t_\rho| <
\rho$ contains no elements of $F^{\alg}$.
Then we can evaluate each of the $f_i(t_\rho)$ to get a convergent
sequence, and thus evaluate $f(t_\rho)$.

Let us turn this geometric thinking into algebra.
Let $L$ be the completion of $K(t_\rho)$
for the $\rho$-Gauss norm.
For any $\lambda \in (0,\rho)$, embed $K[t]$ into
$L \langle (t-t_\rho)/\lambda \rangle$
by mapping $t$ to $t_\rho + (t-t_\rho)$. 
Note that the image of $f$ has constant term $f(t_\rho)$, whose norm
is strictly greater than that of the remaining terms of the image.
Thus each nonzero element of
$K[t]$ maps to a unit in $L \langle (t-t_\rho)/\lambda \rangle$,
so we can extend to isometric embeddings of $K(t)$ and $F_\rho$.

By tensoring with each $L \langle (t-\rho)/\lambda\rangle$, we obtain
a finite differential module $V'$ on the open disc of radius $\rho$
centered at $t_\rho$.
We can speak of the radius of convergence of this module
using the previous definition.

We already made the previous observation, but let us now formalize it.
\begin{lemma} \label{L:same norm}
For any $f \in F_\rho$ and any $\lambda< \rho$, $|f|_\rho$ equals the
$\lambda$-Gauss norm of $f$ within $L \langle (t-t_\rho)/\lambda \rangle$.
\end{lemma}
\begin{proof}
This holds because the constant term $f(t_\rho)$ is dominant.
\end{proof}

\begin{prop} \label{P:generic radius}
The generic radius of convergence of $V$ is equal to the radius of convergence
of $V'$.
\end{prop}
\begin{proof}
Let $G_\lambda$ be the 
completion of $L(t-t_\rho)$ for the $\lambda$-Gauss norm.
If we now compute $|D|_{\tspect,V}$ in terms of some basis
and apply Lemma~\ref{L:same norm} to every matrix entry in the
calculation, we get the same norms whether we work in $F_\rho$
or $G_\lambda$. In other words,
\[
|D|_{\tspect, V \otimes G_\lambda} = \max\{|d|_{\spect, G_\lambda},
|D|_{\tspect,V} \} = \max\{p^{-1/(p-1)} \lambda^{-1}, |D|_{\tspect,V}\}.
\]
On one hand, this implies $R(V) \leq R(V')$ by applying 
Theorem~\ref{T:transfer} to $V \otimes L \langle (t-t_\rho)/\lambda \rangle$
for a sequence of values of $\lambda$ converging to $\rho$.

On the other hand,
pick any $\lambda < R(V \otimes L \llbracket t - t_\rho \rrbracket)$;
then $V \otimes G_\lambda$ is a trivial differential module,
so the truncated spectral norm of $D$ on it is $p^{-1/(p-1)} \lambda^{-1}$.
We thus have
\[
|D|_{\tspect,V} \leq p^{-1/(p-1)} \lambda^{-1},
\]
so $R(V) \geq \lambda$. This yields $R(V) \geq R(V')$.
\end{proof}

Here is an example illustrating both the use of the geometric interpretation
and a good transformation property of the intrinsic normalization.
\begin{prop} \label{P:tame}
Let $m$ be a positive integer coprime to $p$, and let $f_m: F_\rho \to F_{\rho^m}$
be the map $t \mapsto t^m$. Then for any finite differential module $V$
over $F_\rho$, $IR(V) = IR(V \otimes F_{\rho^m})$.
\end{prop}
\begin{proof}
This follows from the geometric interpretation plus the fact that
\begin{equation} \label{eq:ineq}
|t-t_\rho| < c \rho \Leftrightarrow
|t^m - t_\rho^m| < c \rho^m \qquad (c \in (0,1)),
\end{equation}
whose proof is left as an exercise.
\end{proof}

\section{Subsidiary radii}

Let $V$ be a finite differential module over $F_\rho$.
Let $V_1, \dots, V_m$ be the Jordan-H\"older constituents of $V$;
that is, take a maximal filtration of $V$ by differential submodules,
and let $V_1, \dots, V_m$ be the successive quotients. (It is a standard
algebra exercise that the resulting list does not depend on the filtration
up to ordering.)

We define the multiset of \emph{generic radii of subsidiary convergence},
or for short \emph{subsidiary radii},
to be the multiset consisting of $R(V_i)$ with multiplicity $\dim V_i$
for $i=1,\dots, m$.
We also have \emph{intrinsic (generic) radii of subsidiary convergence}
obtained by multiplying the subsidiary radii by $\rho^{-1}$.
You may want to think of the product of the subsidiary radii (or the
intrinsic subsidiary radii) as an analogue of irregularity over $\CC((z))$;
this analogy will be further supported later.

It is not yet clear how to interpret the subsidiary radii as the radii
of convergence of anything. We will give this interpretation in a later unit.

\notes

The notion of considering a ``generic disc'' as above originates in the
work of Dwork [Dwo73]. Our definition of the 
generic radius of convergence is taken from Christol and Dwork [CD94].
The intrinsic radius of convergence was introduced in [Ked07], where it
is called the ``toric normalization''; it also figures prominently
in the coordinate-free treatment of generic radii of convergence
given in [BdV07].

Much of the terminology used here is original: this includes the
qualifier ``intrinsic'', and the name ``subsidiary radii''.

A theory of exponents for differential modules on an annulus with
intrinsic radius of convergence 1 everywhere was developed
by Christol and Mebkhout [CM97, \S 4--5]; an alternate development
was later given by Dwork [Dwo97] (see also [DGS94, \S 6]).
The exponents are elements of the quotient
$\ZZ_p/\ZZ$; this makes the construction somewhat complicated, as
one must use \emph{archimedean} considerations to identify a $p$-adic 
number, in a manner we will not elaborate further here. 
When the differences between exponents
satisfy a $p$-adic non-Liouville condition (that is, they cannot be 
approximated unusually well by integers), one obtains a decomposition
into modules of rank 1 [CM97, \S 6]. This is automatic in case of
a Frobenius structure, as then the set of exponents is invariant under
the operation $x \to x^p$ and so all of the exponents are forced to be
rational numbers, whose differences are always non-Liouville.
See [Loe97] (or a promised later unit) for a detailed exposition.

\exercises

\begin{enumerate}
\item
Prove Proposition~\ref{P:free}. (Hint:
first prove that $K \langle \alpha/t, t/\beta \rangle$ has no nonzero
differential ideals. Then given a finite differential module over $K \langle
\alpha/t, t/\beta \rangle$, consider the annihilator of the torsion
submodule.)
\item
Exhibit an example showing that the cokernel of $\frac{d}{dt}$ on
$K \langle \alpha/t, t/\beta \rangle$ is not spanned over $K$
by $t^{-1}$. That is, antidifferentiation with respect to $t$ 
is not well-defined.
\item
Prove Proposition~\ref{P:example1}. Optional: give an explicit formula
for $IR(V_\lambda)$ in terms of $\rho$ and the minimum distance from
$\lambda$ to an integer.
\item
Prove Proposition~\ref{P:example2}. (Hint: for (a), consider the cases
$\lambda \in \ZZ_p$, $\lambda \in \gotho_K - \ZZ_p$,
and $\lambda \notin \gotho_K$ separately. For (b), use (a) to reduce
to the case  $\lambda \in \ZZ_p$.)
\item
Prove that there exists $a \in \ZZ_p$ satisfying
\eqref{eq:monsky}.
\item
Prove \eqref{eq:ineq}.
\end{enumerate}

\end{document}