%! TEX root = DA.tex
% vim: tw=50
% 20/11/2024 12PM

\begin{theorem*}
  Let $d \ge 3$. Let $F(X, Y) \in \Zbb[X, Y]$ be a
  homogeneous polynomial of degree $d$ without
  repeated factors. Let $G(X, Y) \in \Zbb[X, Y]$
  be of degree $\le d - 1$. Assume $F - G$ is
  irreducible. Then
  \[
    F(X, Y) = G(X, Y)
    \qquad X, Y \in \Zbb
  \]
  has at most finitely many solutions.
\end{theorem*}

Schinzel proved this only assuming that $F \neq
aQ^n$ for some irreducible $Q$ of degree $\le 2$.
He used Siegel's theorem on integral points.
If an algebraic curve has infinitely many points,
then it has genus $D$ and at most $2$ points at
infinity.
Our proof is based on an argument of Corvaja and
Zannier for proving Siegel's theorem.

\textbf{Subspace theorem:}
Let $V$ be a vector space of dimension $n$ over
$\ol{\Qbb}$. Let $e_1^{(0)}, \ldots, e_n^{(0)}$
and $e_1, \ldots, e_n$ be two bases of $V$. Then
for all $\eps > 0$, there exists a finite number
of elements $f_1, \ldots, f_m \in V$ such that all
$\varphi \in V^*$ that solves:
\[
  \prod_{i = 1}^{n} |\varphi(e_i)| \le
  H(\varphi(e_1^{(0)}), \ldots,
  \varphi(e_n^{(0)}))^{-\eps}
  \tag{$*$}
  \label{lec18_eq}
\]
with $\varphi(e_i^{(0)}) \in \Zbb$ for all $i = 1,
\ldots, n$, $\varphi$ satisfies $\varphi(f_j) = 0$
for some $j \in \{1, \ldots, n\}$.

$\exists \alpha_{i, j} \in \ol{\Qbb}$ such that
\[
  e_i = \sum_{j} \alpha_{ij} e_j^{(0)}
\]
and $L_i = \alpha_{i1} X_1 + \cdots + \alpha_{in}
X_n$. $\varphi$ satisfies \eqref{lec18_eq} if and
only if
\[
  (x_1, \ldots, x_n)
  = (\varphi(e_1^{(0)}), \ldots, \varphi(e_n^{(0)}))
  \in \Zbb^n
\]
satisfies
\[
  \prod_{i = 1}^{\infty} |L_i(x_1, \ldots, x_n)| <
  H(x_1, \ldots, x_n)^{-\eps}
.\]

Let $F, G$ be as in the theorem, and write $P = F
- G$.

We assume that $Y \nmid F$.

Then there exists $\alpha_1, \ldots, \alpha_d \in
\ol{\Qbb}$ distinct such that
\[
  F(X, Y)
  = (X - \alpha_1 Y) \cdots (X - \alpha_d Y)
.\]
Write $\Gamma$ for the set of $(x, y) \in \Cbb^2$
with $P(x, y) = 0$. Then for $(x, y) \in \Gamma$
we have
\[
  F(x, y) \le C(|x| + |y|)^{d - 1}
.\]
By a similar argument to the lemma for Thue's
equation, for all $\eps > 0$ there exists $R =
R(P, \eps)$ such that $(x, y) \in \Gamma$ with
$|x| + |y| > R$, then $\left| \frac{x}{y} -
\alpha_j \right| < \eps$ for some $j$.

We pick a small $\eps > 0$, in particular
$|\alpha_i - \alpha_j| > 2\eps$ for $i \neq j$. We
define
\begin{align*}
  \Gamma_0
  &= \{(x, y) \in \Gamma : |x| + |y| < R\} \\
  \Gamma_j
  &= \left\{(x, y) \in \Gamma : |x| + |y| \ge R,
  \left| \frac{x}{y} - \alpha_j \right| < \eps \right\}
\end{align*}
for $j = 1, \ldots, d$.
\begin{center}
  \includegraphics[width=0.6\linewidth]{images/23bc79f7834d467d.png}
\end{center}
$\Gamma_0$ is bounded so only has finitely many
integer points. We want to show this also for
$\Gamma_1, \ldots, \Gamma_j$. Write $I = P\Qbb[X,
Y]$ for the ideal generated by $P$. Take some $D
\in \Zbb_{\ge 1}$ and large enough. Write
$\ol{\Qbb}[X, Y]^{(D)}$ for polynomials of degree
$\le D$. We will apply the subspace theorem in the
vector space
\[
  V = \ol{\Qbb}[X, Y]^{(D)} / (I \cap \ol{\Qbb}[I,
  Y]^{(D)})
.\]
Elements $f \in V$ can be evaluated on $\Gamma$.

In particular, for $(x, y) \in \Gamma$, the map $f
\mapsto f(x, y)$ is an element of $V^*$. Reference
basis: the monomials $X^k Y^m$ for $k + m \le D$
span $V$. Pick a linearly independent family for
$e_1^{(0)}, \ldots, e_n^{(0)}$, where $n = \dim
V$.

If $(x, y) \in \Gamma \cap \Zbb^2$, then
$e_i^{(0)}(x, y) \in \Zbb$. Also,
\[
  H(e_1^{(0)}(x, y), \ldots, e_n^{(0)}(x, y))
  < C |y|^D
.\]
We need to find some $l_j$'s that decay on a fixed
$\Gamma_i$.

For $j = 1, \ldots, d$ we introduce a symbol $p_j$
and call these the ``points of $\Gamma$ at
infinity''. We define for $f \in V$:
\[
  \ord_{p_j}(f) = \sup \{m \in \Zbb : f(x, y)
  \cdot y^m \text{ is bounded on $\Gamma_j$}\}
.\]
Note $\ord_{p_j}(f) \ge -D$.

\begin{lemma*}
  Let $f \in V$ and let $j \in \{1, \ldots, d\}$.
  If $\ord_{p_j}(f) < \infty$, then the limit
  \[
    \lim_{(x, y) \in \Gamma, |y| \to \infty} f(x,
    y) y^{\ord_{p_j}(f)}
  \]
  exists and $\neq 0$. In addition, we have
  \[
    \lim_{(x, y) \in \Gamma_j, |y| \to \infty} (X
    - \alpha Y) Y^{-1}
    = \alpha_j - \alpha
  \]
  for all $\alpha \in \ol{\Qbb}$.
\end{lemma*}

Can be proved that $\ord_{p_j}(f) = \infty$ if and
only if $f = 0$.

$\frac{Z}{Y}$ is a local uniformiser at $p_j =
(\alpha_i, 1, 0)$.