%! TEX root = DA.tex
% vim: tw=50
% 06/11/2024 12PM

\[
  \frac{\partial}{\partial X_2} G^{(2)} P -
  G^{(2)} \frac{\partial}{\partial X_2} P =
  F^{(1)} \left( G^{(1)} \frac{\partial}{\partial
  X_2} G^{(2)} - \frac{\partial}{\partial X_2}
  G^{(1)} \cdot G^{(2)} \right)
\]
We will later have to worry about whether the
resulting polynomial is $0$.

For any $h$:
\begin{align*}
  &~~~\begin{vmatrix}
    P
    & G^{(2)}
    & \cdots
    & G^{(h)} \\
    \frac{\partial}{\partial X_2} P
    & \frac{\partial}{\partial X_2} G^{(2)}
    & \cdots
    & \frac{\partial}{\partial X_2} G^{(h)} \\
    \vdots
    & \vdots
    & \ddots
    & \vdots \\
    \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} P
    & \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(2)}
    & \cdots
    & \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(h)}
  \end{vmatrix} \\
  &= \begin{vmatrix}
    F_1 G^{(1)}
    & G^{(2)}
    & \cdots
    & G^{(h)} \\
    F_1 \frac{\partial}{\partial X_2} G^{(1)}
    & \frac{\partial}{\partial X_2} G^{(2)}
    & \cdots
    & \frac{\partial}{\partial X_2} G^{(h)} \\
    \vdots
    & \vdots
    & \ddots
    & \vdots \\
    F_1 \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(1)}
    & \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(2)}
    & \cdots
    & \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(h)}
  \end{vmatrix} \\
  &= F_1 \begin{vmatrix}
    G^{(1)}
    & G^{(2)}
    & \cdots
    & G^{(h)} \\
    \frac{\partial}{\partial X_2} G^{(1)}
    & \frac{\partial}{\partial X_2} G^{(2)}
    & \cdots
    & \frac{\partial}{\partial X_2} G^{(h)} \\
    \vdots
    & \vdots
    & \ddots
    & \vdots \\
    \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(1)}
    & \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(2)}
    & \cdots
    & \frac{\partial^{h - 1}}{\partial X_2^{(h -
    1)}} G^{(h)}
  \end{vmatrix}
\end{align*}
The degree increases $h$-fold, but not the index.
\begin{align*}
  &~~~\begin{vmatrix}
    P_{0, 0} & P_{0, 1} & \cdots & P_{0, h - 1} \\
    \vdots & \vdots & \ddots & \vdots \\
    P_{h - 1, 0} & P_{h - 1, 1} & \cdots & P_{h -
    1, h - 1}
  \end{vmatrix} \\
  &= \begin{vmatrix}
    F^{(1)} & F^{(2)} & \cdots & F^{(h)} \\
    F_1^{(1)} & F_1^{(2)} & \cdots & F_1^{(h)} \\
    \vdots & \vdots & \ddots & \vdots \\
    F_{h - 1}^{(1)} & F_{h - 1}^{(2)} & \cdots &
    F_{h - 1}^{(h)}
  \end{vmatrix}
  \cdot
  \begin{vmatrix}
    G^{(1)} & G_1^{(1)} & \cdots & G_{h - 1}^{(1)}
    \\
    G^{(2)} & G_1^{(2)} & \cdots & G_{h - 1}^{(2)}
    \\
    G^{(h)} & G_1^{(h)} & \cdots & G_{h - 1}^{(h)}
  \end{vmatrix}
\end{align*}
where $P_{ij} = \frac{1}{i! j!} \frac{\partial^{i
+ j}}{\partial X_1^i \partial X_2^j} P$, $F_i =
\frac{1}{i!} \frac{\partial^i}{\partial X_1^i} F$.

\begin{lemma*}
  Let $F^{(1)}, F^{(2)}, \ldots, F^{(h)}$ be
  $\Qbb$-linearly independent polynomials in
  $\Zbb[X]$. Then
  \[
    \begin{vmatrix}
      F^{(1)} & F^{(2)} & \cdots F^{(h)} \\
      F_1^{(1)} & F_1^{(2)} & \cdots & F_1^{(h)}
      \\
      \vdots & \vdots & \ddots & \vdots \\
      F_{h - 1}^{(1)} & F_{h - 1}^{(2)} & \cdots &
      F_{h - 1}^{(h)}
    \end{vmatrix}
    \neq 0
  .\]
  (Wronskian)
\end{lemma*}

\begin{proof}[Proof of Proposition assuming the
  lemma]
  Suppose to the contrary that the proposition
  does not hold for some $P, \frac{p_1}{q_1},
  \frac{p_2}{q_2}$. Write $P = F^{(1)} G^{(1)} +
  \cdots + F{(h)} G^{(h)}$ such that $h$ is
  minimal. Then $h \le n_2 + 1$ and the $F^{(1)},
  \cdots, F^{(k)}$ and $G^{(1)}, \ldots, G^{(h)}$
  are $\Qbb$-linearly independent. Then consider
  \[
    \mathcal{P} = \begin{vmatrix}
      P_{0,0} & \cdots & P_{0, h - 1} \\
      \vdots & \ddots & \vdots \\
      P_{h - 1, 0} & \cdots & P_{h - 1, h - 1}
    \end{vmatrix}
  \]
  and
  \[
    \mathcal{F} = \begin{vmatrix}
      F_{0,0} & \cdots & F_{0, h - 1} \\
      \vdots & \ddots & \vdots \\
      F_{h - 1, 0} & \cdots & F_{h - 1, h - 1}
    \end{vmatrix}
    \mathcal{G} = \begin{vmatrix}
      G_{0,0} & \cdots & G_{0, h - 1} \\
      \vdots & \ddots & \vdots \\
      G_{h - 1, 0} & \cdots & G_{h - 1, h - 1}
    \end{vmatrix}
  \]
  Then $\mathcal{P}(X_1, X_2) = \mathcal{F}(X_1)
  \mathcal{G}(X_2) \neq 0$ by the above Lemma.

  Note $\deg_{X_j} \mathcal{P} \le h n_j$, $\deg
  \mathcal{F} \le n_1$, $\deg \mathcal{G} \le
  n_2$. Also
  \begin{align*}
    H(\mathcal{P})
    &\le \ub{h!}_{\text{ways to
    multiply entries}}
    (\ub{(n_1 + 1)(n_2 + 1)}_{\text{monomials in
    the entries}})^h
    \ub{(2^{n_1 + n_2} HP)^h}_{\text{coefficients of entries}}
    \\
    &\le 2^{(n_1 + n_2) h} 2^{(n_1 + n_2)h} q_j^{h
    n_j / C}
  \end{align*}
  for $j = 1, 2$.

  $H(\mathcal{P}) = H(\mathcal{F})
  H(\mathcal{G})$. Then
  \begin{align*}
    H(\mathcal{F})
    &\le (8^{n_1 + n_2} q_j^{n_j / C}) \\
    &\le (q_j^{h n_1 / C})^h
  \end{align*}
  \begin{align*}
    H(\mathcal{G})
    &\le (8^{n_1 + n_2} q_2^{n_2 / C}) \\
    &\le (q_j^{h n_2 / C})^h
  \end{align*}
  $I_{P_{i, j}} \ge I_P - i \log q_1 - j \log
  q_2$. If $j \le \frac{\eps h}{10} + 1$, $\log q_1 <
  \frac{\eps}{10} \log q_2$. By the
  indirect assumption
  \[
    I_P \ge \eps(n_1 \log q_1 + n_2 \log q_2)
  ,\]
  \[
    I_{P_{i, j}} \ge \frac{\eps}{2} n_2 \log q_2 +
    \frac{\eps}{2} n_1 \log q_1
  .\]
