%! 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 \[ I_P \ge \eps(n_1 \log q_1 + n_2 \log q_2) ,\] \[ I_{P_{i, j}} \ge \frac{\eps}{2} n_2 \log q_2 + \frac{\eps}{2} n_1 \log q_1 .\]