\begin{aligned} F = (P, Q): \mathcal{S} \to \mathbb{C}^2 \qu...

Prompt

\begin{aligned} F = (P, Q): \mathcal{S} \to \mathbb{C}^2 \quad : \quad & \begin{cases} \mathcal{S} = \left\{ (x, y, w) \in \mathbb{C}^3 \;\middle|\; y = w^2 + \dfrac{32}{9} x^{15} w^9 \right\} \\[12pt] P(x, w) = x + x^{16} w^7 \\[6pt] Q(x, w) = w^2 \end{cases} \\ \wedge\quad & \mathrm{d}P \wedge \mathrm{d}Q \equiv \mathrm{d}x \wedge \mathrm{d}y \iff \frac{\partial P}{\partial x}\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial Q}{\partial x} \equiv \mathbf{1 \quad (\text{egzakt azonosság})} \\ \wedge\quad & P(0,0) = Q(0,0) = 0 \;\wedge\; \begin{pmatrix} \frac{\partial P}{\partial x}(0,0) & \frac{\partial P}{\partial y}(0,0) \\ \frac{\partial Q}{\partial x}(0,0) & \frac{\partial Q}{\partial y}(0,0) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \wedge\quad & [\mathbb{C}(\mathcal{S}) : \mathbb{C}(x, y)] = d = 9 \ge 5 \implies \exists z_1 \ne z_2 \in \mathcal{S} : F(z_1) = F(z_2) \implies \mathbf{\neg\operatorname{Inj}(P, Q)} \\ \wedge\quad & S_{P,Q} = \big\{ (u, v) \in \mathbb{C}^2 \mid v^2 - u^7 = 0 \big\} \ne \emptyset \implies \mathbf{\neg\operatorname{Proper}(F)} \\ \wedge\quad & \pi_1\big(\mathbb{C}^2 \setminus S_{P,Q}\big) \cong \langle u, v \mid u^2 = v^7 \rangle \cong \mathbb{Z}_2 *_{\mathbb{Z}} \mathbb{Z}_7 \implies \text{\textbf{Nem-abeli monodrómia}} \\ \wedge\quad & X_P := P_x \partial_y - P_y \partial_x \implies \operatorname{depth}(\mathcal{T}_{\text{BCW}}) = \infty \implies \mathbf{\neg\operatorname{LND}(X_P) \;\wedge\; \neg\operatorname{Inv}(P, Q)} \end{aligned}