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

Prompt

$$ \begin{aligned} F = (P, Q): \mathbb{C}^2 \to \mathbb{C}^2 \quad : \quad & \begin{cases} P(x, y) = x + x^2 y + x^{16} y^{16} & (\deg P = 32) \\ Q(x, y) = y + \frac{7}{2} x^{42} y^{41} + x^{56} y^{56} & (\deg Q = 112) \end{cases} \\ \wedge\quad & \begin{cases} A(x, y) = x^2 y & (\deg A = 3 \le 31) \\ B(x, y) = \frac{7}{2} x^{42} y^{41} & (\deg B = 83 \le 111) \end{cases} \\ \wedge\quad & 2\left(\frac{\partial B}{\partial x}\frac{\partial H}{\partial y} - \frac{\partial B}{\partial y}\frac{\partial H}{\partial x}\right) \equiv 7 H^5 \left(\frac{\partial A}{\partial x}\frac{\partial H}{\partial y} - \frac{\partial A}{\partial y}\frac{\partial H}{\partial x}\right) \quad \text{ahol} \quad H = x^8 y^8 \\ \wedge\quad & \frac{\partial P}{\partial x}\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial Q}{\partial x} \equiv 1 \\ \wedge\quad & P(0,0) = Q(0,0) = 0 \;\wedge\; \begin{pmatrix} \frac{\partial P}{\partial x}(0,0) & \frac{\partial P}{\partial y}(0,0) \\ \frac{\partial Q}{\partial x}(0,0) & \frac{\partial Q}{\partial y}(0,0) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \wedge\quad & \det \begin{pmatrix} \frac{\partial}{\partial x}\left(\frac{P+Q}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{P+Q}{\sqrt{2}}\right) \\ \frac{\partial}{\partial x}\left(\frac{Q-P}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{Q-P}{\sqrt{2}}\right) \end{pmatrix} \equiv 1 \\ \wedge\quad & S_{P,Q} \supseteq \big\{ (u, v) \in \mathbb{C}^2 \;\big|\; v^2 - u^7 = 0 \big\} \ne \emptyset \implies \neg\operatorname{Proper}(F) \\ \wedge\quad & [\mathbb{C}(x,y) : \mathbb{C}(P, Q)] \ge 5 \implies \exists z_1 \ne z_2 \in \mathbb{C}^2 : \big(P(z_1), Q(z_1)\big) = \big(P(z_2), Q(z_2)\big) \\ \wedge\quad & \neg\exists R, S \in \mathbb{C}[x, y] : \Big( R(P(x,y), Q(x,y)) = x \;\wedge\; S(P(x,y), Q(x,y)) = y \Big) \end{aligned} $$