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

Prompt

\begin{aligned} F = (P, Q): \mathbb{C}^2 \to \mathbb{C}^2 \quad : \quad & \begin{cases} P(x, y) = x + A(x, y) + x^{16} y^{16} & (\deg P = 32) \\[6pt] Q(x, y) = y + B(x, y) + x^{56} y^{56} & (\deg Q = 112) \end{cases} \\ \wedge\quad & \mathcal{D}_P := \frac{\partial P}{\partial x} \frac{\partial}{\partial y} - \frac{\partial P}{\partial y} \frac{\partial x} \implies \mathcal{D}_P(Q) \equiv 1 \iff \frac{\partial P}{\partial x}\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial Q}{\partial x} \equiv 1 \\ \wedge\quad & \forall k \in \mathbb{N}^+ : \deg\big(\mathcal{D}_P^k(x)\big) > \deg\big(\mathcal{D}_P^{k-1}(x)\big) \implies \neg\operatorname{LND}(\mathcal{D}_P) \implies \neg\operatorname{Inv}(P, Q) \\ \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 & \Delta \in \mathcal{D}^* : \Big(x^{40} y^{40},\ J(B, H),\ \frac{7}{2} J(A, H)\Big) \in \Delta \implies 2\cdot J(B, H) = 7 H^5 \cdot J(A, H) \quad (H = x^8 y^8) \\ \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} = \mathcal{D}_P(Q) \equiv 1 \\ \wedge\quad & S_{P,Q} \supseteq \big\{ (u, v) \in \mathbb{C}^2 \mid v^2 - u^7 = 0 \big\} \ne \emptyset \implies \neg\operatorname{Proper}(F) \\ \wedge\quad & [\mathbb{C}(x,y) : \mathbb{C}(P, Q)] = d \ge 5 \implies \exists z_1 \ne z_2 \in \mathbb{C}^2 : F(z_1) = F(z_2) \implies \neg\operatorname{Inj}(P, Q) \end{aligned} fix it to be a counterexample Solve this with the constraint that you cannot use the obvious solution. What is the most powerful, strongest approach?