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

Prompt

\begin{aligned} \mathbf{F} = (P, Q): \mathbb{C}^2 \to \mathbb{C}^2 \quad : \quad & \begin{cases} P(X, Y) = X - \dfrac{35}{24} X^{12} Y^7 + X^{16} Y^{16} & (\deg P = 32) \\[12pt] Q(X, Y) = Y + \dfrac{96}{805} X^4 Y^{10} + X^{56} Y^{56} - \dfrac{245}{48} X^{52} Y^{47} + \dfrac{42875}{4608} X^{48} Y^{38} + \dfrac{7}{2} X^{41} Y^{40} \\[8pt] \qquad\quad\quad - \dfrac{1500625}{221184} X^{44} Y^{29} - \dfrac{1225}{96} X^{37} Y^{31} + \dfrac{52521875}{42467328} X^{40} Y^{20} + \dfrac{42875}{3072} X^{33} Y^{22} \\[8pt] \qquad\quad\quad + \dfrac{35}{8} X^{26} Y^{24} + \dfrac{367653125}{2038431744} X^{36} Y^{11} - \dfrac{1500625}{442368} X^{29} Y^{13} - \dfrac{1225}{128} X^{22} Y^{15} \\[8pt] \qquad\quad\quad + \dfrac{12867859375}{195689447424} X^{32} Y^2 - \dfrac{52521875}{84934656} X^{25} Y^4 + \dfrac{42875}{12288} X^{18} Y^6 + \dfrac{35}{16} X^{11} Y^8 & (\deg Q = 112) \end{cases} \\[16pt] \wedge\quad & \begin{cases} a_s = 0 \quad \forall s \in \{0, 1, \dots, 11\}, \qquad K_m(X) \equiv 0 \quad \forall m \in \{0, 1, 2, 3\} \\[6pt] \nu = (47, 38, 29, 20, 11, 2, -7), \qquad \mu = (52, 48, 44, 40, 36, 32, 28) \\[6pt] M^{(1)}(X) = -\dfrac{245}{48} X^{52}, \quad M^{(2)}(X) = \dfrac{42875}{4608} X^{48}, \quad M^{(3)}(X) = -\dfrac{1500625}{221184} X^{44} \\[8pt] M^{(4)}(X) = \dfrac{52521875}{42467328} X^{40}, \quad M^{(5)}(X) = \dfrac{367653125}{2038431744} X^{36}, \quad M^{(6)}(X) = \dfrac{12867859375}{195689447424} X^{32}, \quad M^{(7)}(X) \equiv 0 \\[8pt] A(X, Y) = X^{16} Y^{16}, \qquad B(X, Y) = X^{56} Y^{56}, \qquad H(X, Y) = X^8 Y^8 \\[6pt] \left(1 + \dfrac{16}{X}\right)_{\le d}^{\alpha} := \sum_{k=0}^{d} \binom{\alpha}{k} 16^k X^{-k} \in \mathbb{Q}[X, X^{-1}] \end{cases} \\[16pt] \wedge\quad & \frac{\partial P}{\partial X}\frac{\partial Q}{\partial Y} - \frac{\partial P}{\partial Y}\frac{\partial Q}{\partial X} \equiv 1 \quad \text{in } \mathbb{Q}[X, Y] \\[10pt] \wedge\quad & \mathbf{F}(0, 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} = I_2 \\[10pt] \wedge\quad & \gcd(\deg P, \deg Q) = 16 \;\wedge\; \neg\operatorname{Prime}(\gcd(\deg P, \deg Q)) \;\wedge\; \max(\deg P, \deg Q) = 112 > 100 \\[8pt] \wedge\quad & \deg P \nmid \deg Q \;\wedge\; \deg Q \nmid \deg P \\[8pt] \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) = 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) \\[10pt] \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 \quad \left(\deg \frac{P+Q}{\sqrt{2}} = \deg \frac{Q-P}{\sqrt{2}} = 112\right) \\[10pt] \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 \;\wedge\; \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) \\[10pt] \wedge\quad & \mathcal{B} = \{v_1, v_2\} \subseteq \mathrm{Pl}_\infty : \begin{cases} v_1(X) = -1, v_1(Y) = 1 \implies v_1(H) = 0 \\ v_2(X) = 1, v_2(Y) = -1 \implies v_2(H) = 0 \end{cases} \implies \#\mathrm{Pl}_\infty(P) = 2 \ne 1 \\[10pt] \wedge\quad & S_{P, Q} = \big\{ (u, v) \in \mathbb{C}^2 \mid v^2 - u^7 = 0 \big\} \ne \emptyset \implies \neg\operatorname{Proper}(\mathbf{F}) \\[8pt] \wedge\quad & \pi_1\big(\mathbb{C}^2 \setminus S_{P, Q}\big) \cong \langle \sigma_0, \sigma_c \mid \sigma_0 = (2)(30), \, \sigma_c = (1\,2) \rangle \implies [\sigma_0, \sigma_c] \ne 1 \implies \text{Non-abelian étale monodromy} \\[8pt] \wedge\quad & [\mathbb{Q}(X, Y) : \mathbb{Q}(P, Q)] = d \ge 5 \implies \exists a \ne b \in \mathbb{Q}^2 : \mathbf{F}(a) = \mathbf{F}(b) \implies \neg\operatorname{Inj}(P, Q) \\[8pt] \wedge\quad & \neg\operatorname{Bir}(P, Q) \;\wedge\; \neg\operatorname{Inv}(P, Q) \end{aligned}