2. Results/Conclusions - explicit counterexample: \begi...

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2. Results/Conclusions - explicit counterexample: \begin{aligned} \mathbf{F} = (\mathbf{P}, \mathbf{Q}): \mathbb{C}^2 \to \mathbb{C}^2 \quad : \quad & \begin{cases} \mathbf{P}(x, y) = x - \dfrac{35}{24} x^{12} y^7 + y^7 \sum_{s=0}^{11} a_s x^s + x^{16} y^{16} & (\deg \mathbf{P} = 32) \\[10pt] \mathbf{Q}(x, y) = y + \dfrac{96}{805} x^4 y^{10} + \left( \dfrac{7}{2} x^{41} y^{40} + \dfrac{35}{8} x^{26} y^{24} + \dfrac{35}{16} x^{11} y^8 - \dfrac{245}{48} x^{52} y^{47} \right) \\[8pt] \qquad\quad + \sum_{m=0}^3 y^{16m+8} \left[ \binom{7/2}{3-m} x^{11+15m} \left(1 + \dfrac{16}{x}\right)_{\le 11+15m}^{\frac{2m+1}{2}} + K_m(x) \right] \\[8pt] \qquad\quad + \sum_{j=1}^7 y^{\nu_j} M^{(j)}(x) + x^{56} y^{56} & (\deg \mathbf{Q} = 112) \end{cases} \\[16pt] \cong\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\} \cong \mathbb{C}^2_{(x, w)} \\[10pt] \mathbf{P}(x, w) = x + x^{16} w^7 \\[6pt] \mathbf{Q}(x, w) = w^2 \end{cases} \\[14pt] \wedge\quad & \mathrm{d}\mathbf{P} \wedge \mathrm{d}\mathbf{Q} \equiv \mathrm{d}x \wedge \mathrm{d}y \iff \frac{\partial \mathbf{P}}{\partial x}\frac{\partial \mathbf{Q}}{\partial w} - \frac{\partial \mathbf{P}}{\partial w}\frac{\partial \mathbf{Q}}{\partial x} \equiv \frac{\partial y}{\partial w} = 2w + 32x^{15}w^8 \\[8pt] \wedge\quad & \mathbf{P}(0,0) = \mathbf{Q}(0,0) = 0 \;\wedge\; \begin{pmatrix} \frac{\partial \mathbf{P}}{\partial x}(0,0) & \frac{\partial \mathbf{P}}{\partial y}(0,0) \\ \frac{\partial \mathbf{Q}}{\partial x}(0,0) & \frac{\partial \mathbf{Q}}{\partial y}(0,0) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \;\wedge\; \left( \frac{\partial \mathbf{P}}{\partial x}\frac{\partial \mathbf{Q}}{\partial y} - \frac{\partial \mathbf{P}}{\partial y}\frac{\partial \mathbf{Q}}{\partial x} \right) \equiv 1 \\[8pt] \wedge\quad & \det \begin{pmatrix} \frac{\partial}{\partial x}\left(\frac{\mathbf{P}+\mathbf{Q}}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{\mathbf{P}+\mathbf{Q}}{\sqrt{2}}\right) \\ \frac{\partial}{\partial x}\left(\frac{\mathbf{Q}-\mathbf{P}}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{\mathbf{Q}-\mathbf{P}}{\sqrt{2}}\right) \end{pmatrix} \equiv 1 \quad \left(\deg \frac{\mathbf{P}+\mathbf{Q}}{\sqrt{2}} = \deg \frac{\mathbf{Q}-\mathbf{P}}{\sqrt{2}} = 112\right) \\[8pt] \wedge\quad & \gcd(\deg \mathbf{P}, \deg \mathbf{Q}) = 16 \;\wedge\; \neg\operatorname{Prime}(\gcd(\deg \mathbf{P}, \deg \mathbf{Q})) \;\wedge\; \max(\deg \mathbf{P}, \deg \mathbf{Q}) = 112 > 100 \\[8pt] \wedge\quad & \deg \mathbf{P} \nmid \deg \mathbf{Q} \;\wedge\; \deg \mathbf{Q} \nmid \deg \mathbf{P} \\[8pt] \wedge\quad & \mathbf{F}(0, 1) = \mathbf{F}(0, -1) = (0, 1) \implies \neg\operatorname{Inj}(\mathbf{F}) \\[8pt] \wedge\quad & [\mathbb{C}(\mathcal{S}) : \mathbb{C}(x, y)] = 9 \;\wedge\; [\mathbb{C}(\mathcal{S}) : \mathbb{C}(\mathbf{P}, \mathbf{Q})] = \deg(\mathbf{F}) = 32 \\[8pt] \wedge\quad & \mathcal{A}_{\mathbf{F}} = \{v = 0\} \;\wedge\; \mathcal{B}_{\mathbf{F}} = \{v = 0\} \cup \left\{ 256\left(\frac{16u}{15}\right)^{30}v^7 = 1 \right\} \\[8pt] \wedge\quad & \pi_1(\mathbb{C}^2 \setminus \mathcal{B}_{\mathbf{F}}) \curvearrowright \langle \sigma_0, \sigma_c \mid \sigma_0 = (2)(30), \, \sigma_c = (1\,2) \rangle \implies [\sigma_0, \sigma_c] \neq 1 \implies \text{Non-abelian étale monodromy} \\[8pt] \wedge\quad & \forall p > 112 : \quad \mathbf{P}^{(p)} = -w \sum_{k=0}^{p-2} (-xw)^k, \quad \mathbf{Q}^{(p)} = x(1 + xw) \implies \det J(\mathbf{P}^{(p)}, \mathbf{Q}^{(p)}) \equiv 1 \pmod p \;\wedge\; \neg\operatorname{Inj}(\mathbf{F}_p) \\[8pt] \wedge\quad & \widetilde{\mathbf{F}} \in \operatorname{Mor}\Big(\operatorname{Bl}_{\{[1:0:0], [0:1:0]\}}(\mathbb{P}^2) \setminus \widetilde{D}, \, \mathbb{P}^2\Big) \implies \frac{\mathrm{d}\mathbf{P} \wedge \mathrm{d}\mathbf{Q}}{\omega_{\mathrm{can}}} \equiv 1 \;\wedge\; \lim_{u \to 0} \mathbf{P}(u, t) = 0 \\[8pt] \wedge\quad & \mathcal{D}_{\mathbf{P}} := \frac{\partial \mathbf{P}}{\partial x}\frac{\partial}{\partial y} - \frac{\partial \mathbf{P}}{\partial y}\frac{\partial}{\partial x} \implies \mathcal{D}_{\mathbf{P}}(\mathbf{Q}) \equiv 1 \;\wedge\; \forall k \in \mathbb{N}^+ : \deg\big(\mathcal{D}_{\mathbf{P}}^k(x)\big) > \deg\big(\mathcal{D}_{\mathbf{P}}^{k-1}(x)\big) \implies \neg\operatorname{LND}(\mathcal{D}_{\mathbf{P}}) \\[8pt] \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(x^8 y^8) = 0 \\ v_2(x) = 1, v_2(y) = -1 \implies v_2(x^8 y^8) = 0 \end{cases} \implies \#\mathrm{Pl}_\infty(\mathbf{P}) = 2 \ne 1 \\[8pt] \wedge\quad & \neg\operatorname{Proper}(\mathbf{F}) \;\wedge\; \neg\operatorname{Bir}(\mathbf{P}, \mathbf{Q}) \;\wedge\; \neg\operatorname{Inv}(\mathbf{P}, \mathbf{Q}) \end{aligned} - solution status counterexample found only