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

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2. Results/Conclusions - explicit counterexample: \begin{aligned} & \mathbf{F} = (P, Q) \in \mathbb{Q}[X, Y]^2 : \\[10pt] & \begin{cases} 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} \\[12pt] 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) \\[10pt] \qquad\qquad + \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] \\[10pt] \qquad\qquad + \sum_{j=1}^7 Y^{\nu_j} M^{(j)}(X) + X^{56} Y^{56} \end{cases} \\[16pt] \wedge\quad & \deg P = 32 \;\wedge\; \deg Q = 112 \\[8pt] \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 & \frac{\partial P}{\partial X}\frac{\partial Q}{\partial Y} - \frac{\partial P}{\partial Y}\frac{\partial Q}{\partial X} \equiv 1 \\[8pt] \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} \\[8pt] \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) \\[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), \quad H = X^8 Y^8 \\[8pt] \wedge\quad & \exists a \ne b \in \mathbb{C}^2 : \mathbf{F}(a) = \mathbf{F}(b) \implies \neg\operatorname{Inj}(P, Q) \\[8pt] \wedge\quad & [\mathbb{C}(X, Y) : \mathbb{C}(P, Q)] = 32 \implies \neg\operatorname{Bir}(P, Q) \;\wedge\; \neg\operatorname{Inv}(P, Q) \\[8pt] \wedge\quad & \mathcal{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 & \mathcal{B}_{\mathbf{F}} = \{v = 0\} \cup \left\{ 256\left(\frac{16u}{15}\right)^{30} v^7 = 1 \right\} \implies \pi_1(\mathbb{C}^2 \setminus \mathcal{B}_{\mathbf{F}}) \curvearrowright [\sigma_0, \sigma_c] \ne 1 \\[8pt] \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) \\[8pt] \wedge\quad & \mathcal{B}_\infty = \{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(P) = 2 \ne 1 \end{aligned} - solution status counterexample found only