\[ H: \mathbb{C}^2 \longrightarrow \mathbb{C}^2, \qquad H(x,...

Prompt

\[ H: \mathbb{C}^2 \longrightarrow \mathbb{C}^2, \qquad H(x, y) = \begin{pmatrix} F_1(x, y) \\ F_2(x, y) \end{pmatrix} \] \[ \det JH(x, y) = \frac{\partial F_1}{\partial x}\frac{\partial F_2}{\partial y} - \frac{\partial F_1}{\partial y}\frac{\partial F_2}{\partial x} \equiv 1 \quad \forall (x, y) \in \mathbb{C}^2 \] \[ D_{F_1}(F_2) = -\frac{\partial F_1}{\partial y}\frac{\partial F_2}{\partial x} + \frac{\partial F_1}{\partial x}\frac{\partial F_2}{\partial y} \equiv 1 \] \[ \boxed{9F_2(x, y)^2 - 8F_1(x, y)^3 = -8x^3 - 3y^2} \] \[ \boxed{\mathbb{C}(F_1, F_2) \cap \mathbb{C}[x, y] = \mathbb{C}[9F_2^2 - 8F_1^3]} \] \[ F_1(x, y) = \sum_{i=0}^d P_i(x)y^i, \qquad F_2(x, y) = \sum_{j=0}^d Q_j(x)y^j \] \[ \sum_{i+j=n+1} \left( j P_i'(x) Q_j(x) - i P_i(x) Q_j'(x) \right) = \delta_{n0} \] \[ (n+1)\bigl(f'(x) Q_{n+1}(x) - g'(x) P_{n+1}(x)\bigr) = \delta_{n0} - R_n(x) \] \[ R_n(x) = \sum_{i=1}^n \left( (n+1-i) P_i'(x) Q_{n+1-i}(x) - i P_i(x) Q_{n+1-i}'(x) \right) \] \[ P_{n+1}(x) \longmapsto P_{n+1}(x) + f'(x) t_n(x), \qquad Q_{n+1}(x) \longmapsto Q_{n+1}(x) + g'(x) t_n(x), \qquad t_n(x) \in \mathbb{C}[x] \] \[ P_i(x) = Q_i(x) \equiv 0 \quad (\forall i > d), \qquad \deg P_i, \; \deg Q_i \le d - i \] \[ \mathbf{p_+} = \left( \frac{-11 + \sqrt{33}}{4}, \;\; \frac{11 - 3\sqrt{33}}{2} \right), \qquad \mathbf{p_-} = \left( \frac{-11 - \sqrt{33}}{4}, \;\; \frac{11 + 3\sqrt{33}}{2} \right) \] \[ \mathbf{p_+} \neq \mathbf{p_-} \] \[ -8\left(\frac{-11+\sqrt{33}}{4}\right)^3 - 3\left(\frac{11-3\sqrt{33}}{2}\right)^2 = -11 \] \[ -8\left(\frac{-11-\sqrt{33}}{4}\right)^3 - 3\left(\frac{11+3\sqrt{33}}{2}\right)^2 = -11 \] \[ 9\left(\frac{4}{3}\right)^2 - 8\left(\frac{3}{2}\right)^3 = 9\cdot\frac{16}{9} - 8\cdot\frac{27}{8} = 16 - 27 = -11 \] \[ H(\mathbf{p_+}) = H(\mathbf{p_-}) = \begin{pmatrix} \frac{3}{2} \\[6pt] \frac{4}{3} \end{pmatrix} \] fix the weak parts