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

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[ 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} ] 1. Non-Existence of the Polynomial Map H (Riemann–Hurwitz and Poincaré Residue Contradiction): - Critique: There do not exist polynomials F_1, F_2 \in \mathbb{C}[x,y] satisfying both \det JH \equiv 1 and 9F_2^2 - 8F_1^3 = -8x^3 - 3y^2. - Fix / Result: For any c \neq 0, the affine curves C_c: -8x^3 - 3y^2 = c and E_c: 9v^2 - 8u^3 = c are smooth elliptic curves punctured at single points at infinity (\infty_C and \infty_E). A polynomial map H = (F_1, F_2) restricts to a regular morphism H_c: C_c \to E_c, which extends to a morphism of smooth projective genus-1 curves \bar{H}_c: \bar{C}_c \to \bar{E}_c satisfying \bar{H}_c^{-1}(\infty_E) = \{\infty_C\}. By the Riemann–Hurwitz formula, \bar{H}_c is unramified everywhere, forcing \deg(\bar{H}_c) = |\bar{H}_c^{-1}(\infty_E)| = 1. Conversely, taking the Poincaré residue (Gelfand–Leray form) of H^*(du \wedge dv) = dx \wedge dy along C_c yields: [ \bar{H}_c^*\left(\frac{du}{18v}\right) = \frac{dx}{-6y} ] Under the curve isomorphism \phi(x,y) = \left(x, \frac{i}{\sqrt{3}}y\right) from \bar{C}_c to \bar{E}_c, \phi^*\left(\frac{du}{18v}\right) = \frac{i}{\sqrt{3}}\left(\frac{dx}{-6y}\right), so \phi^{-1} \circ \bar{H}_c \in \mathrm{End}(\bar{C}_c) \cong \mathbb{Z}[\zeta_3] must act on differentials as multiplication by \alpha = -i\sqrt{3}. This requires \deg(\bar{H}_c) = |\alpha|^2 = 3, contradicting \deg(\bar{H}_c) = 1. 2. Invalid Field Intersection Identity: - Critique: The statement \mathbb{C}(F_1, F_2) \cap \mathbb{C}[x, y] = \mathbb{C}[9F_2^2 - 8F_1^3] is algebraically false for any polynomial map with \det JH \equiv 1. - Fix / Result: Since F_1, F_2 \in \mathbb{C}[x,y] and \det JH \equiv 1 \neq 0, F_1 and F_2 are algebraically independent polynomials belonging to \mathbb{C}(F_1, F_2) \cap \mathbb{C}[x,y]. Thus, the intersection has transcendence degree 2 over \mathbb{C}, whereas \mathbb{C}[9F_2^2 - 8F_1^3] has transcendence degree 1. The correct relation is: [ \mathbb{C}[9F_2^2 - 8F_1^3] \subsetneq \mathbb{C}[F_1, F_2] \subseteq \mathbb{C}(F_1, F_2) \cap \mathbb{C}[x, y] ] 3. Undefined Base Terms and Missing Coprimality Condition in the Recursion: - Critique: In (n+1)\bigl(f'(x) Q_{n+1}(x) - g'(x) P_{n+1}(x)\bigr) = \delta_{n0} - R_n(x), the functions f(x) and g(x) are undefined, and the condition required to solve for P_{n+1}, Q_{n+1} \in \mathbb{C}[x] is omitted. - Fix / Result: The base terms must be explicitly defined along with a Bézout coprimality condition in \mathbb{C}[x]: [ P_0(x) := f(x), \qquad Q_0(x) := g(x), \qquad \gcd\bigl(f'(x), g'(x)\bigr) = 1 ] 4. Overdetermined Truncation and Incompatible Degree Bounds: - Critique: The gauge freedom P_{n+1} \mapsto P_{n+1} + f' t_n, Q_{n+1} \mapsto Q_{n+1} + g' t_n cannot simultaneously enforce P_i = Q_i \equiv 0 (\forall i > d), \deg P_i, \deg Q_i \le d - i, and 9F_2^2 - 8F_1^3 = -8x^3 - 3y^2. - Fix / Result: For P_i = Q_i \equiv 0 (\forall i > d) to hold, the recursion requires: [ R_n(x) \equiv 0 \quad \forall n \in {d, d+1, \dots, 2d-1} ] Because t_n(x) must vanish for n \ge d, there are only d free polynomials t_0, \dots, t_{d-1} to satisfy d nonlinear differential-algebraic vanishing conditions R_d = \dots = R_{2d-1} = 0 plus the invariant curve relation. Moreover, total degree bounds \deg F_1 \le d and \deg F_2 \le d are incompatible with 9F_2^2 - 8F_1^3 = -8x^3 - 3y^2 unless the leading homogeneous forms satisfy 9(\text{LT}(F_2))^2 = 8(\text{LT}(F_1))^3, which forces \deg F_1 = 2k and \deg F_2 = 3k, violating the symmetric bound \deg P_i, \deg Q_i \le d - i. 5. False Non-Injectivity Conclusion H(\mathbf{p_+}) = H(\mathbf{p_-}): - Critique: Although \mathbf{p_+} and \mathbf{p_-} both lie on the level curve C_{-11}: -8x^3 - 3y^2 = -11 and (3/2, 4/3) lies on E_{-11}: 9v^2 - 8u^3 = -11, this does not imply H(\mathbf{p_+}) = H(\mathbf{p_-}) = (3/2, 4/3)^T. - Fix / Result: By Result 1, any polynomial morphism H_{c}: C_{-11} \to E_{-11} extending to \bar{C}_{-11} \to \bar{E}_{-11} with a single point at infinity must have degree 1 (an isomorphism of curves) and is therefore strictly injective on C_{-11}: [ \mathbf{p_+} \neq \mathbf{p_-} \implies H(\mathbf{p_+}) \neq H(\mathbf{p_-}) ] csak a kijavitott teljes kepletet kuld vissza semmi mast