Find explicit polynomials $F_1(x, y), F_2(x, y) \in \mathbb{...

Prompt

Find explicit polynomials $F_1(x, y), F_2(x, y) \in \mathbb{C}[x, y]$ and two distinct points $P = (x_1, y_1), Q = (x_2, y_2) \in \mathbb{C}^2$ ($P \neq Q$) that disprove the two-dimensional Jacobian Conjecture over $\mathbb{C}$ by satisfying: $$\det JF = \frac{\partial F_1}{\partial x}\frac{\partial F_2}{\partial y} - \frac{\partial F_1}{\partial y}\frac{\partial F_2}{\partial x} \equiv 1 \quad \text{on all } \mathbb{C}^2$$ $$F(P) = F(Q)$$ ### Technical Constraints and Boundaries 1. **Mathematical Domain:** Must be global polynomials $F_1, F_2 \in \mathbb{C}[x, y]$ over characteristic 0. No rational functions, no Laurent polynomials, no meromorphic functions, no punctured domains, and no dimension $n \ge 3$. 2. **Standard Normalization:** Under standard affine transformation, the collision points and 1-jet can be fixed without loss of generality as: - $P = (0, 0)$ and $Q = (1, 0)$ - $F_1(x, y) = x + P(x, y)$ and $F_2(x, y) = y + Q(x, y)$, where $P, Q \in (x, y)^2$ - Collision conditions: $1 + P(1, 0) = 0$ and $Q(1, 0) = 0$ - Keller identity: $P_x + Q_y + P_x Q_y - P_y Q_x \equiv 0$ 3. **Excluded / Proven-Void Directions (do not revisit):** - Full non-homogeneous degree $\le 3$ (Gröbner basis proves the collision ideal is $(1)$ via explicit Nullstellensatz certificates). - Homogeneous quartic leading ansatz (proven to generate the unit ideal). - Hamiltonian/Poisson transformations $F = (x + H_y, y - H_x)$ with $\det \operatorname{Hess}(H) = 0$ (all reduce to triangular polynomial automorphisms). - Denominator-cancellation coordinate lenses on rational maps where $\det J$ vanishes on an exceptional divisor (e.g., $\det J \propto s$). - Finite field constructions (e.g., $\mathbb{F}_2$). 4. **Target Space:** Full non-homogeneous systems of degree $d \ge 4$ (starting with the 24-variable, 29-equation degree-4 coefficient ideal or its 15-variable linear reduction) and asymptotic non-proper Puiseux/Laurent tracts lifting to global polynomial maps. 5. **Execution Directive:** Compute and search autonomously until a valid counterexample is found. Do not quit, do not output status updates, and do not provide conditional alternatives or theoretical impossibility statements. 6. **Required Output Format:** Output exclusively the final mathematical formulas for $F_1(x, y)$, $F_2(x, y)$, $P$, and $Q$. Do not include any explanations, documentation, proofs, derivations, verification steps, or additional text.