Find explicit polynomials \(F_1(x,y)=x+A(x,y)\), \(F_2(x,y)=...

Prompt

Find explicit polynomials \(F_1(x,y)=x+A(x,y)\), \(F_2(x,y)=y+B(x,y)\) over \(\mathbb{C}\) with \(A,B\in(x,y)^2\) such that \(\det J_F\equiv 1\) but \(F(0,0)=F(1,0)\) (counterexample to 2D Jacobian conjecture). Constraints: \(A(1,0)=-1\), \(B(1,0)=0\), and \(A_x+B_y+A_xB_y-A_yB_x\equiv 0\). Degrees \(\le4\) ruled out (known); degrees \(\ge5\) open. **Exhausted negative results (no solution found):** 1. Full coefficient matching \(d=3,4,5\): 9/15/22 free params, 11/20/32 equations, Gröbner+Newton+simulated annealing, no solution. 2. Sparse supports (size 2–4, degree \(\le4\)): 300+ supports tested, zero solutions. 3. Weighted-homogeneous Keller maps: forces triangular automorphism, no collision. 4. Drużkowski/homogeneous degree-3: all give automorphisms (Lemma: binary homogeneous \(\mathrm{Jac}(f,g)=0\) forces proportionality). 5. Elimination parameterization: \(J_1\) determines \(B_2\) from \(A_2\); \(J_2\) determines \(B_3\); etc. Top constraint \(J_{2d-3}=\mathrm{Jac}(A_d,B_d)=0\). Case analysis for \(d=3\) (including \(A_3=cL^3,B_3=c'L^3\)) yields no solution after substitution. 6. Factorized collision \(A=-x^2+x^2(x-1)v(x)+yS\), \(B=x^2(x-1)w(x)+yT\): symbolic solve empty, numerical objective stalls at 2.9–4.8. 7. Low \(x\)-degree (1 or 2) maps: ODE forces triviality or no numerical solution. 8. Top-proportional \(B_d=kA_d\): no convergence. 9. \(y\)-degree \(\le2\) (checkpoint): ODE system on \(p_i(x),q_i(x)\) coefficients in \(y\) yields contradiction with normalization. Must have \(\deg_y\ge3\) in at least one component. 10. \(y\)-degree 3 reduction: from \(2p_3'r_2-3p_3r_2'=0\), get \(r_2=\alpha h^2\), \(p_3=\beta h^3\). Remaining system on \(p_0,p_1,r_0,r_1,h\). Low-degree \(h\) (linear–quadratic) with cubic–quartic \(p_0,r_0\): residual stalls at 0.4. Quintic \(h\), quartic \(p_0,r_0\): residual 0.67. 11. Degree-pair filtering: \(\gcd(\deg F_1,\deg F_2)\le2\) excluded; dividing degrees give triangular automorphism. Smallest non-excluded pair: \((9,6)\), total degree \(\ge9\). **Established structural facts:** - Writing \(F_i\) as polynomials in \(y\) with \(x\)-coefficients \(p_j(x),q_j(x)\): Jacobian splits into \(2m+1\) ODEs. Lowest: \(p_0'q_1-p_1q_0'=1\). - Top binary Jacobian: \(\mathrm{Jac}(A_d,B_d)=0\) forces \(A_d=cL^d\), \(B_d=c'L^d\). - Substitution \(R=Q-\lambda P\) reduces \(y\)-degree-3 system via the \(y^4\) equation. - All numerical objectives plateau at 0.3–386 (never 0), consistent with no counterexample in searched ranges. **Computational tools available:** SymPy 1.14.0 (Gröbner, solve, nsolve, lambdify); Newton with random starts (tol \(10^{-14}\), maxsteps 200–300, prec 50–80); simulated annealing (5000–30000 iterations, exponential cooling); combinatorial support enumeration. Budget: exact Gröbner to ~25 unknowns; numerical to ~40 unknowns/~60 equations. **Directions not yet tried (must be pursued):** - Full systems at \(d=6,7\) (30+ unknowns, 50+ equations) via Gröbner (Singular/Magma) or homotopy continuation (Bertini). - Multi-linear-form compositions \(A_k=\sum c_{k,i}L_1^aL_2^b\). - \(y\)-degree 3 with quartic/quintic \(h(x)\), or \(y\)-degree \(\ge4\) with proportional top \(r_m=\alpha h^m\). - Novel algebraic structures (Poisson brackets, non-commutative identities). - Systematic sparse supports at \(d\ge5\). - At least 5 structurally distinct new families after each failure. **Deliverable:** Explicit \(F_1(x,y)\), \(F_2(x,y)\) as exact polynomials over \(\mathbb{Q}\) or a number field, with \(p=(0,0)\), \(q=(1,0)\), and machine-checkable verification that \(\det J_F\equiv 1\), \(A,B\in(x,y)^2\), \(A(1,0)=-1\), \(B(1,0)=0\), \(F(p)=F(q)\). Output only the polynomials and points when verified.