Find an explicit counterexample to the two-dimensional Jacob...

Prompt

Find an explicit counterexample to the two-dimensional Jacobian conjecture over \(\mathbb{C}\). We seek global polynomials \(F_1(x,y)=x+A(x,y)\) and \(F_2(x,y)=y+B(x,y)\) with \(A,B\in(x,y)^2\subset\mathbb{C}[x,y]\), such that \(\det J_F\equiv 1\) identically (making \(F\) a Keller map) yet \(F\) is not injective on \(\mathbb{C}^2\). Normalize the collision to \(F(0,0)=F(1,0)\), i.e., \(A(1,0)=-1\) and \(B(1,0)=0\). The Jacobian condition reduces to the exact polynomial identity \(A_x+B_y+A_xB_y-A_yB_x\equiv 0\). The problem is open for all degrees \(\ge 5\) (in total degree of \(F\)); degrees \(\le 4\) in \(\mathbb{C}^2\) are known to admit no counterexample. **Prior attempts and verified negative results (no counterexample found in any of these families):** 1. **Full coefficient matching** for \(A,B\) of total degree \(d=3,4,5\): via homogeneous decomposition, the conditions \(J_k=0\) at each total degree \(k\) become linear in the top coefficients at each step. At degree \(d=3\) (9 free parameters, 11 equations after elimination), \(d=4\) (15 free parameters, 20 equations), and \(d=5\) (22 free parameters, 32 equations), exact Gröbner basis computation and numerical continuation (Newton with random starts, simulated annealing) found no solution. The top-degree condition \(J_{2d-3}=\mathrm{Jac}(A_d,B_d)=0\) forces \(A_d=cL^d, B_d=c' L^d\) for some linear form \(L\) (by the binary Jacobian lemma: \(\mathrm{Jac}(f,g)=0\) for binary forms of equal degree forces proportionality). 2. **Sparse supports** of monomials (support size 2–4) in total degree \(\le 4\): over 300 enumerated and randomly sampled supports, zero solutions. 3. **Weighted-homogeneous** Keller maps: if \(A,B\) are weighted-homogeneous of the same weighted degree with distinct weights on \(x,y\), then \(A_x\) and \(B_y\) have different weighted degrees and must vanish individually, giving a triangular automorphism. No counterexample. 4. **Structured compositions / Drużkowski-type maps** (binary cubic linear forms): \(F_1=x+(ax+by)^3, F_2=y+(cx+dy)^3\). The Keller conditions force \(d=bc/a\), and all resulting maps are polynomial automorphisms. More generally, all homogeneous top-degree-3 Keller maps in 2D are automorphisms (Lemma 4, proved by case analysis on whether \(A_3,B_3\) are zero and proportionality arguments). 5. **Elimination-based parameterizations**: setting \(J_1=0\) determines \(B_2\) from \(A_2\) (e.g., \(b_{11}=-2a_{20}, b_{02}=-a_{11}/2\)); \(J_2=0\) determines \(B_3\) linearly from \(A_3\) and \(\mathrm{Jac}(A_2,B_2)\); similarly for higher \(A_k,B_k\). The remaining constraints are only the top-degree equations \(J_d,\ldots,J_{2d-3}\). Case analysis for \(d=3\): the subcase \(A_3=cL^3, B_3=c'L^3, B_2=kA_2+eL^2\) with \(L=x+ty\) gives a one-parameter family; direct substitution shows \(J_3\neq 0\) in all cases. 6. **Factorized collision form**: \(A(x,0)=-x^2+x^2(x-1)v(x)+yS\), \(B(x,0)=x^2(x-1)w(x)+yT\) with \(S,T\in(x,y)\). For \(S,T\) of degree \(\le 3\) and \(v,w\) low-degree, exact symbolic solving (sympy) returns empty solution sets; numerical search (10000-step simulated annealing) stalls at objective \(\sim 2.9\) or higher. Increasing \(S,T\) degree to 3 (22 unknowns, 27 equations) likewise stalls at objective \(\sim 4.8\). 7. **Low \(x\)-degree maps** \(A=p_0(y)+xp_1(y)+\cdots\): if \(x\)-degree is 1, the Jacobian ODE forces the map to be trivial. If \(x\)-degree is 2 with \(y\)-degree \(\le 3\), no solution found numerically. 8. **Top-degree proportional family** \(B_4=kA_4\): 15 unknowns, 17 equations; 20 numerical restarts fail to converge. 9. **Degree-pair filtering / structural obstructions**: if one component has \(y\)-degree \(\le 2\), the system decomposes into 4 ODEs in \(x\) and a contradiction arises with the collision normalization. If one component's degree divides the other's, a triangular reduction gives an automorphism. If \(\gcd(\deg P,\deg Q)\le 2\), known results (Magnus, Applegate–Onishi) preclude a counterexample. Consequently every counterexample must have \(\deg F_1, \deg F_2 \ge 3\) with \(\gcd(\deg F_1,\deg F_2)\ge 3\), and the total degree of \(F\) must be at least 9 (with the smallest non-excluded degree pair \((9,6)\)). **Additional structural facts established:** - Writing \(P=F_1, Q=F_2\) as polynomials in \(y\) with coefficients in \(\mathbb{C}[x]\), the condition \(\det J_F\equiv 1\) splits by powers of \(y\) into \(2m+1\) ODEs (where \(m=\max(\deg_y P,\deg_y Q)\)). The lowest equation is \(p_0'q_1-p_1q_0'=1\), a Bezout-type condition on \(x\)-polynomials. - For \(y\)-degree 3, substituting \(R=Q-\lambda P\) and using \(J_{y^4}: 2p_3'r_2-3p_3r_2'=0\) forces \(r_2=\alpha h^2, p_3=\beta h^3\) for some polynomial \(h(x)\). Then \(p_2=(3\beta/(2\alpha))hr_1+\gamma h^2\). The remaining system involves only \(p_0,p_1,r_0,r_1,h\). Numerical search with low-degree ansatz for \(h\) (linear–quadratic) and moderate-degree \(p_0,r_0\) (cubic–quartic) does not reach zero residual (best residual \(\sim 0.4\)), suggesting higher-degree \(h\) is needed. - All numerical evidence (simulated annealing objectives plateauing at \(0.3\)–\(386\) rather than reaching \(0\)) is consistent with the conjecture being true, but does not constitute a proof. **What remains open and must be pursued:** - Total degree \(d\ge 6\) for \(F\) (equivalently \(\deg A,\deg B\ge 5\)), with degree pair \((\deg F_1,\deg F_2)\) satisfying \(\gcd\ge 3\), e.g. \((9,6),(9,9),(6,9),(10,8)\), etc. - Ternary or higher-composition families: \(A_k=\sum c_{k,i}L_1^a L_2^b\) for two independent linear forms \(L_1,L_2\). - Higher-degree \(y\)-degree-3 reduction with quartic or quintic \(h(x)\), or \(y\)-degree \(\ge 4\) with the proportional-top structure \(r_{m}=\alpha h^m, p_{m}=\beta h^m\). - Exact Gröbner basis computation (Singular/Magma) for \(d=6,7\) full systems. - Homotopy continuation (Bertini/HomotopyContinuation.jl) over \(\mathbb{C}\) starting from known near-solutions or degenerate limits. - Novel algebraic structures (Poisson brackets, non-commutative generalizations) that might yield polynomial identities encoding both Keller and collision conditions. **Computational infrastructure available:** - SymPy 1.14.0 for symbolic manipulation, Gröbner bases (greveraels method), exact solving, and numerical `nsolve` with tolerance \(10^{-14}\), `maxsteps` 200–300, precision 50–80 digits. - Simulated annealing / hill climbing over \(\mathbb{C}^N\) with exponential cooling (step size \(0.5\to0.18\)), 5000–30000 iterations, using `cmath` and `lambdify` for speed. - Combinatorial support enumeration via `itertools.combinations`. - Budget: enough for systems up to ~40 unknowns / ~60 equations numerically; exact Gröbner for \(\le 25\) unknowns in reasonable time. **Deliverable:** Explicitly specify \(F_1(x,y)\), \(F_2(x,y)\), \(p=(0,0)\), \(q=(1,0)\) as exact polynomials over \(\mathbb{Q}\) or a specified number field, with a machine-checkable verification that (i) \(\det J_F\equiv 1\), (ii) \(A,B\in(x,y)^2\), (iii) \(A(1,0)=-1\), \(B(1,0)=0\), and (iv) \(F(0,0)=F(1,0)\) but \(F\) is not the identity, together with a proof (or at least convincing exact algebraic argument) that \(F\) is not injective on \(\mathbb{C}^2\). If no such counterexample exists at the first degree tried, systematically increase the degree and the structural complexity of the families explored, maintaining a complete record of every family attempted, the equations derived, and the reason each family was abandoned.