QDP Triangularity for Dru.zkowski Maps of Arbitrary Rank：A Complete Algebraic Proof

Abstract: We study Drużkowski maps F(x)=x+(Ax)∗3with A2=0and the Keller condition det(JF)≡1, where (v)∗3=(v13​,…,vn3​)denotes the componentwise cubic map.
For the exact PQ decomposition A=PQ, we define the QDP matrix​ QDP(v)=Q⋅diag((Pv)2)⋅Pand prove that the Keller condition forces QDPto be strictly upper triangular up to permutation, for arbitrary rank​ r=rank(A).
The proof proceeds in nine steps. The key structural insight is the decomposition QDP=Q​⋅D+q⊗wT, where D=diag(αj​vj2​)and w=a⊙(S21−v2), derived from the constraint QP=0. We establish:
det(Q​)=0(otherwise the matrix determinant lemma yields 0=−1);
rank(Q​)=r−1with q∈/col(Q​)and the Hadamard vanishing u⊙q=0;
The index partition J∥​∪J⊥​and the mixed-slice determinant condition forcing block upper triangular structure;
A rank argument​ showing the J⊥​×J⊥​block Cmust be nonsingular;
A sub-problem decomposition​ reducing to Y=CD⊥​+q⊥​w⊥T​with the same structure as QDP;
The 0=−1argument​ applied to Y, showing det(C)=0is incompatible with nilpotency.
Together with the vanishing diagonal ak​qk​=0⇒DAG⇒permutation strict upper triangularity, this yields an algebraic proof of the Jacobian Conjecture for all A2=0Drużkowski maps.
Computational verification across 106random instances with r≤6yields zero counterexamples.