Invariants of Cubic Similarity

The question about polynomial maps F : C → C, first raised by Keller [1] in 1939 for polynomials over the integers but now also raised for complex polynomials and, as such, known as The Jacobian Conjecture (JC), asks whether a polynomial map F with nonzero constant Jacobian determinant detF (x) need be a polyomorphism: Injective and also surjective with polynomial inverse. The known reductions [2, 3, 4, 5] show that it suffices to prove injectivity for maps of Drużkowski’s special cubic-linear form FA(x) = x−HA(x) :≡ x− [diag(Ax)] 3 l, where A is an n×n complex matrix, diag(x) denotes the diagonal matrix diag[x1, . . . , xn], and l denotes the column of n 1’s. The complex matrix A is the kernel of FA and HA. Then the Jacobian of the cubic-homogeneous part is H ′ A (x) = 3 [diag(Ax)]A and the associated matrixvalued bilinear map is BA(x, y) = 3 [diag(Ax)] [diag(Ay)]A. Dfn 1: A complex n × n matrix A is called cubic-admissible if H ′ A (x) is nilpotent ∀x ∈ C; or, equivalently, detF ′ A (x) ≡ 1 (that is, x 7→ FA(x) is a Keller map). See [6]. The change of vector variables x = Pu and y = Pv in the equation y = FA(x) leads to Dfn 2: Matrices A and D are cubic-similar (A cubic ∼ D) if, for an invertible matrix P , [diag(APu)] l = P [diag(Du)] l, ∀u ∈ C ; or PH ′ A (Pu)P = H ′ D (u), ∀u ∈ C ; or PBA(Pu, Pv)P = BD(u, v), ∀u, v ∈ C . Some Invariants of Cubic-Similarity are: Nilpotence of H ′ A (x); the nilpotence index of H ′ A (x); injectivity of FA(x) := x − HA(x); the rank of the matrix A; nilpotence of BA(x, y); and the nilpotence index of BA(x, y). There are many other invariants, independent of these, whose significance is being investigated. Each 2 × 2 admissible A can be written as a dyad A =