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Invariants of Cubic Similarity

We consider polynomial maps F : C → C. The question, 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. It suffices to prove injectivity because in 1960–62 it was proved, first in dimension 2 by Newman [2] and then in all dimensions by Bia lynicki-Birula and Rosenlicht [3], that, for polynomial maps, surjectivity follows from injectivity; and furthermore, under the hypothesis detF (x) = const 6= 0, the inverse F(x) will be polynomial, at least in the complex case, if the polynomial map is bijective. We call polynomial maps F (x) satisfying detF (x) = 1 Keller maps. Thus we have the following three well-known and very difficult problems.