Polyomorphisms Conjugate to Dilations

Abstract

Consider polynomial maps f : ℂ n → ℂ n and their dilations sf(x) by complex scalars s. That is, maps f whose components f i are polynomials with complex coefficients in the n variables (x 1, x 2, ... , x n ) = x ∈ ℂ n . The question, first raised by O.-H. Keller in 1939 [10] 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 det f′(x) need be a polyomorphism: I.e., bijective with polynomial inverse. It suffices to prove injectivity because in 1960–62 it was proved, first in dimension 2 by Newman [19] and then in all dimensions by Białynicki-Birula and Rosenlicht [4], that, for polynomial maps, surjectivity follows from injectivity; and furthermore, under Keller’s hypothesis, the inverse f −1(x) will be polynomial, at least in the complex case, if the polynomial map is bijective. The group of all polyomorphisms of ℂ n is denoted GA n (ℂ). It is isomorphic to the group Aut C[x] of automorphisms σ of the polynomial ring ℂ[x] by means of the correspondence ø(f) = σ where σ(x i ) = f i (x). Polynomial maps f(x) satisfying det f′(x) = const ≠ 0 are called Keller maps. We can and do assume that f(0) = 0 and f′(0) = I. Five main problems arise: