The Jacobian Conjecture: Linear triangularization for homogeneous polynomial maps in dimension three

Let k be a field of characteristic zero and F: k(3) -> k(3) a polynomial map of the form F = x + H, where H is homogeneous of degree d >= 2. We show that the Jacobian Conjecture is true for such mappings. More precisely, we show that if JH is nilpotent there exists an invertible linear map T such that T-1 HT = (0, h(2)(x(1)), h(3)(x(1), x(2))), where the hi are homogeneous of degree d. As a consequence of this result, we show that all generalized Druzkowski mappings F = x + H (x(1) + L-1(d),..., x(n) + L-n(d)) where L-i are linear forms for all i and d >= 2, are linearly triangularizable if J H is nilpotent and rk JH <= 3. (c) 2005 Elsevier Inc. All rights reserved.