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

Let k be a field of characteristic zero and F:k3→k3 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−1HT=(0,h2(x1),h3(x1,x2)), where the hi are homogeneous of degree d. As a consequence of this result, we show that all generalized Drużkowski mappings F=x+H=(x1+L1d,…,xn+Lnd), where Li are linear forms for all i and d⩾2, are linearly triangularizable if JH is nilpotent and rkJH⩽3.