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

Abstract

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 H T = ( 0 , h 2 ( x 1 ) , h 3 ( x 1 , x 2 ) ) , where the h i are homogeneous of degree d. As a consequence of this result, we show that all generalized Drużkowski 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 JH is nilpotent and rk J H ⩽ 3 .