Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture

Abstract

Let H : k n → k n be a polynomial map. It is shown that the Jacobian matrix JH is strongly nilpotent if and only if JH is linearly triangularizable if and only if the polynomial map F = X + H is linearly triangularizable. Furthermore it is shown that for such maps F, sF is linearizable for almost all s ∈ k (except a finite number of roots of unity).