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 (definition 1.1) 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).

Unipotent Jacobian matrices and univalent maps

“…The Jacobian matrix matrix J(F ) was called strongly nilpotent if all products of n matrices of the form J(F )(a i ), for possibly distinct points a i ∈ R n , are zero. In [23] the Jacobian matrix of a polynomial map F : k n → k n was defined to be strongly nilpotent if the matrix product J(F )(a 1 )J(F )(a 2 ) · · · J(F )(a n ) is zero, where the a i are distinct sequences of independent variables of length n (that is, "generic points"). If k is infinite, this is equivalent to the obvious generalization of the first definition to k. The following somewhat more general definition can be used in both these cases.…”

Classification of Quadratic Homogeneous Automorphisms in Dimension Five

“…[7] showed that a polynomial map F = X + H is linearly triangularizable if and only if JH is strongly nilpotent, i.e., JH Y 1 JH Y 2 · · · JH Y n = 0 for vector variables Y 1 Y n . By this result, we may determine whether F is linearly triangularizable in each case by checking that whether JH is strongly nilpotent.…”

The non-Keller mapping with one zero at infinity

“…It is shown that the Jacobian of non-Keller mapping being constant must vanish. It is a new mapping unlike with Keller maps that were studied over the last fifteen years [3][4][5][6].…”