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

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

Let k be a field of characteristic zero and F : k → k 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 r...

Triangularization properties of power linear maps and the Structural Conjecture

In this paper, we discuss several additional properties a power linear Keller map may have. The Structural Conjecture by Drużkowski in [Dru] asserts that two such properties are equivalent, but we show that one of this properties is stronger than the other. We even show that the property of linear triangularizability is strictly in between. Furthermore, we give some positive results for small d...

Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian Conjecture

Polynomial Automorphisms and the Jacobian Conjecture

In this paper we give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed. At the end of the paper, we discuss the recent counter-examples, in all dimensions greater than two, to the Markus-Yamabe conjecture (Global asymptotic Jacobian conjecture). Résumé Dans ce papier nous pr...

Endomorphisms of Polynomial Rings ; the Jacobian Conjecture

The Jacobian Conjecture is established : If f1, · · · , fn be elements in a polynomial ring k[X1, · · · , Xn] over a field k of characteristic zero such that det(∂fi/∂Xj) is a nonzero constant, then k[f1, · · · , fn] = k[X1, · · · , Xn]. Let k be an algebraically closed field, let k be an affine space of dimension n over k and let f : k −→ k be a morphism of algebraic varieties. Then f is given...