Power Linear Keller Maps

The famous generalized Jacobian Conjecture, which was first formulated by O. H. Keller in 1939, the involved subjects are geometry, algebra, analytics, analysis-situs and so on. Many associated conjectures and problems have been formulated, it have greatly promoted the development of mathematics. Although the Jacobian Conjecture does not be solved now, the proceeding of the research on polynomial mappings have been made great walks, especially the works on the power Keller maps, such as the properties of invertible, tame or wild, linear triangularizable. One of the remarkable results is the work on the Nagata conjecture. Recently on the work of linear triangularizable of power Keller maps, Prof. Arno van den Essen had obtain some nice results in the case of n ≤ 3.The main work on this paper is about the linear triangularizable problems of power linear Keller maps, we begin our work from a new ideal, that is the relationship between the structure of matrice A and corank A.Firstly, we show that in the case of corank A ≤ 2, the power linear Keller maps F is linear triangularizable, so it is also tame and the Jacobian Conjecture is true at the same time.Second, in the cases of corank A = 3,4,5, undering some condictions we will obtain the property that a_i = qa_s for some different rows of the matrice A. By such property, we point out the structure of the matrice A, and the property of linear triangularizable of the correspending mapping F. Furthermore, the relationship and the differences have been also pointed out, base on such analyse we do some guess on the case of corank A ≥ 6.After this, we extend our work to the case of non-homogeneous power linear Keller maps, and obtain some similar results while corank A ≤ 5.Finally, we do some thinking about the Jacobian Conjecture, the linear triangularizable, nilpotent and strong nilpotent, which may do some good help on the...