Nilpotent Jacobians in dimension three
It is known that if \(F=x+H:k^{n}\rightarrow k^{n}\) (\(k\) - a field of characteristic zero) is a polynomial Keller's mapping (i.e. \(\text{Jac}~F= \text{const}.\neq 0\)) then the Jacobian matrix \(JH\) of \(H\) is nilpotent. It is known that to solve the Jacobian conjecture it suffices to study the case of homogeneous \(H.\) This, in turn, is related to the homogeneous dependence problem (which is still open): if \(H\) is homogeneous with \(JH\) nilpotent and \(H(0)=0,\) then \(H_{1},\ldots ,H_{n}\) are linearly dependent. In the inhomogeneous case the answer to the above problem is negative due to a counterexample given by the second author. The paper under review studies the polynomial mappings \(H\) with \(JH\) nilpotent in dimension \(n=3.\) The authors completely classify such mappings of the form \(H=(u(x,y),v(x,y),h(u(x,y),v(x,y)))\) with \(JH\) nilpotent for which components are linear independent.
scientific article; zbMATH DE number 5004397