The Jacobian conjecture for symmetric Jacobian matrices.
Let \(F = (F_1,\ldots,F_n) : \mathbb C^n \longrightarrow \mathbb C^n\) be a polynomial map and \(J(F) = (\partial F_i/\partial x_j)\) the Jacobian matrix of \(F\). The Jacobian conjecture asserts that \(F\) is invertible if det\((J(F)) \in \mathbb C^*\). It is known that it suffices to prove the conjecture for all \(n \geq 2\) and all polynomial mappings of the form \(F = x + H, H=(H_1,\ldots,H_n)\) and \(H_i\) is either zero or homogeneous of degree 3. It is proved that for all \(n \leq 4\) the Jacobian conjecture holds for all polynomial mappings \(F = x + H\) where \(H\) is homogeneous of degree \(> 1\) and \(J(F)\) is symmetric. The same assumptions for \(F : \mathbb R^n \longrightarrow \mathbb R^n\) imply \(F = x\).
scientific article; zbMATH DE number 2078854