This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over ℂ such that satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d ≥ 3.
We discuss several additional properties a power linear Keller map may have. The Structural Conjecture of Drużkowski (1983) asserts that certain two such properties are equivalent, but we show that one of them 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 dimensions and small Jacobian ranks.
Let k be an algebraically closed field of characteristic zero and a Drużkowski mapping of degree ≥ 2 with det JF = 1. We classify all such mappings whose Jacobian matrix JF is symmetric. It follows that the Jacobian Conjecture holds for these mappings.
We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the following:...
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