Classification of Quadratic Homogeneous Automorphisms in Dimension Five

Abstract

In this paper, we present a classification of quadratic homogeneous automorphisms in dimension 5 up to linear conjugation. As a consequence, we give an affirmative answer to the Rusek Conjecture in dimension 5.

The classification of some polynomial maps with nilpotent Jacobians

“…For quadratic homogeneous H, the Dependence Problem has an affirmative answer in the case n ≤ 5, and several authors contributed to that result. See [3, Appendix A] and [21] for the case n = 5. Recently, the second author generalized the condition n ≤ 5 to rank JH ≤ 4 ( [7]).…”

Polarization algebras and their relations

“…It is proved that every finitely generated commutative algebra with the Engel identity x(x(xy)) = 0 is solvable [7] and every finite dimensional commutative algebra with the same identity is nilpotent [15]. The classification of homogeneous quadratic automorphisms in dimension 5 given in [22] can be considered as a classification of 5 dimensional commutative Engel algebras since all quadratic Keller maps are automorphisms [25].…”

Some polynomial maps with Jacobian rank two or three

“…For quadratic homogeneous H, the Dependence Problem has an affirmative answer in the case n ≤ 5, and several authors contributed to that result. See [3, Appendix A] and [16] for the case n = 5.…”