Abstract
This article mainly deals with the Nagata polynomial automorphism. In 2004, I. P. Shestakov and U.U. Umirbaev proved that the Nagata automorphism is wild, that is, it is not a superposition of elementary polynomial automorphisms. Then other simplifications of this result were sought. So far, no such proofs have been found. In 2023, R. K. Kerimbayev showed that the Nagata automorphism satisfies an algebraic equation of degree three. In this regard, there was hope that there was another way to prove the Shestakov-Umirbayev theorem. Moreover, elementary polynomial automorphisms were also algebraic of the second degree. And their superposition was algebraic in some cases and not algebraic in others. Since the Nagata automorphism is algebraic of degree three, we were interested in the superposition of elementary automorphisms that is algebraic of degree three. The result obtained is expressed as follows: If a superposition of elementary automorphisms is algebraic of the third degree, then the trace of the Jacobi matrix of this superposition is a constant polynomial. And the trace of the Jacobi matrix of the Nagata automorphism is not a regular polynomial. This result shows that the Nagata automorphism is not a superposition of elementary automorphisms. The article also lists other wild automorphisms besides the Nagata automorphism.