Inspired by the Cayley–Hamilton theorem for linear operators, we introduce the notion of an algebraic polynomial automorphism. In this framework, a polynomial automorphism is called algebraic if it satisfies a nontrivial polynomial operator identity when regarded as an operator acting on the polynomial algebra; otherwise, it is called non-algebraic. This notion provides a new algebraic perspective on polynomial automorphisms, allowing them to be studied using ideas analogous to those appearing in the theory of algebraic linear operators and operator identities.
We investigate the algebraicity of several important classes of polynomial automorphisms. First, we prove that every elementary automorphism is algebraic. More generally, we show that any finite composition of commuting elementary automorphisms is algebraic as well. Moreover, we obtain explicit bounds for the degree of the corresponding minimal polynomial satisfied by these automorphisms. These results indicate that algebraicity naturally arises within a large and fundamental family of polynomial transformations.
On the other hand, algebraicity does not follow from tameness. To demonstrate this phenomenon, we construct a tame automorphism of k[x,y,z] which is non-algebraic. Finally, we study the classical Nagata automorphism together with a family of higher-dimensional Nagata-type automorphisms. We prove that they satisfy a cubic operator identity and hence are algebraic of degree three.