Locally finite polynomial endomorphisms
The paper is a study of a special class of polynomial endomorphisms of \(\mathbb{C}^N\) that are called ``Locally Finite'' (LF) and have nice dynamical properties. It is proven that such an endomorphism \(F\) may be equivalently defined in three ways: (1) there is a polynomial in one variable \(p(T)\) such that \(p(F)=0,\) where \(F^n\) is the \(n\)th iteration of \(F;\) (2) \(\sup_{n\geq 0} \deg F^n<\infty;\) (3) \(\dim \text{Span}_{n\geq 0} \;r\circ F^n<\infty\) for each \(r\in\mathbb{C}[x_1,\dots,x_N].\) The special type of the vanishing polynomial \(p(T)\) is found. Then the unipotent and semisimple LF polynomial automorphisms are defined and it is shown that any LF polynomial automorphism is a composition of uniquely defined commuting LF polynomial automorphisms \(F_s\) and \(F_u,\) where \(F_s\) is semisimple and \(F_u\) is unipotent. The authors prove also that the exponential defines a bijective map from the set of locally nilpotent derivations on the ring \(\mathbb{C}[x_1,\dots,x_N]\) to the set of unipotent polynomial automorphisms of \(\mathbb{C}^N.\) For \(N=2\) the vanishing polynomial for a LF polynomial automorphism \(F\) of degree \(d\) is found explicitly and the degree of minimal vanishing polynomial is shown to be less than \(d+1.\)
scientific article; zbMATH DE number 5186639
scientific article