Locally finite polynomial endomorphisms

Abstract

We study polynomial endomorphisms F of C N which are locally finite in the following sense: the vector space generated byWe show that such endomorphisms exhibit similar features to linear endomorphisms: they satisfy the Jacobian Conjecture, have vanishing polynomials, admit suitably defined minimal and characteristic polynomials, and the invertible ones admit a Dunford decomposition into "semisimple" and "unipotent" constituents. We also explain a relationship with linear recurrent sequences and derivations. Finally, we give particular attention to the special cases where F is nilpotent and where N = 2.

Quasi-locally finite polynomial endomorphisms

“…It is clear that (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) in the following assertions (see [7] for (i) =⇒ (ii) and prop. 1.5 for (iii) =⇒ (iv)):…”

“…The Nagata automorphism (x − 2y(xz + y 2 ) − z(xz + y 2 ) 2 , y + z(xz + y 2 ), z) (see [10]) is LF (see [7]) but not triangularisable (see [2]). …”

“…In the case where F is an automorphism, this is equivalent to saying that its topological entropy h(F ) is zero (see [13]). A first subclass of dynamically trivial polynomial endomorphisms was introduced in [7]. It is the set of polynomial endomorphisms F which are locally finite (LF for short) in the following sense: the complex vector space generated by the r • F n , n ≥ 0, is finite dimensional for each r ∈ C[X].…”

Untitled

“…If there is an univariate poly- its monic generator will be called the minimal polynomial for F and denoted by µ F . The paper [1] gives many equivalent conditions for F to be locally finite and presents a formula for a polynomial p(T ) such that p(F ) = 0 provided F (0) = 0 (see [1], Th. 1.2).…”

Remarks on a Normal Subgroup ofGAn

“…The set of all locally finite polynomial automorphisms of k n will be denoted LF n . It is not hard to see that the following conditions on G are equivalent (see [3], Th. 1.1):…”