Invariant varieties for polynomial dynamical systems

TL;DR

This paper characterizes invariant varieties in polynomial dynamical systems, providing explicit descriptions, and explores their implications in algebraic dynamics, model theory, and conjectures related to rational points and minimal sets.

Contribution

It offers a nearly canonical decomposition of polynomials into clusters, describes skew-invariant varieties explicitly, and connects these to conjectures in algebraic dynamics and model theory.

Findings

Explicit description of (weakly) skew-invariant varieties

Characterization of invariant curves under polynomial maps

Results supporting conjectures on rational points and minimal sets

Abstract

We study algebraic dynamical systems (and, more generally, $\sigma$-varieties) $\Phi:{\mathbb A}^n_{\mathbb C} \to {\mathbb A}^n_{\mathbb C}$ given by coordinatewise univariate polynomials by refining a theorem of Ritt. More precisely, we find a nearly canonical way to write a polynomial as a composition of "clusters". Our main result is an explicit description of the (weakly) skew-invariant varieties. As a special case, we show that if $f(x) \in {\mathbb C}[x]$ is a polynomial of degree at least two which is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and $X \subseteq {\mathbb A}^2_{\mathbb C}$ is an irreducible curve which is invariant under the action of $(x,y) \mapsto (f(x),f(y))$ and projects dominantly in both directions, then $X$ must be the graph of a polynomial which commutes with $f$ under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius. We also show that in models of ACFA$_0$, a disintegrated set defined by $\sigma(x) = f(x)$ for a polynomial $f$ has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$, and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of $f$ is defined over a fixed field of a power of $\sigma$.