2009/01/15 by Medvedev, Alice, Scanlon, Thomas
#03C60 #11C08 #11Gxx #14Gxx #37F10 #Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.0901.2352
We study algebraic dynamical systems (and, more generally, σ-varieties) Φ:\mathbb An\mathbb C → \mathbb An\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) ∈ \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 ⊆ \mathbb A2\mathbb C is an irreducible curve which is invariant under the action of (x,y) ↦ (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 ACFA0, a disintegrated set defined by σ(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 σ, 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 σ.