Iterated polynomials are dense

TL;DR

The paper addresses the density of iterated polynomial maps in spaces of polynomials by studying the iterated map $P \mapsto P^{\circ r}$ on $\mathbf k[x]$ within ind-topologies. It proves constructively that this power map is dominant in the Euclidean ind-topology for valued fields with a nontrivial absolute value and dominant in the Zariski ind-topology for infinite base fields, with a stronger statement extending to nontrivial words in the free monoid. A central technical tool is the Hasse-derivative framework, culminating in a key interpolation proposition that produces a parametric family $P_{\varepsilon}$ whose $r$-th iterate approximates any given target polynomial as $\varepsilon \to 0$. These results are then extended to endomorphisms of the affine line, linking ind-topological density to polynomial iteration and addressing the finitary (finite-field) case with density-zero asymptotics and open questions on asymptotics and limits.

Abstract

For any infinite field k and any positive integer r, we show constructively that the map sending each polynomial P $\in$ k[x] to its r-th iterate is dominant in various inductive limit topologies on the space of all polynomials.

Theorems & Definitions (23)

• Theorem 1.1: Borel
• Theorem 1.2: HofmannMukherjea
• Definition 1.3
• Theorem
• Remark 1.4
• Definition 2.1: Hasse derivative, see e.g. Goldschmidt
• Proposition 2.2
• Remark 2.3
• Definition 2.4
• Remark 2.5