Dynamical Cancellation of Polynomials

TL;DR

This paper establishes a precise criterion for when a finite set of polynomials over a number field exhibits a dynamical cancellation property after a finite number of iterations, with conditions verifiable through finite computations.

Contribution

It provides a necessary and sufficient condition for polynomial cancellation in dynamical systems over number fields, extending prior work and enabling finite computational verification.

Findings

Characterization of cancellation conditions for polynomial sets

Finite computability of the cancellation criteria

Extension of previous dynamical cancellation results

Abstract

Extending the work of Bell, Matsuzawa and Satriano, we consider a finite set of polynomials $S$ over a number field $K$ and give a necessary and sufficient condition for the existence of a $N \in \mathbb{N}_{> 0}$ and a finite set $Z \subset \mathbb{P}^1_K \times \mathbb{P}^1_K$ such that for any $(a,b) \in (\mathbb{P}^1_K \times \mathbb{P}^1_K) \setminus Z$ we have the cancellation result: if $k>N$ and $\phi_1,\ldots ,\phi_k$ are maps in $S$ such that $\phi_{k} \circ \dots \circ \phi_1 (a) = \phi_k \circ \dots \circ \phi_1(b)$, then in fact $\phi_N \circ \dots \circ \phi_1(a) = \phi_N \circ \dots \circ \phi_1(b)$. Moreover, the conditions we give for this cancellation result to hold can be checked by a finite number of computations from the given set of polynomials.