Dynamical Cancellation of Polynomials

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 $Z\subset {\mathbb{P}}_K^1\times {\mathbb{P}}_K^1$ such that for any $(a,b)\in ({\mathbb{P}}_K^1\times {\mathbb{P}}_K^1)\setminus Z$ we have the cancellation result: if $k>N$ and ${\phi }_1,\dots ,{\phi }_k$ are maps in $S$ such that ${\phi }_k\circ \cdots \circ {\phi }_1(a)={\phi }_k\circ \cdots \circ {\phi }_1(b)$, then in fact ${\phi }_N\circ \cdots \circ {\phi }_1(a)={\phi }_N\circ \cdots \circ {\phi }_1(b)$. 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.