Eventually stable polynomial

Let ${\displaystyle K}$  be a field and ${\displaystyle f\in K[x]}$  be a non-constant polynomial. The polynomial ${\displaystyle f}$  is called stable or dynamically irreducible if, for every natural number ${\displaystyle n}$ , the ${\displaystyle n}$ -fold composition ${\displaystyle f^{n}=f\circ f\circ \ldots \circ f}$  is irreducible over ${\displaystyle K}$ .

A non-constant polynomial ${\displaystyle g\in K[x]}$  is called ${\displaystyle f}$ -stable if, for every natural number ${\displaystyle n\geq 1}$ , the composition ${\displaystyle g\circ f^{n}}$  is irreducible over ${\displaystyle K}$ .

The polynomial ${\displaystyle f}$  is called eventually stable if there exists a natural number ${\displaystyle N}$  such that ${\displaystyle f^{N}}$  is a product of ${\displaystyle f}$ -stable factors. Equivalently, ${\displaystyle f}$  is eventually stable if there exist natural numbers ${\displaystyle N,r\geq 1}$  such that for every ${\displaystyle n\geq N}$  the polynomial ${\displaystyle f^{n}}$  decomposes in ${\displaystyle K[x]}$  as a product of ${\displaystyle r}$  irreducible factors.

• If ${\displaystyle f=(x-\gamma )^{2}+c\in K[x]}$  is such that ${\displaystyle -c}$  and ${\displaystyle f^{n}(\gamma )}$  are all non-squares in ${\displaystyle K}$  for every ${\displaystyle n\geq 2}$ , then ${\displaystyle f}$  is stable. If ${\displaystyle K}$  is a finite field, the two conditions are equivalent.^[3]

• Let ${\displaystyle f=x^{d}+c\in K[x]}$  where ${\displaystyle K}$  is a field of characteristic not dividing ${\displaystyle d}$ . If there exists a discrete non-archimedean absolute value on ${\displaystyle K}$  such that ${\displaystyle |c|<1}$ , then ${\displaystyle f}$  is eventually stable. In particular, if ${\displaystyle K=\mathbb {Q} }$  and ${\displaystyle c}$  is not the reciprocal of an integer, then ${\displaystyle x^{d}+c\in \mathbb {Q} [x]}$  is eventually stable.^[4]

Let ${\displaystyle K}$  be a field and ${\displaystyle \phi \in K(x)}$  be a rational function of degree at least ${\displaystyle 2}$ . Let ${\displaystyle \alpha \in K}$ . For every natural number ${\displaystyle n\geq 1}$ , let ${\displaystyle \phi ^{n}(x)={\frac {f_{n}(x)}{g_{n}(x)}}}$  for coprime ${\displaystyle f_{n}(x),g_{n}(x)\in K[x]}$ .

We say that the pair ${\displaystyle (\phi ,\alpha )}$  is eventually stable if there exist natural numbers ${\displaystyle N,r}$  such that for every ${\displaystyle n\geq N}$  the polynomial ${\displaystyle f_{n}(x)-\alpha g_{n}(x)}$  decomposes in ${\displaystyle K[x]}$  as a prodcut of ${\displaystyle r}$  irreducible factors. If, in particular, ${\displaystyle r=1}$ , we say that the pair ${\displaystyle (\phi ,\alpha )}$  is stable.

R. Jones and A. Levy proposed the following conjecture in 2017.^[1]

Conjecture: Let ${\displaystyle K}$  be a field and ${\displaystyle \phi \in K(x)}$  be a rational function of degree at least ${\displaystyle 2}$ . Let ${\displaystyle \alpha \in K}$  be a point that is not periodic^{[disambiguation needed]} for ${\displaystyle \phi }$ .

1. If ${\displaystyle K}$  is a number field, then the pair ${\displaystyle (\phi ,\alpha )}$  is eventually stable.
2. If ${\displaystyle K}$  is a function field and ${\displaystyle \phi }$  is not isotrivial, then ${\displaystyle (\phi ,\alpha )}$  is eventually stable.

Several cases of the above conjecture have been proved by Jones and Levy,^[1] Hamblen et al.^[4], and DeMark et al.^[5]