Eventually stable polynomial

A non-constant polynomial with coefficients in a field is said to be eventually stable if the number of irreducible factors of the -fold iteration of the polynomial is eventually constant as a function of . The terminology is due to R. Jones and A. Levy, who generalized the seminal notion of stability first introduced by R. Odoni.

Definition

Let be a field and be a non-constant polynomial. The polynomial is called stable or dynamically irreducible if, for every natural number , the -fold composition is irreducible over .

A non-constant polynomial is called -stable if, for every natural number , the composition is irreducible over .

The polynomial is called eventually stable if there exists a natural number such that is a product of -stable factors. Equivalently, is eventually stable if there exist natural numbers such that for every the polynomial decomposes in as a product of irreducible factors.

Examples

• If is such that and are all non-squares in for every , then is stable. If is a finite field, the two conditions are equivalent.
• Let where is a field of characteristic not dividing . If there exists a discrete non-archimedean absolute value on such that , then is eventually stable. In particular, if and is not the reciprocal of an integer, then is eventually stable.

Generalization to rational functions and arbitrary basepoints

Let be a field and be a rational function of degree at least . Let . For every natural number , let for coprime .

We say that the pair is eventually stable if there exist natural numbers such that for every the polynomial decomposes in as a product of irreducible factors. If, in particular, , we say that the pair is stable.

R. Jones and A. Levy proposed the following conjecture in 2017.

Conjecture: Let be a field and be a rational function of degree at least . Let be a point that is not periodic for .

1. If is a number field, then the pair is eventually stable.
2. If is a function field and is not isotrivial, then is eventually stable.

Several cases of the above conjecture have been proved by Jones and Levy, Hamblen et al., and DeMark et al.

References