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,[1] who generalized the seminal notion of stability first introduced by R.
be a non-constant polynomial.
is called stable or dynamically irreducible if, for every natural number
is irreducible over
A non-constant polynomial
-stable if, for every natural number
, the composition
is irreducible over
is called eventually stable if there exists a natural number
Equivalently,
is eventually stable if there exist natural numbers
decomposes in
irreducible factors.
ϕ ∈
be a rational function of degree at least
α ∈
For every natural number
ϕ
( ϕ , α )
is eventually stable if there exist natural numbers
( x ) − α
decomposes in
irreducible factors.
( ϕ , α )
R. Jones and A.
Levy proposed the following conjecture in 2017.
[1] Several cases of the above conjecture have been proved by Jones and Levy,[1] Hamblen et al.[4], and DeMark et al.[5]