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 ${\displaystyle n}$-fold iteration of the polynomial is eventually constant as a function of ${\displaystyle n}$. The terminology is due to R. Jones and A. Levy ^[1] , who generalized the seminal notion of stability first introduced by R. Odoni ^[2] .

Definition

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.

Examples

• 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] .

Generalization to rational functions and arbitrary basepoints

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 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]

References

1. Jones, Rafe; Levy, Alon (2017). "Eventually stable rational functions". International Journal of Number Theory. 13 (9): 2299--2318.
2. Odoni, R.W.K. (1985). "The Galois theory of iterates and composites of polynomials". Proceedings of the London Mathematical Society. 51 (3): 385--414.
3. Jones, Rafe (2012). "An iterative construction of irreducible polynomials reducible modulo every prime". Journal of Algebra. 369: 114--128.
4. Hamblen, Spencer; Jones, Rafe; Madhu, Kalyani (2015). "The density of primes in orbits of ${\displaystyle z^{d}+c}$". IMRN International Mathematics Research Notices (7): 1924--1958.
5. DeMark, David; Hindes, Wade; Jones, Rafe; Misplon, Moses; Stoll, Michael; Stoneman, Michael (2020). "Eventually stable quadratic polynomials over ${\displaystyle \mathbb {Q} }$". New York Journal of Mathematics. 26: 526--561.