## Irreducibility of iterates of post-critically finite quadratic polynomials over $$\mathbb{Q}$$.(English)Zbl 1450.11112

The author studies integer polynomials $$f$$ having the following properties: (A) monic and quadratic; (B) the orbit of the critical point under the iteration of $$f$$ is finite; (C) an iterate of $$f$$ is reducible modulo every prime, but all the iterates are irreducible over $$\mathbb{Q}$$.
The main theorem gives a characterization for polynomials $$f$$ having all these properties assuming $$f\ne (x-m^2)^2+m^2-1$$ for $$m=3^2$$, $$99^2$$, $$577^2$$. This result provides infinitely many examples that are not produced with the help of criteria given by R. Jones [J. Algebra 369, 114–128 (2012; Zbl 1302.11086)].
An essential step in the proof consists of showing that the irreducibility of first few iterates of a polynomial satisfying (A) and (B) implies the irreducibility of all the iterates.
The exceptions appear in a reasoning based on Diophantine equations, among which the Pell equation $$X^2-2Y^2=1$$ and the famous $$X^2+1=2Y^4$$. The proof method rests valid for polynomials with rational coefficients of the form $$(x+a)^2-a$$ or $$(x+a)^2-a-1$$. This fact encourages the author to put forward a rigidity conjecture for stability of polynomials with coefficients in a field of characteristic different from $$2$$ that satisfy (A) and (B).

### MSC:

 11R09 Polynomials (irreducibility, etc.) 37P15 Dynamical systems over global ground fields

### Citations:

Zbl 1302.11086; Zbl 0945.11020
Full Text:

### References:

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