Irreducibility of iterates of post-critically finite quadratic polynomials over \(\mathbb{Q}\).

*(English)*Zbl 1450.11112The 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).

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).

Reviewer: Mihai Cipu (Bucureşti)

##### References:

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