an:06038819
Zbl 1243.11115
Jones, Rafe; Boston, Nigel
Settled polynomials over finite fields
EN
Proc. Am. Math. Soc. 140, No. 6, 1849-1863 (2012); erratum ibid. 148, No. 2, 913-914 (2020).
00299998
2012
j
11T06 11C08 37P25 60J99
irreducible polynomials; polynomial iterates; settled polynomials
If \(K\) be a field and \(f,g\in K[X]\), then \(g\) is \(f\)-stable, if the composition \(g\circ f^n\) (\(f^n\) denoting the \(n\)-th iterate of \(f\)) is irreducible over \(K\). Moreover let \(s_n\) be the sum of degrees of \(f\)-stable polynomials dividing \(f^n\) (according to their multiplicity as factors of \(f^n)\). The polynomial \(f\) is called \textit{settled} if the ratio \(s_n/\deg f^n\) tends to \(1\). It is conjectured that if \(K\) is a finite field of odd characteristic and \(f=aX^2+bX+c\), (\(a\neq0, f\neq X^2\)), then \(f\) is settled. The authors show that such \(f\) is stable if and only if the adjusted critical orbit of \(f\), i.e. the set \(\{-f(\gamma),f^2(\gamma),f^3(\gamma),\dots\}\) with \(\gamma=-b/(2a)\), contains no squares. They show also that if \(f\) is quadratic with all iterates separable, then factorizations of the sequence of iterates of \(f\) can be described by an irreducible absorbing Markov process. A conjecture which makes this description precise is presented (Conjecture 3.6) and computational evidence of it is given.
W??adys??aw Narkiewicz (Wroc??aw)