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Settled polynomials over finite fields. (English) Zbl 1243.11115
Proc. Am. Math. Soc. 140, No. 6, 1849-1863 (2012); erratum ibid. 148, No. 2, 913-914 (2020).
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 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.

##### MSC:
 11T06 Polynomials over finite fields 11C08 Polynomials in number theory 37P25 Dynamical systems over finite ground fields 60J99 Markov processes
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