## 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
Full Text:

### References:

 [1] Nidal Ali, Stabilité des polynômes, Acta Arith. 119 (2005), no. 1, 53 – 63 (French). · Zbl 1088.11078 [2] Mohamed Ayad and Donald L. McQuillan, Irreducibility of the iterates of a quadratic polynomial over a field, Acta Arith. 93 (2000), no. 1, 87 – 97. · Zbl 0945.11020 [3] Mohamed Ayad and Donald L. McQuillan, Corrections to: ”Irreducibility of the iterates of a quadratic polynomial over a field” [Acta Arith. 93 (2000), no. 1, 87 – 97; MR1760091 (2001c:11031)], Acta Arith. 99 (2001), no. 1, 97. · Zbl 0945.11020 [4] Nigel Boston and Rafe Jones, Arboreal Galois representations, Geom. Dedicata 124 (2007), 27 – 35. · Zbl 1206.11069 [5] John J. Cannon and Derek F. Holt, The transitive permutation groups of degree 32, Experiment. Math. 17 (2008), no. 3, 307 – 314. · Zbl 1175.20004 [6] Lynda Danielson and Burton Fein, On the irreducibility of the iterates of \?$$^{n}$$-\?, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1589 – 1596. · Zbl 1007.12001 [7] Burton Fein and Murray Schacher, Properties of iterates and composites of polynomials, J. London Math. Soc. (2) 54 (1996), no. 3, 489 – 497. · Zbl 0865.12003 [8] Rafe Jones, Iterated Galois towers, their associated martingales, and the \?-adic Mandelbrot set, Compos. Math. 143 (2007), no. 5, 1108 – 1126. · Zbl 1166.11040 [9] Rafe Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 523 – 544. · Zbl 1193.37144 [10] -, An iterative construction of irreducible polynomials reducible modulo every prime, arXiv:1012.2857v1 (2010). To appear in J. Algebra. [11] Alina Ostafe and Igor E. Shparlinski, On the length of critical orbits of stable quadratic polynomials, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2653 – 2656. · Zbl 1268.11155 [12] E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006. Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544]. · Zbl 0471.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.