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Stable polynomials over finite fields. (English) Zbl 1319.11089
Let \(f\) be a polynomial of degree at least 2 with coefficients in a field \(\mathbb K\). We call \(f\in{\mathbb K}[X]\) stable if all of its iterates \[ f^{(0)}(X)=X,\quad f^{(n)}(X)=f^{(n-1)}(f(X)),\quad n\geq 1 \] are irreducible over \(\mathbb K\). In the paper under review, the authors are concerned with the case of \({\mathbb K}\) being a finite field \({\mathbb F}_q\) with \(q\) elements, where \(q=p^s\) and \(p\) is an odd prime.
The authors first give a summary on discriminants and resultants, and Weil’s bound for character sums. Then, in Section 3, they give a necessary condition for the stability of arbitrary polynomials over \({\mathbb F}_q\). This partially generalizes the quadratic polynomial case by R. Jones and N. Boston [Proc. Am. Math. Soc. 140, No. 6, 1849–1863 (2012; Zbl 1243.11115)]. In Section 4, they prove the non-existence of certain cubic polynomial for \(p=3\). In the last section, using Weil’s bound for multiplicative character sums, the authors obtain an upper bound \(O(q^{d+1-1/{2\log(2d)}})\) for the number of stable polynomials of degree \(d\).
Reviewer: Ke Gong (Kaifeng)

MSC:
11T06 Polynomials over finite fields
11L40 Estimates on character sums
11T55 Arithmetic theory of polynomial rings over finite fields
11R09 Polynomials (irreducibility, etc.)
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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