An estimate on the number of stable quadratic polynomials (Q609358)

Let \(\mathbb F_q\) be a finite field of odd characteristic. A polynomial \(f(x)\in\mathbb F_q[x]\) is called stable if for all \(n \geq 0\) the polynomials \(f^{(n)}(x)\) defined by \[ f^{(0)}(x) = x\quad\text{and}\quad f^{(n)}(x) = f^{(n-1)}(f(x)) \] are irreducible over \(\mathbb F_q\). The authors estimate the number \(S_q\) of quadratic polynomials in \(\mathbb F_q[x]\), \(f(x) = ax^2+bx+c, b,c\in\mathbb F_q,a\in\mathbb F_q^*\), which are stable. As main results the authors show that \(S_q \geq (q-1)^2/4\) and \(S_q = O(q^{5/2}\log q)\). This also improves the bound conjectured by \textit{A. Ostafe} and \textit{I. Shparlinski} [Proc. Am. Math. Soc. 138, No. 8, 2653--2656 (2010; Zbl 1268.11155)].