We use the theory of resultants to study the stability, that is, the property of having all iterates irreducible, of an arbitrary polynomial over a finite field . This result partially generalizes the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for , we show that certain polynomials of degree three are not stable. We also use the Weil bound for multiplicative character sums to estimate the number of stable polynomials over a finite field of odd characteristic.
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Domingo Gómez-Pérez, Alejandro P. Nicolás, Alina Ostafe, Daniel Sadornil, Stable polynomials over finite fields. Rev. Mat. Iberoam. 30 (2014), no. 2, pp. 523–535