Revista Matemática Iberoamericana

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Volume 30, Issue 2, 2014, pp. 523–535
DOI: 10.4171/RMI/791

Published online: 2014-07-08

Stable polynomials over finite fields

Domingo Gómez-Pérez[1], Alejandro P. Nicolás[2], Alina Ostafe[3] and Daniel Sadornil[4]

(1) Universidad de Cantabria, Santander, Spain
(2) Universidad de Valladolid, Spain
(3) University of New South Wales, Sydney, Australia
(4) Universidad de Cantabria, Santander, Spain

We use the theory of resultants to study the stability, that is, the property of having all iterates irreducible, of an arbitrary polynomial $f$ over a finite field $\mathbb{F}_q$. This result partially generalizes the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for $p=3$, 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.

Keywords: Finite fields, irreducible polynomial, iterations of polynomials, discriminant

Gómez-Pérez Domingo, Nicolás Alejandro, Ostafe Alina, Sadornil Daniel: Stable polynomials over finite fields. Rev. Mat. Iberoam. 30 (2014), 523-535. doi: 10.4171/RMI/791