Abstract: 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 Fq. 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.

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