Stable polynomials over finite fields

Abstract

Abstract. We use the theory of resultants to study the stability of an arbitrary polynomial f over a finite field F q , that is, the property of having all its iterates irreducible. This result partially generalises 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 arbitrary polynomials over finite fields of odd characteristic.

On irreducible divisors of iterated polynomials

“…We also note that in [8] an upper bound is given on the number of stable polynomials of degree d 2 over F q .…”

The arithmetic of consecutive polynomial sequences over finite fields

“…For proving our results, we treat the discriminant of a general polynomial f as a multivariate polynomial in the coefficients of f and study for which substitutions of the variables the discriminant is a square. This technical result has been given in [8,Lemma 5.2], which in fact implies an explicit result. Here, we reproduce the proof briefly.…”

“…, Y d ) depends on some variable Y i . The result in [8,Lemma 5.2] comes from the sum of three upper bounds…”

“…In the following, we want to improve the upper bound in Theorem 5.1 when q is much larger than N . To give such an improvement on bounding I N , we use the same technique as in [8,Theorem 5.5]. For this we need the following lemma.…”

A note on the stability of trinomials over finite fields

“…We also noted in the introduction that Odoni [11] studied the stability of additive polynomial f (x) = x p − x − 1 over F p . His method is different from the methods of the current paper and those of [1,7]. Using Capelli's lemma he showed that the Galois group of f f (x) over F p is the cyclic group of order p and hence f f (x) is not stable.…”