Polynomial Distribution and Sequences of Irreducible Polynomials over Finite Fields

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Let k = GF ( q ) be the finite field of order q . Let f 1 ( x ), f 2 ( x )∈ k [ x ] be monic relatively prime polynomials satisfying n =deg f 1 >deg f 2 ⩾0 and f 1 ( x )/ f 2 ( x )≠ g 1 ( x p )/ g 2 ( x p ) for any g 1 ( x ), g 2 ( x )∈ k [ x ]. Write Q ( x )= f 1 ( x )+ tf 2 ( x ) and let K be the splitting field of Q ( x ) over k ( t ). Let G be the Galois group of K over k ( t ). G can be regarded as a subgroup of S n . For any cycle pattern λ of S n , let π λ ( f 1 , f 2 , q ) be the number of square-free polynomials of the form f 1 ( x )− αf 2 ( x ) ( α ∈ k ) with factor pattern λ (corresponding in the natural way to cycle pattern). We give general and precise bounds for π λ ( f 1 , f 2 , q ), thus providing an explicit version of the estimates for the distribution of polynomials with prescribed factorisation established by S. D. Cohen in 1970. For an application of this result, we show that, if q ⩾4, there is a (finite or infinite) sequence a 0 , a 1 , …∈ k , whose length exceeds 0.5 log q /log log q , such that for each n ⩾1, the polynomial f n ( x )= a 0 + a 1 x +…+ a n x n ∈ k [ x ] is an irreducible polynomial of degree n . This resolves in one direction a problem of Mullen and Shparlinski that is an analogue of an unanswered number-theoretical question of A. van der Poorten.