Polynomial distribution and sequences of irreducible polynomials over finite fields

Let k = GF(q) be the finite field of order q. Let f(1)(x), f(2)(x) is an element of k[x] be monic relatively prime polynomials satisfying n = deg f(1) > deg f(2) greater than or equal to 0 and f(1)(x)/f(2)(x) not equal g(1)(x(p))/g(2)(x(p)) for any g(1)(x), g(2)(x) is an element of 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 lambda of S-n, let pi(lambda)(f(1),f(2), q) be the number of square-free polynomials of the form f(1)(x) - alpha f(2)(x) (alpha is an element of k) with factor pattern lambda (corresponding in the natural way to cycle pattern). We give general and precise bounds for pi(lambda)(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 greater than or equal to 4, there is a (finite or infinite) sequence a(0), a(1),... is an element of k, whose length exceeds 0.5 log q/log log q, such that for each n greater than or equal to 1, the polynomial f(n)(x) = a(0) + a(1) x +...+ a(n)x(n) is an element of 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. (C) 1999 Academic Press.