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 el ement 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). Le t 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 t he number of square-free polynomials of the form f(1)(x) - alpha f(2)(x) (a lpha is an element of k) with factor pattern lambda (corresponding in the n atural way to cycle pattern). We give general and precise bounds for pi(lam bda)(f(1), f(2), q), thus providing an explicit version of the estimates fo r the distribution of polynomials with prescribed factorisation established by S. D. Cohen in 1970. For an application of this result, we show that, i f 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 polynom ial of degree n. This resolves in one direction a problem of Mullen and Shp arlinski that is an analogue of an unanswered number-theoretical question o f A. van der Poorten. (C) 1999 Academic Press.