On the large sieve method in GF (q, x)

The arithmetic of consecutive polynomial sequences over finite fields

“…For a non-zero polynomial f ∈ F q [X], we denote by rad (f ) the product of all distinct monic irreducible factors of f . To obtain an upper bound for the number of consecutive irreducible sequences of fixed length, we need the following result due to Johnsen [16,Corollary 2] on the number of irreducible polynomials over F q in an arithmetic progression.…”

Sieving by Almost-Primes

“…When the elements of 2 are not necessarily pairwise coprime the situation becomes more complicated. Here Johnsen [6] (see also [3] for a simpler account) has shown how to extend (1.1) to the case when 2 consists of a set of primes and their powers; but, as far as we know, the general situation has not been considered up to now. We shall prove a result (Theorem 1) which yields, under some fairly natural conditions ((1.3) and (1.4) below) on 2 and on the associated residue classes, a non-trivial upper estimate of Z.…”

The values of a polynomial over a finite field

“…where, since n ^ q, b{q, n) is given by (6). In general, we therefore expect v(n, t) to be approximately b(q, ri)q.…”

“…The object of this paper is to derive, using a version of the large sieve for function fields due to J. Johnsen [6], explicit lower bounds for the average number of distinct values taken by a polynomial over a finite field.…”

“…We now cite a particular case of the large sieve inequality contained in the Corollary to Theorem 5 of [6]. Let Sf be a set consisting of Z distinct polynomials of degree ^ N in A: [JC], so that ZSq N+l -Let H' be a set of monic square-free polynomials of degree not exceeding X = [\(N+1)] with the property that, to every monic irreducible P dividing a member of W, there exists a set of w(P) ( > 0) residue classes (mod P) such that all members of Sf belong to one of these residue classes (mod P).…”