Irreducible Polynomials of given Forms

We survey under a unified approach on the number of irreducible polynomials of given forms: xn + g(x) where the coefficient vector of g comes from an affine algebraic variety over Fq . For instance, all but 2 log n coefficients of g(x) are prefixed. The known results are mostly for large q and little is know when q is small or fixed. We present computer experiments on several classes of polynomials over F2 and compare our data with the results that hold for large q. We also mention some related applications and problems of (irreducible) polynomials with special forms. 1. The general problem and known results Let Fq denote a finite field with q elements and V an affine algebraic variety over Fq, say defined by r polynomials f1, . . . , fr ∈ Fq[x1, · · · , xn]. Let Vq be the Fq-rational points in V , i.e. Vq = {(t1, · · · , tn) ∈ F n q : fi(t1, · · · , tn) = 0, 1 ≤ i ≤ r}. (1) We define In(Vq) to be the number of points (t1, · · · , tn) ∈ Vq such that F (x) = x + t1x n−1 + · · ·+ tn−1x+ tn (2) is irreducible in Fq[x]. We also denote by P (Vq) the set of all polynomials in (2) with (t1, · · · , tn) ∈ Vq. For example, when the polynomials f1, . . . , fr are linear in x1, . . . , xn, V is a coset of a linear subspace. If the linear subspace has dimensionm then V is called a linear variety of dimensionm. For a linear variety V of dimension m, P (Vq) can be rewritten as P (Vq) = {x n + g0(x) + a1g1(x) + · · ·+ amgm(x) : (a1, . . . , am) ∈ F m q }, (3) where gi ∈ Fq[x] has degree at most n−1 for 0 ≤ i ≤ m, and g1, . . . , gm are linearly independent over Fq. Problem 1.1. Let V be an affine variety over Fq. Determine In(Vq). When V is a linear variety, we require that not all the constants in g0(x), g1(x), . . . , gm(x) are zero and that the polynomials x n + g0(x), g1(x), . . . , gm(x) are relatively prime; otherwise In(Vq) = 0 trivially. When Vq = F n q , In(Vq) is just the number of monic irreducible polynomials of degree n in Fq[x] and there is a well–known formula for it. Generally one would not expect to find an explicit formula for In(Vq). In practice, it often suffices to have a good lower bound or an asymptotic formula for it. We are most interested in the asymptotic behaviour of In(Vq). We note that counting the number of points in Vq Date: June 16, 1998 (revised). 1991 Mathematics Subject Classification. Primary 11T55; Secondary 12Y05.