On the enumeration of irreducible polynomials over GF(q) with prescribed coefficients

We present an efficient deterministic algorithm which outputs exact expressions in terms of n for the number of monic degree n irreducible polynomials over IF, of characteristic p for which the first l < p coefficients are prescribed, provided that n is coprime to p. Each of these counts is l/n(q(n-1)+O(q(n/2))). The main idea behind the algorithm is to associate to an equivalent problem a set of Artin-Schreier curves defined over F-q whose number of F(q)n-rational affine points must be combined. This is accomplished by computing their zeta functions using a p-adic algorithm due to Lauder and Wan. Using the computational algebra system Magma one can, for example, compute the zeta functions of the arising curves for q = 5 and l = 4 very efficiently, and we detail a proof-of-concept demonstration. Due to the failure of Newton's identities in positive characteristic, the l >= p cases are seemingly harder. Nevertheless, we use an analogous algorithm to compute example curves for q = 2 and l <= 7, and for q = 3 and l = 3. Again using Magma, for q = 2 we computed the relevant zeta functions for l = 4 and l = 5, obtaining explicit formulae for these open problems for n odd, as well as for subsets of these problems for all n, while for q = 3 we obtained explicit formulae for l = 3 and n coprime to 3. We also discuss some of the computational chal- lenges and theoretical questions arising from this approach in the general case and propose some natural open problems. (C) 2019 Elsevier Inc. All rights reserved.