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

Abstract

We present an efficient deterministic algorithm which outputs exact expressions in terms of n for the number of monic degree n irreducible polynomials over Fq of characteristic p for which the first l<p coefficients are prescribed, provided that n is coprime to p. Each of these counts is 1n(qn−l+O(qn/2)). The main idea behind the algorithm is to associate to an equivalent problem a set of Artin–Schreier curves defined over Fq whose number of Fqn-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 challenges and theoretical questions arising from this approach in the general case and propose some natural open problems.