Irreducible polynomials with several prescribed coefficients

We study the number of monic irreducible polynomials of degree n over F-q having certain preassigned coefficients, where we assume that the constant term (if preassigned) is nonzero. Hansen and Mullen conjectured that for n >= 3, one can always find an irreducible polynomial with any one coefficient preassigned (regardless of the ground field F-q). Their conjecture was established in all but finitely many cases by Wan, and later resolved in full in work of Ham and Mullen. In this note, we present a new, explicit estimate for the number of irreducibles with several preassigned coefficients. One consequence is that for any epsilon > 0, and all large enough n depending on e, one can find a degree n monic irreducible with any left perpendicular(1 - epsilon)root nright perpendicular coefficients preassigned (uniformly in the choice of ground field F-q). For the proof, we adapt work of Katai and Harman on rational primes with preassigned digits. (C) 2013 Elsevier Inc. All rights reserved.