Irreducible polynomials with several prescribed coefficients

We study the number of monic irreducible polynomials of degree n over Fq 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 Fq). 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 ϵ>0, and all large enough n depending on ϵ, one can find a degree n monic irreducible with any ⌊(1−ϵ)n⌋ coefficients preassigned (uniformly in the choice of ground field Fq). For the proof, we adapt work of Kátai and Harman on rational primes with preassigned digits.