Irreducible polynomials with several prescribed coefficients

We study the number of irreducible polynomials over F-q with some coefficients prescribed. Using the technique developed by Bourgain, we show that there is an irreducible polynomial of degree n with r coefficients prescribed in any location when r <= [(1/4 - epsilon) n] for any epsilon > 0 and q is large; and when r <= delta n for some delta > 0 and for any q. The result improves earlier work of Pollack stating that a similar result holds for r <= [(1 - epsilon) root n. (C) 2016 Elsevier Inc. All rights reserved.