Irreducible polynomials with several prescribed coefficients

We study the number of irreducible polynomials over $\mathbf{F}_{q}$ 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 $r \leq \left[\left(1/4 - ε\right)n \right]$ for any $ε>0$ and $q$ is large; and when $r\leqδn$ for some $δ>0$ and for any $q$. The result is improved from the earlier work of Pollack that the similar result holds for $r\leq\left[(1-ε)\sqrt{n}\right]$.

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