On irreducible polynomials of small height over finite fields (Q1925009)

For a prime \(p\), let \(\mathbb{F}_p\) be the finite field of \(p\) elements and \(P_n(p)\) be the set of all monic irreducible polynomials of degree \(n\) over \(\mathbb{F}_p\). Also, let \(h_n(p)= \min_{f\in P_n(p)}H(f)\), where \(H(f)=\) greatest absolute value of the coefficients of \(f\) (here, for \(p\geq 3\), \(\mathbb{F}_p= \{0,\pm1,\dots, \pm\frac{p-1}{2}\}\)). The author proves the following result. For fixed \(n\geq 2\) and an arbitrary function \(q(x)\to\infty\), the bound \(h_n(p)\leq f(p)\) holds for all but \(o(H/\log X)\) primes \(p\) in the interval \([X-H,X]\), provided \(H\geq X^{7/12+ \varepsilon}\). The proof is based on some arguments which were used in proving Artin's conjecture on average.