On a Conjecture on Irreducible Polynomials over Finite Fields with Restricted Coefficients

Let q be a prime power, F-q be the finite field of order q and let n, d be positive integers. Munemasa and Nakamura conjectured at WAIFI 2016 that there exist f is an element of F-q[x] of degree n and alpha is an element of F-qd not lying in any proper subfield such that f - alpha is irreducible in F-qd[x]. In this paper, we prove that the conjecture holds true for every triple (q, n, d) such that d is larger than a constant that depends only on n. As a subproduct of our proofs we deduce that if F is an element of F-q[x] is a polynomial such that F - t(0) has a certain special factorization pattern for some t(0) is an element of F-q, then the statistics of all the factorization patterns of F - t(1), where t1 ranges in F-qd, are entirely determined up to an explicit error term independent of the size of the base field. At the end of the paper we provide some experimental results to show how sharp our statistics are.