The distance to an irreducible polynomial. II (Q2894521)

A classical question of Turán asks for the existence of an absolute constant \(C\) such that for every polynomial \(f\) having integer coefficients there exists an irreducible polynomial \(g\) having integer coefficients, for which \(\deg(g)\leq\deg(f)\) and \(L(f-g)\leq C\). Here \(L(h)\) stands for the sum of the absolute values of the coefficients of a polynomial \(h\). The authors show that \(C=5\) suffices for all \(f\) as above with \(\deg(f)\leq 40\). This is a considerable extension of earlier related results from the literature. Their strategy is to investigate analogous questions in \(\mathbb{F}_p[x]\) for small primes \(p\). Further, it is also proved that a positive proportion of the polynomials in \(\mathbb{F}_2[x]\) have distance at least \(4\) to all irreducible polynomials.NEWLINENEWLINEFor Part I, see the second author, Contemp. Math. 517, 275--288 (2010; Zbl 1227.11049).