On shifted Eisenstein polynomials (Q2517642)

A polynomial \(f\in Z[X]\) is called a shifted Eisenstein polynomial, if for some integral \(s\) the polynomial \(f(X+s)\) is an Eisenstein polynomial with respect to some prime. First, the authors show (Th. 3.3) that if \(\deg f=n\) and \(f(x+s)\) is not Eisenstein for \(0\leq s\leq n^{n/(n-1)}L(f)^2\), where \(L(f)\) is the sum of absolute values of coefficients of \(f\), then \(f\) is not a shifted Eisenstein polynomial. If \(E_n(x),\bar E_n(x)\) denotes the number of Eisenstein, resp. shifted Eisenstein polynomials of degree \(n\) having height \(\leq x\), then one has (Th. 4.5) \[ \liminf_{x\to\infty}{\bar E_n(x)\over E_n(x)}\geq 1 +\gamma_n \] with explicitly given \(\gamma_n>0\). The authors conjecture (Question 4.6) that \[ \bar E_n(x) = (2^{n+1}+o(1))x^{n+1}, \] and their last theorem (Th.5.1) shows that there are infinitely many irreducible polynomials of any degree which are not shifted Eisenstein polynomials.