On Lehmer's question for integer-valued polynomials (Q6642859)

The Mahler measure of a complex polynomial \( P(x) = a_n \prod_{i=1}^n (x - \alpha_i) \) is defined as\N\[\NM(P) = |a_n| \prod_{i=1}^n \max\{1, |\alpha_i|\}.\N\]\NThe well-known Lehmer's problem asks whether the quantity \( M(P) \), for \( P \in \mathbb{Z}[x] \), can be made arbitrarily close to but larger than 1. The problem is still open, and the smallest known value of the Mahler measure, other than 1, is \( M(P) = 1.176280818\ldots \) for \( P(x) = x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 \).\N\NIn this paper, the author extends Lehmer's problem to integer-valued polynomials in \( \mathbb{Q}[x] \), i.e., \( p(x) \in \mathbb{Q}[x] \) such that \( p(k) \in \mathbb{Z} \) for all \( k \in \mathbb{Z} \). More concretely, the question is asked whether \( M(P) \) can be made arbitrarily close to but larger than 1 when \( P(x) \) is an irreducible integer-valued polynomial.\N\NThe main result of this paper is an affirmative answer to Lehmer's problem for integer-valued polynomials. The author considers the family of integral-valued polynomials given by\N\[\Nf_p(x) = \frac{x^p - x}{p} + x^{(p+1)/2} + 1\N\]\Nfor odd primes \( p \). The first result gives that \( f_p(x) \) is irreducible for primes \( p \equiv 3 \pmod{4} \). It is then shown that\N\[\NM(f_p(x)) \sim \frac{1 + \sqrt{1 + \frac{4}{p^2}}}{2}\N\]\Nas \( p \to \infty \) and \( M(f_p(x)) \) is strictly decreasing. In particular,\N\[\N\lim_{p \to \infty} M(f_p(x)) = 1.\N\]