The integer Chebyshev constant of Farey intervals (Q1021026)

Let \(I\) be an interval, and let \(N\) be a positive integer. Then \(t_N(I)\) is defined as the minimum of the quantity \(\sum_{x \in I} |P(x)|^{1/\deg P}\), where \(P\) is a nonzero polynomial with integer coefficients such that \(\deg P \leq N\). The constant \(t_{\mathbb{Z}}(I) = \lim_{N \to \infty} t_N(I)\) is called the integer Chebyshev constant of the interval \(I\). For \(|I| \geq 4\), we have \(t_{\mathbb{Z}}(I)=|I|/4\), but no value of this constant is known for \(|I|<4\) In this paper, the authors prove some new bounds for this constant for some intervals \(I = [p/q, r/s]\) satisfying \(qr-ps = 1\). As an application, they prove a new bound for the so called Schur-Siegel-Smyth trace problem: if \(\alpha\) is a totally positive algebraic integer of degree \(d\) which is not a root of some explicitly given five polynomials then \(\text{Trace}(\alpha)> 1.783622 \deg(\alpha)\).