Chebyshev polynomials with integer coefficients (Q2708944)

Let \({\mathcal P}_n(\mathbb Z)\) denote the set of polynomials of degree at most \(n\) with integer coefficients. Given an interval \([a,b] \subset \mathbb R\), an integer Chebyshev polynomial \(q_n \in {\mathcal P}_n(\mathbb Z) \neq 0\) is one that minimizes the uniform norm \(\|q_n\|_{[a,b]}\) over non-zero polynomials in \({\mathcal P}_n(\mathbb Z)\). The integer Chebyshev constant is defined by \(\text{inch}([a,b]) = \lim_{n\to \infty}\|q_n\|_{[a,b]}^{1/n}\). Let \(Q_n\) denote the integer Chebyshev polynomials for \([0,1]\). \textit{E. Aparicio Bernardo} [J. Approximation Theory 55, No. 3, 270-278 (1988; Zbl 0663.41008)] showed that \(Q_n\) has certain factors that occur with high multiplicity: NEWLINE\[NEWLINEQ_n(x) = (x(1-x))^{[\alpha_1 n]}(2x-1)^{[\alpha_2 n]} (5x^2 - 5x + 1)^{[\alpha_3 n]}R_n(x),NEWLINE\]NEWLINE where \(\alpha_1 \geq 0.1456\), \(\alpha_2 \geq 0.0166\) and \(\alpha_3 \geq 0.0037\). These lower bounds were improved by \textit{V. Flammang, G. Rhin} and \textit{C. J. Smyth} [J. Théor. Nombres Bordx. 9, No. 1, 137-168 (1997; Zbl 0892.11033)] and further factors were shown to occur. The author's main result concerns \(\alpha_1\) and \(\alpha_2\). He shows that the pair \((\alpha_1,\alpha_2)\) must lie in a certain explicit region in the plane. As a consequence \(.2961 \leq \alpha_1 \leq 0.3634\) and \(.0952 \leq \alpha_2 \leq 0.1767\), signifigantly improving the known lower bounds and giving the first non-trivial upper bounds on \(\alpha_1\) and \(\alpha_2\). The interesting proof uses the recently developed weighted potential theory [see \textit{E. B. Saff} and \textit{V. Totik}, Logarithmic potentials with external fields (1977; Zbl 0881.31001)]. As the author observes, this powerful method seems likely to be able to give more information about further factors of \(Q_n\).NEWLINENEWLINEFor the entire collection see [Zbl 0947.00027].