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Small polynomials with integer coefficients. (English) Zbl 1091.11009
Let \(E\) be a compact subset of the plane and let \(\| \cdot\| _E\) denote the supremum norm on \(E\). The integer Chebyshev constant \(t_{\mathbb Z}(E)\) is the limit as \(n \to \infty\) of the \(n\)th root of the infimum of \(\| P\| _E\) over all non-zero polynomials of degree at most \(n\) with integer coefficients. The extremal polynomials \(Q_n\) for this problem are called integer Chebyshev polynomials. The interesting case is that of \(\text{ cap}(E) < 1\) since otherwise \(t_{\mathbb Z}(E) = 1\). The author computes the exact value of \(t_{\mathbb Z}(E)\) for a certain class of lemniscates. One of the author’s main results is that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors (over the integers). The case \(E = [0,1]\) has been much studied. The author improves many of the known results in this case, for example showing that \[ Q_n(x) = (x(1-x))^{[\alpha_1 n]} (2x-1)^{[\alpha_2 n]} (5x^2 - 5x + 1)^{[\alpha_3 n]} R_n(x), \] where \(0.31 \leq \alpha_1 \leq 0.34, 0.11 \leq \alpha_2 \leq 0.14\) and \(0.035 \leq \alpha_3 \leq 0.057\). He obtains the following improvement on previous estimates: \(0.4213 < t_{\mathbb Z}([0,1]) < 0.4232\). The techniques used are from weighted potential theory [E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren der Mathematischen Wissenschaften. 316, Berlin: Springer (1997; Zbl 0881.31001)].

MSC:
11C08 Polynomials in number theory
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