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A new exceptional polynomial for the integer transfinite diameter of \([0,1]\). (English) Zbl 1071.11019

The integer Chebyshev polynomials \(P_k\) on \(I=[0,1]\) are defined as polynomials of degree \(k\) with integral coefficients whose maximal absolute value \(M_k\) in \(I\) is minimal among all non-zero polynomials of degree \(k\) in \(Z[X]\). The integer transfinite diameter \(t_Z(I)\) of \(I\) is defined as the limit of \(M_k^{1/k}\) for \(k\to\infty\). A list of the polynomials \(P_k\) for \(k\leq75\) has been given by L. Habsieger and B. Salvy [Math. Comput. 66, 763–770 (1997; Zbl 0911.11033)], and now the author lists them for \(76\leq k\leq100\). It has been asked by P. Borwein and T. Erdelyi [Math. Comput. 65, 661–681 (1996; Zbl 0859.11044)] whether all zeros of the \(P_k\)’s lie in \([0,1]\). A negative answer with \(k=70\) was provided in the paper of Habsieger and Salvy quoted above, and now the author shows that \(P_{95}\) also has non-real zeros. Finally the bound \(t_Z(I)<0.423164171\) is established, improving slightly the previous bound \(t_Z(I)<0.4232\), due to I. E. Pritsker (to appear).

MSC:

11C08 Polynomials in number theory
41A10 Approximation by polynomials

References:

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