A new exceptional polynomial for the integer transfinite diameter of \([0,1]\)
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 \textit{L. Habsieger} and \textit{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 \textit{P. Borwein} and \textit{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).
scientific article; zbMATH DE number 2184628