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The Monic Integer Chebyshev Problem

Kevin Hare · Department of Pure Mathematics, University of Waterloo

We study the problem of finding monic integer polynomials $P$ with small supremum norm $||P||_I$ on an interval $I$. The monic integer transfinite diameter $t_{\mathrm{M}}(I)$ is defined as \[ t_{\mathrm{M}}(I) := {\inf_P ||P||_I^{1/\deg(P)}, \] where the infimum is taken over all non-constant monic polynomials with integer coefficients. We show that if $I$ has length $1$ then $t_{\mathrm{M}}(I) = 1/2$.

We also consider the problem of determining $t_{\mathrm{M}}(I)$ when $I$ is a Farey interval. We give some partial results, which support a conjecture of Borwein, Pinner and Pritsker concerning this value.