CECM Home > Events > CECM Annual Summer Meeting 2005 > Program > Abstract: Kevin Hare
Abstract
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.