# The Integer Chebyshev Problem

Borwein, Peter and Erdelyi, Tamas (1994) The Integer Chebyshev Problem. [Preprint]

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We are concerned with the problem of minimizing the supremum norm on an interval of a non-zero polynomial of degree at most $n$ with integer coefficients. This is an old and hard problem that cannot be exactly solved in any non-trivial cases. We examine the case of the interval [0,1] in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal integer Chebyshev'' polynomials, showing for example, that on small intervals $[0, \delta]$ and for small degrees $d$, $x^d$ achieves the minimal norm. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.