The integer Chebyshev problem

Borwein, Peter; Erdélyi, Tamás

doi:10.1090/S0025-5718-96-00702-8

The integer Chebyshev problem
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by Peter Borwein and Tamás Erdélyi PDF

Math. Comp. 65 (1996), 661-681 Request permission

Abstract:

We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most $n$ with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial 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 intevals $[0, \delta ]$ and for small degrees $d$, $x^{d}$ achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys and others, as to what the “integer transfinite diameter” of $[0,1]$ should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.

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