The integer Chebyshev problem Academic Article

We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at mostnnwith 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][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,][0, delta ]and for small degreesdd,xdx^{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][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.