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Abstract : |
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,]andforsmall 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. 1., |
