The integer Chebyshev problem Academic Article uri icon

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 δ] and for small degrees d, xd 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.

author list (cited authors)

  • Borwein, P., & Erdélyi, T.

citation count

  • 16

publication date

  • April 1996