Abstract: The study of polynomials with integer coefficients and small norms
is an old problem originated in the work of Hilbert, Polya, Schur, Fekete, and others. We shall discuss the history of this problem and some applications to distribution of primes and approximation by integer polynomials. Integer Chebyshev problem remains open for the supremum norm on intervals of the real line, but we shall present its solution for some special lemniscates defined by irreducible polynomials.
Moreover, we generalize these results to other heights of polynomials that include an analogue of the Mahler measure (or Weil height).
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