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On the asymptotic structure of the polynomials on minimal diophantic deviation from zero. (English) Zbl 0663.41008

Let \(N\) be a sufficiently large integer and let \(P\) be a polynomial over \(\mathbb Z\) of degree \(N\) for which the maximum of \(| P(x)|\) in \([0,1]\) attains the minimal value in the family of all polynomials over \(\mathbb Z\) of degree \(\leq N\). The author shows that \(P\) must be divisible by certain large powers (exceeding \(N/200\)) of each of the polynomials \(x(1-x)\), \(2x-1\) and \(5x^ 2-5x+1\). This is a consequence of a more general result, ensuring the vanishing of \(P\) at zeros of polynomials satisfying certain rather complicated conditions.

MSC:

41A10 Approximation by polynomials
11C08 Polynomials in number theory
11J99 Diophantine approximation, transcendental number theory
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