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

Aparicio Bernardo, Emiliano

J. Approximation Theory 55, No. 3, 270-278 (1988).

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.

Reviewer: Władysław Narkiewicz (Wrocław)

MSC:

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

Keywords:

approximation by integral polynomials; diophantic deviation

References:

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