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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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##### References:
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