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.


41A10 Approximation by polynomials
11C08 Polynomials in number theory
11J99 Diophantine approximation, transcendental number theory
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[1] Andria, D. George, Approximation of continuous functions by polynomials with integral coefficients, J. Approx. Theory, 4, 357-362 (1971) · Zbl 0223.41007
[2] Bernardo, Emiliano Aparicio, On Extremal Properties of Polynomials with Integral Coefficients and Approximation Functions and Their Polynomials, (Dissertation (1954), Moscow State University: Moscow State University Moscow), 1-4, [Abstract, Russian]
[3] Bernardo, Emiliano Aparicio, Sobre la aproximación de las funciones mediante polinomios de coeficientes enteros, (Actas de la Octava Reunión Anual de Matemáticos Españoles. Actas de la Octava Reunión Anual de Matemáticos Españoles, Madrid (1967)), 21-33
[4] Bernardo, Emiliano Aparicio, Estructura asintótica de los polinomios con coeficientes enteros de desviación minima a cero en el segmento [0, 1], (Actas de las III Jornadas Matemáticas Hispano-Lusitanas, Sevilla, 1974, Vol. I (1975), Univ. de Sevilla), 1.19-1.32
[5] Bernardo, Emiliano Aparicio, Sobre unos sistemas de números enteros algebraicos de D. S. Gorshkov y sus aplicaciones al cálculo, Rev. Mat. Hisp-Amer., 41, No. 1-No. 2, 3-17 (1981), (4)
[6] Bernardo, Emiliano Aparicio, Trans. Amer. Math. Soc., Issue 4, 7-10 (1986) · Zbl 0657.41007
[7] Ferguson, Le Baron O., Approximation by polynomials with integral coefficients, (Math. Surveys, Vol. 17 (1980), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0441.41003
[8] Gelfond, A. O., On uniform approximation by polynomials with rational integral coefficients, Uspehi Mat. Nauk, 10, No. 1(63), 41-65 (1955), [Russian]
[9] Hewitt, Edwin; Zuckerman, H. S., Approximation by polynomials with integral coefficients, a reformulation of the Stone-Weierstrass theorem, Duke Math. J., 26, 305-324 (1959) · Zbl 0087.05802
[10] Natanson, I. P., (Constructive Function Theory, Vol. I-Vol. II (1964-1965), Ungar: Ungar New York) · Zbl 0133.31101
[11] Sanov, I. N., Functions with integral parameters, deviating the least from zero, Leningrad. Gos. Univ. Uchen. Zap. Ser. Mat. Nauk, 111, 32-46 (1949)
[12] Trigub, R. M., Approximation of functions by polynomials with integral coefficients, Izv. Akad. Nauk SSSR Ser. Mat., 26, 261-280 (1962), [Russian] · Zbl 0145.29202
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