Sur le diamètre transfini entier d’un intervalle à extrémités rationnelles. (On the integer transfinite diameter of intervals with rational end points.). (French) Zbl 0826.41009

Flammang, Valérie

Ann. Inst. Fourier 45, No. 3, 779-793 (1995).

Summary: In this paper we give new upper and lower bounds for the integer transfinite diameter of intervals \(I=[{p\over q},{r\over s}]\) where \(|ps-qr|=1\). We will see how the upper bound of such intervals depends on the lower bound of some measures of monic, totally positive polynomials with integer coefficients. (These measures generalize the usual length.) The lower bounds are obtained by applying a classic resultant’s lemma to a family of totally positive polynomials introduced by C.J. Smyth. These upper and lower bounds improve recent Amoroso’results.

MSC:

41A10	Approximation by polynomials
11J82	Measures of irrationality and of transcendence

Keywords:

integer transfinite diameter; semi-infinite linear programming

References:

[1]	[1] , Sur le diamètre transfini entier d’un intervalle réel, Annales de l’Institut Fourier, 40-4 (1990), 885-911. · Zbl 0713.41004
[2]	[2] , f-Transfinite diameter and number theoretic applications, Annales de l’Institut Fourier, 43-4 (1993), 1179-1198. · Zbl 0790.41007
[3]	[3] , Metodos para el calculo approximado de la desviacion diopantea uniforme minima a cero en un segmento, Revista Matematica Hispano-Americana, 4 Serie, t. XXXVIII, n° 6 (1978), 259-270.
[4]	[4] , New bounds for the uniform Diophantine deviation from zero in [0,1] and [0, 1/4], Proceedings of the sixth conference of Portuguese ad Spanish mathematicians, Part I, Santander (1979), 289-291. · Zbl 0938.11501
[5]	[5] , Number Theoretic Applications of Polynomials with Rational Coefficients Defined by Extremality Conditions, Arithmetic and Geometry, Vol. I, ed. M. Artin and J. Tate, Birkhaüser, Progress in Math., 35 (1983), 61-105. · Zbl 0547.10029
[6]	[6] and , On algebraic equations with integral coefficients whose roots belong to a given point set, Math. Zeit., 63 (1955), 158-172. · Zbl 0066.27002
[7]	[7] , Approximation by polynomials with integral coefficients, Math. Surveys, 17, AMS, Providence, R.I., 1980. · Zbl 0441.41003
[8]	[8] , Sur la longueur des entiers algébriques totalement positifs, J. Number Theory (à paraître). · Zbl 0831.11057
[9]	[9] , On the measure of totally real algebraic numbers I, J. Austral. Math. Soc. (Ser.A), 30 (1980), 137-149. · Zbl 0457.12001
[10]	[10] , On the measure of totally real algebraic numbers II, Math. Comp., 37 (1981), 205-208. · Zbl 0475.12001
[11]	[11] , The mean values of totally real algebraic integers, Math. Comp., 42 (1984), 663-681. · Zbl 0536.12006