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

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.


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