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Mesures de polynômes : application au diamètre transfini entier

Abstract : We are interesting in different measures of polynomials ; especially Mahler measure, lenght and Zhang-Zagier measure. We will see that the study of the spectrum of a measure is closely connected with the optimisation of a function depending on the considered measure. This optimisation problem is solved with the technique of auxiliairy functions. In general, the solution is not exact : a semi-infinte linear programming method gives a good numerical approximation. However, we meet some examples of exact solutions called exact auxiliairy functions. We also associate the upper bound of the integer transfinite diameter of an interval which end points are two consecutive elements of a Farey sequence with the lower bound of some measures of polynomials. At last, we present a procedure to find explicitly all polynomials of fixed degree and small measure and we apply it to the Zhang-Zagier measure and to the lenght
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Valérie Flammang. Mesures de polynômes : application au diamètre transfini entier. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1994. Français. ⟨NNT : 1994METZ057S⟩. ⟨tel-01776861⟩



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