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Dans cet article, nous donnons une minoration de la mesure de Mahler d'un polynôme à coefficients entiers, dont toutes les racines sont d'une part réelles positives, d'autre part réelles, en fonction de la valeur de ce polynôme en zéro. Ces minorations améliorent des résultats antérieurs de A. Schinzel. Par ailleurs, nous en déduisons des inégalités de M.-J. Bertin, liant la mesure d'un nombre algébrique à sa norme.

Let α be a totally positive algebraic integer of degree d, i.e., all of its conjugates $\alpha ₁=\alpha ,...,{\alpha }_d$ are positive real numbers. We study the set ₂ of the quantities ${\left({\prod }_{i=1}^d{(1+\alpha {²}_i)}^{1/2}\right)}^{1/d}$. We first show that √2 is the smallest point of ₂. Then, we prove that there exists a number l such that ₂ is dense in (l,∞). Finally, using the method of auxiliary functions, we find the six smallest points of ₂ in (√2,l). The polynomials involved in the auxiliary function are found by a recursive algorithm.

In this paper we build on some recent work of Amoroso, and Borwein and Erdélyi to derive upper and lower estimates for the integer transfinite diameter of small intervals $[\frac{r}{s},\frac{r}{s}+\delta ]$, where $\frac{r}{s}$ is a fixed rational and $\delta \to 0$. We also study functions $g_{-},g,g^{+}$ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call polynomials, associated to a given interval $I$. We show how to estimate from below the proportion of roots of an integer polynomial which is sufficiently small...