The integer transfinite diameter of intervals and totally real algebraic integers

Flammang, V., Rhin, G., and Smyth, C. J.. "The integer transfinite diameter of intervals and totally real algebraic integers." Journal de théorie des nombres de Bordeaux 9.1 (1997): 137-168. <http://eudml.org/doc/248004>.

@article{Flammang1997,
abstract = {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 $\delta \rightarrow 0$. We also study functions $g_-, g, g^+$ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical 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 on $I$ which must also be roots of the critical polynomial. This generalises now classical work of Aparicio, and extends the techniques of Borwein and Erdélyi from the critical polynomial $x$ for $\mathrm \{[0,1]\}$ to any critical polynomial for an arbitrary interval. As an easy consequence of our results, we obtain an inequality about algebraic integers of independent interest : if $\alpha $ is totally real, with minimum conjugate $\alpha _1$, then, with a small number of explicit exceptions, the mean value of $\alpha $ and its conjugates is at least $\alpha _1 + 1.6$.},
author = {Flammang, V., Rhin, G., Smyth, C. J.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {polynomial; Mahler measure; inequality; integer transfinite diameter; small intervals; Farey intervals; critical polynomials; totally real algebraic integers},
language = {eng},
number = {1},
pages = {137-168},
publisher = {Université Bordeaux I},
title = {The integer transfinite diameter of intervals and totally real algebraic integers},
url = {http://eudml.org/doc/248004},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Flammang, V.
AU - Rhin, G.
AU - Smyth, C. J.
TI - The integer transfinite diameter of intervals and totally real algebraic integers
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 1
SP - 137
EP - 168
AB - 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 $\delta \rightarrow 0$. We also study functions $g_-, g, g^+$ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical 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 on $I$ which must also be roots of the critical polynomial. This generalises now classical work of Aparicio, and extends the techniques of Borwein and Erdélyi from the critical polynomial $x$ for $\mathrm {[0,1]}$ to any critical polynomial for an arbitrary interval. As an easy consequence of our results, we obtain an inequality about algebraic integers of independent interest : if $\alpha $ is totally real, with minimum conjugate $\alpha _1$, then, with a small number of explicit exceptions, the mean value of $\alpha $ and its conjugates is at least $\alpha _1 + 1.6$.
LA - eng
KW - polynomial; Mahler measure; inequality; integer transfinite diameter; small intervals; Farey intervals; critical polynomials; totally real algebraic integers
UR - http://eudml.org/doc/248004
ER -