On an approximation property of Pisot numbers II

Zaïmi, Toufik

Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 239-249.

Soit $q$ un nombre complexe, $m$ un entier positif et $l_m\left(q\right)=inf\{\left|P\left(q\right)\right|,P\in {\mathbb{Z}}_m\left[X\right],P\left(q\right)\ne 0\}$, où ${\mathbb{Z}}_m\left[X\right]$ désigne l’ensemble des polynômes à coefficients entiers de valeur absolue $\le m$. Nous déterminons dans cette note le maximum des quantités $l_m\left(q\right)$ quand $q$ décrit l’intervalle $]m,m+1[$. Nous montrons aussi que si $q$ est un nombre non-réel de module $>1$, alors $q$ est un nombre de Pisot complexe si et seulement si $l_m\left(q\right)>0$ pour tout $m$.

Let $q$ be a complex number, $m$ be a positive rational integer and $l_m\left(q\right)=inf\{\left|P\left(q\right)\right|,P\in {\mathbb{Z}}_m\left[X\right],P\left(q\right)\ne 0\}$, where ${\mathbb{Z}}_m\left[X\right]$ denotes the set of polynomials with rational integer coefficients of absolute value $\le m$. We determine in this note the maximum of the quantities $l_m\left(q\right)$ when $q$ runs through the interval $]m,m+1[$. We also show that if $q$ is a non-real number of modulus $>1$, then $q$ is a complex Pisot number if and only if $l_m\left(q\right)>0$ for all $m$.

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