On an approximation property of Pisot numbers II
Toufik Zaïmi1
1King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia
Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 239-249

Let q be a complex number, m be a positive rational integer and l m (q)=inf{P(q),P∈ℤ m [X],P(q)≠0}, where ℤ m [X] denotes the set of polynomials with rational integer coefficients of absolute value ≤m. We determine in this note the maximum of the quantities l m (q) 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 (q)>0 for all m.

Soit q un nombre complexe, m un entier positif et l m (q)=inf{P(q),P∈ℤ m [X],P(q)≠0}, où ℤ m [X] désigne l’ensemble des polynômes à coefficients entiers de valeur absolue ≤m. Nous déterminons dans cette note le maximum des quantités l m (q) 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 (q)>0 pour tout m.

Published online:
DOI: 10.5802/jtnb.446

Toufik Zaïmi  1

1 King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia 
Toufik Zaïmi. On an approximation property of Pisot numbers II. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 239-249. doi: 10.5802/jtnb.446
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