Coxeter polynomials of Salem trees

Charalampos A. Evripidou¹

¹ Department of Mathematics and Statistics University of Cyprus P.O. Box 20537 1678 Nicosia, Cyprus

Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 209-226

Résumé

We compute the Coxeter polynomial of a family of Salem trees, and also the limit of the spectral radii of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if $z$ is a root of multiplicities $m_1,\ldots ,m_k$ for the Coxeter po\discretionary {-}{}{}ly\discretionary {-}{}{}no\discretionary {-}{}{}mials of the trees $\mathcal {T}_1,\ldots ,\mathcal {T}_k$ respectively, then $z$ is a root for the Coxeter po\discretionary {-}{}{}ly\discretionary {-}{}{}no\discretionary {-}{}{}mial of their join, of multiplicity at least $\min\{m-m_1,\ldots ,m-m_k\}$ where $m=m_1+\cdots +m_k$.

Détail

DOI : 10.4064/cm141-2-6

Keywords: compute coxeter polynomial family salem trees limit spectral radii their coxeter transformations number their vertices tends infinity prove root multiplicities ldots coxeter polynomials trees mathcal ldots mathcal respectively root coxeter polynomial their join multiplicity least min m m ldots m m where cdots