The smallest Perron numbers

Abstract

A Perron number is a real algebraic integer α \mathbf {\alpha } of degree d ≥ 2 d \geq 2 , whose conjugates are α i \mathbf {\alpha } _{i} , such that α > max 2 ≤ i ≤ d | α i | \mathbf {\alpha } >\max _{2 \leq i \leq d} \vert \mathbf {\alpha } _{i} \vert . In this paper we compute the smallest Perron numbers of degree d ≤ 24 d \leq 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C \mathbb {C}