Boyd's conjecture

Abstract

We determine the limit of the rate $\frac{{\nu }_{n,a}}{n}$ between the number ${\nu }_{n,a}$ of roots of the trinomial $x^n-ax-1$, $a\in (0,2]$, which are greater than 1 in modulus, and degree $n$. The analogue of Boyd's Conjecture (C) for Perron numbers is a consequence of the limit, under the assumption that the conjecture of Lind-Boyd is valid. The product of these ${\nu }_{n,a}$ roots has also a limit when $n\to \infty$. The explicit expression of the limit by an integral is presented. The computing of the rate and the product for $n=100,150$ as well as of its limits is presented.

arXiv comment19 pages, 3 figures added, some references added, Conjecture (CP) added and proved under the assumption that the conjecture of Lind-Boyd is valid, some remarks added