Let \( k\ge 2 \) be a fixed inteer. Then, \( f_k(X)=X^k-X^{k-1}-\cdots -X-1 \) is the \( k \)-generalized Fibonacci polynomial. This polynomial has exactly one root outside the unit disk. It is real and larger than \( 1 \), and it is denoted by \( \alpha_k \). Let \( \rho_1>\rho_2>\cdots>\rho_K \) be the absolute values of the remaining roots of \( f_k(X) \) which are at most \( 1 \). Let \( \alpha_j=\rho_j e^{i\theta_j}, ~\theta_j\in(0,\pi] \) for \( j=1,2,\ldots, K \) be the roots of \( f_k(X) \) which are inside the upper half \( Im (Z)\ge 0 \) of the disk \( |z|\le 1 \).
In the paper under review, the authors prove the following theorems, which are the main results in the paper.
Theorem 1. The inequality \begin{align*} \frac{\rho_i}{\rho_j}>1+\frac{1}{10k^{9.6}(\pi/e)^{k}} \end{align*} holds for all \( 1\le i<j\le K \) and for all \( k\ge 4 \).
Theorem 2. The inequality \begin{align*} |\alpha_i-\alpha_j|>1+\frac{1}{k^{6.6}(\pi/e)^{k}} \end{align*} holds for all \( 1\le i<j\le K \) and for all \( k\ge 4 \).
The proofs of Theorem 1 and Theorem 2 follow from a clever combination of techniques in number theory and the usual properties of the \( k \)-generalized Fibonacci sequences. All computations are done with the help of a simple computer program in Mathematica.