On the Distance Between Two Algebraic Numbers

On the separation of the roots of the generalized Fibonacci polynomial

“…To prove Theorem 1 we need a good separation result on the α i 's themselves. Before presenting our result on the root separation of f k (X), with which we will prove Theorem 1, we will show how we can obtain a better result than the one proved by Dubickas [2] in our particular case by using results of Mahler [5] and Mignotte [6]. Namely, let us show that the inequality…”

“…The above Theorem 1 is a separation result concerning the absolute values of the differences of the roots of f k (X). Quite general separation results of this kind appear in [2] but they are much worse (the denominator of the analogous expression from the right-hand side in (2) in [2] is exponential in k 2 ). To prove Theorem 1 we need a good separation result on the α i 's themselves.…”

“…and Dubickas [2] improved the right-hand side above to 1 + 1.454 −k 3 . Our first result is improving this bound.…”

Identities for the k-generalized Fibonacci sequence with negative indices and its zero-multiplicity

“…The next lemma is Theorem 1 in [6] and represents an improvement over the analogous inequality in [2].…”