Bounds for zeros of Collatz polynomials, with necessary and sufficient strictness conditions.

In a previous paper, we introduced the Collatz polynomials $ P_N\left (z \right) $ P N (z) , whose coefficients are the terms of the Collatz sequence of the positive integer N. Our work in this paper expands on our previous results, using the Eneström-Kakeya Theorem to tighten our old bounds of the roots of $ P_N\left (z \right) $ P N (z) and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $ \ensuremath {\{z\in \ensuremath {\mathbb {C}}:\left | z \right | = 2\}} $ { z ∈ C : | z | = 2 } are rare: the set of N such that $ P_N\left (z \right) $ P N (z) has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study. [ABSTRACT FROM AUTHOR]