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

Abstract: In a previous paper, we introduced the Collatz polynomials $P_N(z)$, 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 Enestr\"om-Kakeya Theorem to tighten our old bounds of the roots of $P_N(z)$ and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle ${z\in\mathbb{C}: |z| = 2}$ are rare: the set of $N$ such that $P_N(z)$ has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study.