On the average stopping time of the Collatz map in $\mathbb{F}_2[x]$

Abstract

Define the map $T_1$ on $\mathbb{F}_2[x]$ by $T_1(f)=\frac{f}{x}$ if $f(0)=0$ and $T_1(f)=\frac{(x+1)f+1}{x}$ if $f(0)=1$. For a non-zero polynomial $f$ let $\tau_1(f)$ denote the least natural $k$ number for which $T_1^{k}(f)=1$. Define the average stopping time to be $\rho_1(n)=\frac{\sum_{f\in \mathbb{F}_2[x], \text{deg}(f)=n }\tau_1(f)}{2^n}$. We show that $\lim_{n\rightarrow\infty}\frac{\rho_1(n)}{n}=2$, confirming a conjecture of Alon, Behajaina, and Paran. Furthermore, we give a new proof that $\tau_1(f)\in O(\text{deg}(f)^{1.5})$ for all $f\in\mathbb{F}_2[x]\setminus\{0\}$.