Two-adic valuation pattern for normalized quadratic resultants

Determine, for every n=2^r k with r≥0 and k odd, the exact value of v_2(\hat α_n) according to the three cases specified by the conjecture: establish its divisibility by φ(k); show that it is zero when r=0; show that for r=1 it is zero precisely when k=2^s−1 and otherwise equals φ(k); and show that when k=2^s−1 it equals ((r−2)2^{r−1}+1)φ(k).

Background

For the quadratic family F_λ(z)=λz(1−z), the paper defines ν_n=v_2(a_n(0)), proves that v_2(a_n)=ν_n, and introduces the normalized polynomial \hat a_n=2{−ν_n}a_n and normalized resultant \hat α_n=Res(Φ_n,\hat a_n). Consequently, v_2(α_n)=v_2(\hat α_n)+ν_nφ(n).

The authors report computational evidence for n up to 243 and formulate a detailed conjectural pattern for v_2(\hat α_n) when n=2r k with k odd. Parts of the pattern are established later: the cases corresponding to r=0 and r=1 with k=2s−1 are proved, while the full divisibility and valuation assertions remain conjectural in the stated passage.

References

If $n = 2r k\in N$ with $r\in N$ and $k$ odd, then $\varphi(k)$ divides $v_2(\hat \alpha_n)$. Additionally,\begin{enumerate} \item if $r=0$ then $v_2(\hat \alpha_n)=0$; \item if $r=1$ then $ v_2(\hat \alpha_n)=\begin{cases} 0&\text{if }k=2s-1\text{ with }s\in N\ast\ \varphi(k)&\text{otherwise};\end{cases}$ \item if $r\in N$ and $k=2s-1$ with $s\in N\ast$, then $v_2(\hat \alpha_n) = \bigl((r-2)2{r-1}+1\bigr) \varphi(k)$. \end{enumerate}

— An arithmetic approach to parabolic multiplicity in complex dynamics  (2608.20008 - Buff et al., 20 Aug 2026) in Conjecture 2, Section 6, immediately after the definition of $\hat\alpha_n$