LCM in $\mathbb F_q[T]$

Question

Let $q$ be a power of a prime $p$. We work in $\mathbb F_q[T]$. Put $L_n=\prod_{j=1}^n(T^{q^i}-T)$. Does one have $$\deg(\mathrm{LCM}(L_{n+1};L^q_n))=\frac{q^{n+2}}{q-1}+o(q^n)$$ when $n\to\infty$?

Answer 1

Let $$h_n = \prod_{d|n} (T^{q^{n/d}}-T)^{\mu(d)} = \prod_{a\in \Bbb{F}_{q^n}, \Bbb{F}_q(a)=\Bbb{F}_{q^n}} (T-a)$$

They are coprime and $$L_n = \prod_{j=1}^n h_j^{\lfloor n/j\rfloor}$$ Whence $$lcm(L_{n+1},L_n^q)=L_n^q h_{n+1}$$ has degree $$q\sum_{j=1}^n q^j + \sum_{d|n+1} \mu(d) q^{(n+1)/d}$$