The Least Common Multiple of Polynomial Values over Function Fields

Cilleruelo conjectured that for an irreducible polynomial $f\in \mathbb{Z}[X]$ of degree $d\ge 2$ one has $$\log [\mathrm{l}\mathrm{c}\mathrm{m}(f(1),f(2),\dots f(N))]\sim (d-1)N\log N$$ as $N\to \mathrm{\infty }$. He proved it in the case $d=2$ but it remains open for every polynomial with $d>2$. We investigate the function field analogue of the problem by considering polynomials over the ring ${\mathbb{F}}_q[T]$. We state an analog of Cilleruelo’s conjecture in this setting: denoting by $$L_f(n):=\mathrm{l}\mathrm{c}\mathrm{m}(f\left(Q\right):Q\in {\mathbb{F}}_q[T]monic,\mathrm{d}\mathrm{e}\mathrm{g}Q=n)$$ we conjecture that \begin{equation}\label{eq:conjffabs}$\mathrm{d}\mathrm{e}\mathrm{g}$, L_f(n) \sim c_f \left(d-1\right) nq^n,\ n \to \infty\end{equation} ($c_f$ is an explicit constant dependent only on $f$, typically $c_f=1$). We give both upper and lower bounds for $L_f(n)$ and show that the conjectured asymptotic holds for a class of ``special" polynomials, initially considered by Leumi in this context, which includes all quadratic polynomials and many other examples as well. We fully classify these special polynomials. We also show that $\mathrm{d}\mathrm{e}\mathrm{g}{L}_f(n)\sim \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}(L_f(n))$ (in other words the corresponding LCM is close to being squarefree), which is not known over $\mathbb{Z}$.