The least common multiple of polynomial values over function fields
Tom 217 / 2025
Streszczenie
Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb Z[X]$ of degree $d \geq 2$ one has $$ \log \mathrm{lcm} (f(1),\ldots , f(N))\sim (d-1)N\log N $$ as $N \to \infty $. He proved it in the case $d=2$ but it remains open for every polynomial with $d \gt 2$.
We investigate the function field analogue of the problem by considering polynomials over the ring $\mathbb F_q[T]$. We state an analogue of Cilleruelo’s conjecture in this setting: denoting$$L_f(n) := \mathrm{lcm} (f(Q) : Q \in \mathbb F_q[T]\ \mbox{monic},\, \deg Q = n)$$ we conjecture that \[ \deg L_f(n) \sim c_f (d-1) nq^n,\quad n \to \infty \tag{1} \] ($c_f$ is an explicit constant depending only on $f$, typically $c_f=1$). We give both upper and lower bounds for $L_f(n)$ and show that the asymptotic (1) 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 $\deg L_f(n) \sim \deg \mathrm{rad}\,L_f(n)$ (in other words, the corresponding LCM is close to being squarefree), which is not known over $\mathbb Z $.
