The Least Common Multiple of Polynomial Values over Function Fields

TL;DR

This work develops a function field analogue of the Cilleruelo problem for the least common multiple of polynomial values, defining L_f(n) over \mathbb{F}_q[T] and analyzing its degree. The authors prove the expected upper bound deg L_f(n) ≤ c_f (d-1) n q^n and a matching-order lower bound deg L_f(n) ≥ (d-1)/d n q^n (with improvements under extra hypotheses), and establish that deg ell_f(n) (the radical) has the same growth as deg L_f(n). They identify a broad class of special polynomials for which the conjecture holds in full, and they provide a complete classification of these polynomials. The paper also develops a rigorous framework around the space V_f, roots modulo prime powers, and a function-field Chebotarev approach to control contributions from primes, highlighting a deep connection between algebraic structure and LCM growth in function fields. The results offer insights that illuminate the polynomial-value LCM problem and its potential implications for the integer setting, aided by the function-field toolkit such as Poonen-type squarefree results and, conditionally, the ABC conjecture.

Abstract

Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $d \geq 2$ one has $$\log\left[\mathrm{lcm}(f(1),f(2),\ldots f(N))\right]\sim(d-1)N\log N$$ as $N \to \infty$. He proved it in the case $d=2$ but it remains open for every polynomial with $d>2$. 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 $$L_f(n) := \mathrm{lcm} \left(f\left(Q\right)\ : \ Q \in \mathbb F_q[T]\mbox{ monic},\, \mathrm{deg}\,Q = n\right)$$ we conjecture that \begin{equation}\label{eq:conjffabs}\mathrm{deg}\, 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 $\mathrm{deg}\, L_f(n) \sim \mathrm{deg}\,\mathrm{rad}\left(L_f(n)\right)$ (in other words the corresponding LCM is close to being squarefree), which is not known over $\mathbb Z$.

Theorems & Definitions (68)

• Conjecture 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Example 1.7
• Theorem 1.8
• Example 1.9
• Theorem 1.10