Lower Bounds on the Least Common Multiple of a Polynomial Sequence and its Radical

Abstract

Cilleruelo conjectured that for an irreducible polynomial $f \in \mathbb{Z}[X]$ of degree $d \geq 2$, denoting $$L_f(N)=\mathrm{lcm}(f(1),f(2),\ldots f(N))$$ one has $$\log L_f(n)\sim(d-1)N\log N.$$ He proved it in the case $d=2$ but it remains open for every polynomial with $d>2$. While the tight upper bound $\log L_f(n)\lesssim (d-1)N\log N$ is known, the best known general lower bound due to Sah is $\log L_f(n)\gtrsim N\log N.$ We give an improved lower bound for a special class of irreducible polynomials, which includes the decomposable irreducible polynomials $f=g\circ h,\,g,h\in\mathbb Z[x],\mathrm{deg}\, g,\mathrm{deg}\, h\ge 2$, for which we show $$\log L_f(n)\gtrsim \frac{d-1}{d-\mathrm{deg}\, g}N\log N.$$ We also improve Sah's lower bound $\log\ell_f(N)\gtrsim \frac 2dN\log N$ for the radical $\ell_f(N)=\mathrm{rad}(L_f(N))$ for all $f$ with $d\ge 3$ and give a further improvement for polynomials $f$ with a small Galois group and satisfying an additional technical condition, as well as for decomposable polynomials.