Cilleruelo conjectured that for an irreducible polynomial f ∈ ℤ[X] of degree d ⩾ 2, denoting \({L_f}(N) = {\text{lcm}}({f(1),f(2), \ldots,f(N)})\) 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 Lf(n) ≲ (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 ◦ h, g, h ∈ ℤ[x], deg g, deg h ≥ 2, for which we show \(\log {L_f}(n) \gtrsim {{d - 1} \over {d - \deg g}}N\log N.\)
We also improve Sah’s lower bound \(\log {\ell_f}(n) \gtrsim {{2} \over {d}} N\log N\) for the radical ℓf (N) = rad(Lf(N)) for all irreducible f with d ≥ 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.
