A lower bound on the least common multiple of polynomial sequences

For an irreducible polynomial f is an element of Z[x] of degree d >= 2, Cilleruelo conjectured that the least common multiple of the values of the polynomial at the first N integers satisfies log lcm(f (1), . . . , f (N)) similar to (d - 1)N log N as N -> infinity. This is only known for degree d >= 2. We give a lower bound for all degrees d > 2 which is consistent with the conjecture: log lcm(f (1), . . . , f (N)) >> N log N.