  We make progress on a conjecture of Cilleruelo on the growth of the least
common multiple of consecutive values of an irreducible polynomial $f$ on the
additional hypothesis that the polynomial be even. This strengthens earlier
work of Rudnick--Maynard and Sah subject to that additional hypothesis when the
degree of $f$ exceeds two. The improvement rests upon a different treatment of
`large' prime divisors of $Q_f(N) = f(1)\cdots f(N)$ by means of certain zero
sums amongst the roots of $f$. A similar argument was recently used by Baier
and Dey with regard to another problem. The same method also allows for further
improvements on a related conjecture of Sah on the size of the radical of
$Q_f(N)$.
