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We make progress on a conjecture of Cilleruelo on the growth of the leastcommon multiple of consecutive values of an irreducible polynomial $f$ on theadditional hypothesis that the polynomial be even. This strengthens earlierwork of Rudnick--Maynard and Sah subject to that additional hypothesis when thedegree 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 zerosums amongst the roots of $f$. A similar argument was recently used by Baierand Dey with regard to another problem. The same method also allows for furtherimprovements on a related conjecture of Sah on the size of the radical of$Q_f(N)$.

Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences