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

Source abstract

Abstract 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)⋯ 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 ).