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

Marc Technau

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 $Q_f(N) = f(1)\cdots f(N)$ 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 $Q_f(N)$.

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

Abstract

We make progress on a conjecture of Cilleruelo on the growth of the least common multiple of consecutive values of an irreducible polynomial 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 exceeds two. The improvement rests upon a different treatment of `large' prime divisors of by means of certain zero sums amongst the roots of . 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 .
Paper Structure (7 sections, 12 theorems, 39 equations)

This paper contains 7 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preparations
  4. The key lemma
  5. Bounding L(Q): Proof of Theorem\ref{['thm:MainResult']}
  6. Bounding ℓ(Q): Proof of Theorem\ref{['thm:MainResult:Variant']}
  7. Improvements for certain cyclotomic polynomials

Key Result

Theorem 1.1

For any irreducible quadratic polynomial $f\in\mathbb{Z}[X]$, where $B_f$ is a certain explicitly computable constant depending on $f$.

Theorems & Definitions (30)

  • Theorem 1.1: Cilleruelo
  • Theorem 1.2
  • Example
  • Theorem 1.3
  • Example
  • Theorem 1.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 20 more