For an irreducible integral polynomial f of degree n, Cilleruelo’s conjecture states an asymptotic formula for the logarithm of the least common multiple of the first M values f(1) to f(M). It’s well-known for n = 1 as a consequence of Dirichlet’s Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for Sn-polynomials with integral coefficients over a fixed extension K=Q by considering the least common multiple of ideals of OK. The case of linear polynomials is dealt with separately by exploiting Dirichlet’s Theorem for primes in arithmetic progression, to get an asymptotic estimate. In our case, to achieve explicit error terms, we want effective versions of the asymptotics. We will state here both a conditional and unconditional results proved by Lagarias and Odlyzko. For degree-2 polynomials, it is possible to obtain explicit asymptotics for the least common multiple, analogously to the ones achieved for polynomials in Z[X]. However, the latter is not a subject of the current paper.
| Published in | Pure and Applied Mathematics Journal (Volume 13, Issue 6) |
| DOI | 10.11648/j.pamj.20241306.12 |
| Page(s) | 84-99 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Analytic Number Theory, Cilleruelo’s Conjecture, Least Common Multiple of Polynomials, Number Fields, Probability Theory
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| [7] | Lagarias, J.C., Odlyzdo, A.M. Effective versions of the chebotarev density theorem. In A. Frohlich, editor, Algebraic Number Fields, L-Functions and Galois Properties, pages 409-464. Academic Press, New York, London, 1977. |
| [8] | Nagel, T. Généralisation d’un théorème de Tchebycheff Journal de mathématiques pures et appliquées 8e série, tome 4 (1921), p. 343-356. |
| [9] | Rudnick, Z., Zehavi, S. On Cilleruelo’s conjecture for the least common multiple of polynomial sequences, arXiv:1902.01102v2 [math.NT] 15 Apr 2019. |
| [10] | Viglino, I. Towards a generalization of the van der Waerden’s conjecture for Sn-polynomials with integral coefficients over a fixed number field extension, arXiv:2212.11608 [math.NT], 22 Dec 2022. |
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APA Style
Viglino, I. (2024). On the Least Common Multiple of Polynomials over a Number Field. Pure and Applied Mathematics Journal, 13(6), 84-99. https://doi.org/10.11648/j.pamj.20241306.12
ACS Style
Viglino, I. On the Least Common Multiple of Polynomials over a Number Field. Pure Appl. Math. J. 2024, 13(6), 84-99. doi: 10.11648/j.pamj.20241306.12
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Download@article{10.11648/j.pamj.20241306.12,
author = {Ilaria Viglino},
title = {On the Least Common Multiple of Polynomials over a Number Field
},
journal = {Pure and Applied Mathematics Journal},
volume = {13},
number = {6},
pages = {84-99},
doi = {10.11648/j.pamj.20241306.12},
url = {https://doi.org/10.11648/j.pamj.20241306.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20241306.12},
abstract = {For an irreducible integral polynomial f of degree n, Cilleruelo’s conjecture states an asymptotic formula for the logarithm of the least common multiple of the first M values f(1) to f(M). It’s well-known for n = 1 as a consequence of Dirichlet’s Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for Sn-polynomials with integral coefficients over a fixed extension K=Q by considering the least common multiple of ideals of OK. The case of linear polynomials is dealt with separately by exploiting Dirichlet’s Theorem for primes in arithmetic progression, to get an asymptotic estimate. In our case, to achieve explicit error terms, we want effective versions of the asymptotics. We will state here both a conditional and unconditional results proved by Lagarias and Odlyzko. For degree-2 polynomials, it is possible to obtain explicit asymptotics for the least common multiple, analogously to the ones achieved for polynomials in Z[X]. However, the latter is not a subject of the current paper.
},
year = {2024}
}
TY - JOUR T1 - On the Least Common Multiple of Polynomials over a Number Field AU - Ilaria Viglino Y1 - 2024/12/18 PY - 2024 N1 - https://doi.org/10.11648/j.pamj.20241306.12 DO - 10.11648/j.pamj.20241306.12 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 84 EP - 99 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20241306.12 AB - For an irreducible integral polynomial f of degree n, Cilleruelo’s conjecture states an asymptotic formula for the logarithm of the least common multiple of the first M values f(1) to f(M). It’s well-known for n = 1 as a consequence of Dirichlet’s Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for Sn-polynomials with integral coefficients over a fixed extension K=Q by considering the least common multiple of ideals of OK. The case of linear polynomials is dealt with separately by exploiting Dirichlet’s Theorem for primes in arithmetic progression, to get an asymptotic estimate. In our case, to achieve explicit error terms, we want effective versions of the asymptotics. We will state here both a conditional and unconditional results proved by Lagarias and Odlyzko. For degree-2 polynomials, it is possible to obtain explicit asymptotics for the least common multiple, analogously to the ones achieved for polynomials in Z[X]. However, the latter is not a subject of the current paper. VL - 13 IS - 6 ER -
Institut de Mathématiques, EPFLÉcole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
