Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913191894712320 |
|---|---|
| author | Technau, Marc |
| author_facet | Technau, Marc |
| contents | 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 $Q_f(N)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences Technau, Marc Number Theory 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 $Q_f(N)$. |
| title | Zero sums amongst roots and Cilleruelo's conjecture on the LCM of polynomial sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.05724 |