Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.05378 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917693264756736 |
|---|---|
| author | Li, Huixi Wang, Biao Wang, Chunlin Yi, Shaoyun |
| author_facet | Li, Huixi Wang, Biao Wang, Chunlin Yi, Shaoyun |
| contents | In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed $10^{16}$. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba developed a versatile method called the distortion method and significantly reduced Hough's bound to $616,000$. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for $\mathbb{F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. Consequently, we successfully resolve the minimum modulus problem for $\mathbb{F}_q[x]$ and disprove a conjecture by Azlin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_05378 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On covering systems of polynomial rings over finite fields Li, Huixi Wang, Biao Wang, Chunlin Yi, Shaoyun Number Theory In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed $10^{16}$. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba developed a versatile method called the distortion method and significantly reduced Hough's bound to $616,000$. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for $\mathbb{F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. Consequently, we successfully resolve the minimum modulus problem for $\mathbb{F}_q[x]$ and disprove a conjecture by Azlin. |
| title | On covering systems of polynomial rings over finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2308.05378 |