On a Conjecture of Erdős over Function Fields
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915608366415872 |
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| author | Xie, Likun |
| author_facet | Xie, Likun |
| contents | Using Katz's equidistribution framework, we show that for any squarefree polynomial $f \in \mathbb{F}_q[t]$ of degree $n \ge 2$, every residue class modulo $f$ can be represented as a product of two monic irreducible polynomials of degree at most $n$, provided $q$ is sufficiently large in terms of $n$. This gives the function-field analogue of a conjecture of Erdős in the large-$q$ regime.
Sawin previously proved this representation with stronger square-root cancellation via a higher-dimensional sheaf-theoretic construction. This note presents a one-dimensional argument that yields a natural $q^{-1/2}$ saving. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17612 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Conjecture of Erdős over Function Fields Xie, Likun Number Theory Algebraic Geometry Using Katz's equidistribution framework, we show that for any squarefree polynomial $f \in \mathbb{F}_q[t]$ of degree $n \ge 2$, every residue class modulo $f$ can be represented as a product of two monic irreducible polynomials of degree at most $n$, provided $q$ is sufficiently large in terms of $n$. This gives the function-field analogue of a conjecture of Erdős in the large-$q$ regime. Sawin previously proved this representation with stronger square-root cancellation via a higher-dimensional sheaf-theoretic construction. This note presents a one-dimensional argument that yields a natural $q^{-1/2}$ saving. |
| title | On a Conjecture of Erdős over Function Fields |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2510.17612 |