Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912804771987456 |
|---|---|
| author | Lin, Chen Tang, Kaihan |
| author_facet | Lin, Chen Tang, Kaihan |
| contents | Previous research on exceptional units has primarily focused on the ring of rational integers or abstract finite rings, often restricted to linear or quadratic constraints. In this paper, we extend the concept of polynomial-type exceptional units to the ring of integers of an arbitrary algebraic number field. We investigate the number of these polynomial-type exceptional units on general algebraic varieties. By employing the Chinese Remainder Theorem and Hensel's lifting technique, we derive an exact counting formula for the number of these exceptional units on a smooth closed subscheme under the assumption of good reduction. Furthermore, using the Lang-Weil inequality, we establish an asymptotic estimate for the counting function. In particular, we prove that for varieties of degree at most two, the error term can be significantly improved, yielding a sharper asymptotic bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_02743 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields Lin, Chen Tang, Kaihan Number Theory 11D79, 14G15 Previous research on exceptional units has primarily focused on the ring of rational integers or abstract finite rings, often restricted to linear or quadratic constraints. In this paper, we extend the concept of polynomial-type exceptional units to the ring of integers of an arbitrary algebraic number field. We investigate the number of these polynomial-type exceptional units on general algebraic varieties. By employing the Chinese Remainder Theorem and Hensel's lifting technique, we derive an exact counting formula for the number of these exceptional units on a smooth closed subscheme under the assumption of good reduction. Furthermore, using the Lang-Weil inequality, we establish an asymptotic estimate for the counting function. In particular, we prove that for varieties of degree at most two, the error term can be significantly improved, yielding a sharper asymptotic bound. |
| title | Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields |
| topic | Number Theory 11D79, 14G15 |
| url | https://arxiv.org/abs/2601.02743 |