Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2507.16138 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866918216927805440 |
|---|---|
| author | Anderson, Theresa C. Asarhasa, Ufuoma V. Bertelli, Adam Gundlach, Fabian O'Dorney, Evan M. |
| author_facet | Anderson, Theresa C. Asarhasa, Ufuoma V. Bertelli, Adam Gundlach, Fabian O'Dorney, Evan M. |
| contents | For a polynomial $f(x) = \sum_{i=0}^n a_i x^i$, we study the double discriminant $DD_{n,k} = \operatorname{disc}_{a_k} \operatorname{disc}_x f(x)$. This object has been well studied in algebraic geometry, but has been brought to recent prominence in number theory by its key role in the proof of the Bhargava--van der Waerden theorem. We bridge the knowledge gap for this object by proving an explicit factorization: $DD_{n,k}$ is the product of a square, a cube, and possibly a linear monomial. Our proof is entirely algebraic. We also investigate other aspects of this factorization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The structure of the double discriminant Anderson, Theresa C. Asarhasa, Ufuoma V. Bertelli, Adam Gundlach, Fabian O'Dorney, Evan M. Number Theory For a polynomial $f(x) = \sum_{i=0}^n a_i x^i$, we study the double discriminant $DD_{n,k} = \operatorname{disc}_{a_k} \operatorname{disc}_x f(x)$. This object has been well studied in algebraic geometry, but has been brought to recent prominence in number theory by its key role in the proof of the Bhargava--van der Waerden theorem. We bridge the knowledge gap for this object by proving an explicit factorization: $DD_{n,k}$ is the product of a square, a cube, and possibly a linear monomial. Our proof is entirely algebraic. We also investigate other aspects of this factorization. |
| title | The structure of the double discriminant |
| topic | Number Theory |
| url | https://arxiv.org/abs/2507.16138 |