Squarefree values of polynomial discriminants II
We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in parti...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2025-01-01
|
| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508625000095/type/journal_article |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850137864566734848 |
|---|---|
| author | Manjul Bhargava Arul Shankar Xiaoheng Wang |
| author_facet | Manjul Bhargava Arul Shankar Xiaoheng Wang |
| author_sort | Manjul Bhargava |
| collection | DOAJ |
| description | We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ‘arithmetic Bertini theorem’ conjectured by Poonen for
${\mathbb {P}}^1_{\mathbb {Z}}$
. |
| format | Article |
| id | doaj-art-98e24901dab14d93b9eec8a1f0e40020 |
| institution | OA Journals |
| issn | 2050-5086 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj-art-98e24901dab14d93b9eec8a1f0e400202025-08-20T02:30:43ZengCambridge University PressForum of Mathematics, Pi2050-50862025-01-011310.1017/fmp.2025.9Squarefree values of polynomial discriminants IIManjul Bhargava0Arul Shankar1Xiaoheng Wang2https://orcid.org/0000-0002-0111-6932Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA; E-mail:Department of Mathematics, Univerity of Toronto, 40 St. George Street, Toronto, ON M5S 2E4, Canada;Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada; E-mail:We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ‘arithmetic Bertini theorem’ conjectured by Poonen for ${\mathbb {P}}^1_{\mathbb {Z}}$ .https://www.cambridge.org/core/product/identifier/S2050508625000095/type/journal_article11N3511C0811R0911H06 |
| spellingShingle | Manjul Bhargava Arul Shankar Xiaoheng Wang Squarefree values of polynomial discriminants II Forum of Mathematics, Pi 11N35 11C08 11R09 11H06 |
| title | Squarefree values of polynomial discriminants II |
| title_full | Squarefree values of polynomial discriminants II |
| title_fullStr | Squarefree values of polynomial discriminants II |
| title_full_unstemmed | Squarefree values of polynomial discriminants II |
| title_short | Squarefree values of polynomial discriminants II |
| title_sort | squarefree values of polynomial discriminants ii |
| topic | 11N35 11C08 11R09 11H06 |
| url | https://www.cambridge.org/core/product/identifier/S2050508625000095/type/journal_article |
| work_keys_str_mv | AT manjulbhargava squarefreevaluesofpolynomialdiscriminantsii AT arulshankar squarefreevaluesofpolynomialdiscriminantsii AT xiaohengwang squarefreevaluesofpolynomialdiscriminantsii |