The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots in Qp${\mathbb {Q}}_p$
Authors
Bhargava, Manjul
Cremona, John
Fisher, Tom
Gajovic, Stevan
Publication Date
2022Journal Title
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
ISSN
0024-6115
Publisher
Wiley
Volume
124
Issue
5
Pages
713-736
Language
en
Type
Article
This Version
AO
VoR
Metadata
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Bhargava, M., Cremona, J., Fisher, T., & Gajovic, S. (2022). The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots in Qp${\mathbb {Q}}_p$. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 124 (5), 713-736. https://doi.org/10.1112/plms.12438
Description
Funder: Simons Investigator Grant
Funder: Heilbronn Institute for Mathematical Research
Abstract
Abstract: We determine the probability that a random polynomial of degree n $n$ over Z p ${\mathbb {Z}}_p$ has exactly r $r$ roots in Q p ${\mathbb {Q}}_p$ , and show that it is given by a rational function of p $p$ that is invariant under replacing p $p$ by 1 / p $1/p$ .
Keywords
11S05, RESEARCH ARTICLE, RESEARCH ARTICLES
Sponsorship
National Science Foundation (DMS‐1001828)
Deutsche Forschungsgemeinschaft (MU 4110/1‐1)
Identifiers
plms12438, 2101.09590
External DOI: https://doi.org/10.1112/plms.12438
This record's URL: https://www.repository.cam.ac.uk/handle/1810/337138
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
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