@article{e2fe5674ee0b45c59d7acacadb752904,

title = "The density of polynomials of degree nn over Zp having exactly r roots in Qp",

abstract = "We determine the probability that a random polynomial of degree (Formula presented.) over (Formula presented.) has exactly (Formula presented.) roots in (Formula presented.), and show that it is given by a rational function of (Formula presented.) that is invariant under replacing (Formula presented.) by (Formula presented.).",

author = "Manjul Bhargava and John Cremona and Tom Fisher and Stevan Gajovi{\'c}",

note = "Funding Information: The first author was supported by a Simons Investigator Grant and NSF grant DMS‐1001828. The second author was supported by the Heilbronn Institute for Mathematical Research. The fourth author was supported in part by DFG‐Grant MU 4110/1‐1. Funding Information: We thank the CMI-HIMR Summer School in Computational Number Theory held at the University of Bristol in June 2019, where this work began. We also thank Xavier Caruso for kindly sharing with us an earlier draft of his paper [4], and Jordan Ellenberg, Hendrik Lenstra, Steffen M{\"u}ller, Bjorn Poonen, Lazar Radi{\v c}evi{\'c}, Arul Shankar, and Jaap Top for many helpful conversations. The first author was supported by a Simons Investigator Grant and NSF grant DMS-1001828. The second author was supported by the Heilbronn Institute for Mathematical Research. The fourth author was supported in part by DFG-Grant MU 4110/1-1. We thank the referees for a careful reading of our paper, and for providing the additional references at the end of Section 1.1. Publisher Copyright: {\textcopyright} 2022 The Authors. Proceedings of the London Mathematical Society is copyright {\textcopyright} London Mathematical Society.",

year = "2022",

month = may,

doi = "10.1112/plms.12438",

language = "English (US)",

volume = "124",

pages = "713--736",

journal = "Proceedings of the London Mathematical Society",

issn = "0024-6115",

publisher = "Oxford University Press",

number = "5",

}