The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots in Qp${\mathbb {Q}}_p$. Issue 5 (17th March 2022)

• Record Type: Journal Article
• Authors: Bhargava, Manjul; Cremona, John; Fisher, Tom; Gajović, Stevan
• Is Part Of: Proceedings of the London Mathematical Society. Volume 124:Issue 5(2022)
• Page Start: 713
• Page End: 736
• Publication Date: 2022-03-17
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12438
• Languages: English
• ISSNs: 0024-6115

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 $1/p$ .