The density of polynomials of degree nn over Zp having exactly r roots in Qp

Manjul Bhargava; John Cremona; Tom Fisher; Stevan Gajović

doi:10.1112/plms.12438

The density of polynomials of degree nn over Z_p having exactly r roots in Q_p

Manjul Bhargava, John Cremona, Tom Fisher, Stevan Gajović

Mathematics

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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.).

Original language	English (US)
Pages (from-to)	713-736
Number of pages	24
Journal	Proceedings of the London Mathematical Society
Volume	124
Issue number	5
DOIs	https://doi.org/10.1112/plms.12438
State	Published - May 2022

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1112/plms.12438

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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.).",

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AU - Fisher, Tom

AU - Gajović, Stevan

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AB - 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.).

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The density of polynomials of degree nn over Z_p having exactly r roots in Q_p

Abstract

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