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

2021/01/23 by Manjul Bhargava, Bhargava, Manjul, J. E. Cremona +5
Mathematics · #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.2101.09590

Abstract

We determine the probability that a random polynomial of degree n over\n\ℤ_p has exactly r roots in \ℚ_p, and show that it is\ngiven by a rational function of p that is invariant under replacing p by\n1/p.\n

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