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arXiv · 2101.09590

The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$

Manjul Bhargava·John Cremona·Tom Fisher·Stevan Gajović
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Abstract

We determine the probability that a random polynomial of degree $n$ over $\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it is given by a rational function of $p$ that is invariant under replacing $p$ by $1/p$.

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Manjul Bhargava, John Cremona, Tom Fisher, Stevan Gajović. 2021-01-23. The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$. https://arxiv.org/abs/2101.09590

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