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arXiv · 2101.03561

The Expected Number of Roots over The Field of p-adic Numbers

Roy Shmueli
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Abstract

We study the roots of a random polynomial over the field of $p$-adic numbers. For a random monic polynomial with i.i.d. coefficients in $\mathbb{Z}_p$, we obtain an estimate for the expected number of roots of this polynomial. In particular, if the coefficients take the values $\pm1$ with equal probability, the expected number of $p$-adic roots converges to $\left(p-1\right)/\left(p+1\right)$ as the degree of the polynomial tends to $\infty$.

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Roy Shmueli. 2021-01-10. The Expected Number of Roots over The Field of p-adic Numbers. https://doi.org/10.1093/imrn%2Frnab308

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