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

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 $\pm 1$ 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 $\infty$.

References in corpus (2)

• Irreducibility of random polynomials of large degree
• Probabilistic enumerative geometry over $p$-adic numbers: linear spaces on complete intersections