Affiliation:
1. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Abstract
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 $(p-1)/(p+1)$ as the degree of the polynomial tends to $\infty $.
Funder
Israel Science Foundation
Publisher
Oxford University Press (OUP)
Reference24 articles.
1. The density of polynomials of degree $n$ over $\mathbb{Z}_p$ having exactly $r$ roots in $\mathbb{Q}_p$;Bhargava,2021
2. On the roots of certain algebraic equations;Bloch;Proc. Lond. Math. Soc. (3),1932
3. Irreducibility of random polynomials of large degree;Breuillard;Acta Math.,2019
4. The probability that a random monic p-adic polynomial splits;Buhler;Experiment. Math.,2006
Cited by
3 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献