论文标题
在$ p $ - adic数字的字段上随机多项式的根数
The Number of Roots of a Random Polynomial over The Field of $p$-adic Numbers
论文作者
论文摘要
我们研究了P-Adic数字领域的随机多项式的根。对于在$ \ mathbb {z} _p $中具有系数的随机多项式，我们获得了该多项式根部数量的阶乘矩时渐近公式。此外，我们还表明，对于某些$ k> 0 $的$ o \ big（n^{ - k} \ big）$ o \ big（n^{ - k} \ big）$ o \ big $ n $ roots的随机多项式的可能性超过$ \ log n $ roots。
We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an asymptotic formula for the factorial moments of the number of roots of this polynomial. In addition, we show the probability that a random polynomial of degree $n$ has more than $\log n$ roots is $O\big(n^{-K}\big)$ for some $K > 0$.
