The Expected Number of Roots over The Field of p-adic Numbers
The Expected Number of Roots over The Field of p-adic Numbers
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p 进数域上的期望根数
DOI：
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Roy Shmueli
中科院分区：
文献类型：
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作者：
Roy Shmueli
研究了p-adic数域上随机多项式的根.对于具有独立同分布的随机monic多项式，系数在$mathbb{Z}_p$，我们得到了这个多项式的期望根数的估计。特别地，如果系数以相等的概率取值pm 1，则期望的p-adic根的数目收敛于left（p-1
right）/left（p+1
8）$作为多项式的次数趋于$infty$。
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 $left(p-1
ight)/left(p+1
ight)$ as the degree of the polynomial tends to $infty$.