The expected number of zeros of a random system of $p$-adic polynomials
The expected number of zeros of a random system of $p$-adic polynomials
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$p$-进多项式随机系统的预期零个数
DOI：
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发表时间：
2006
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通讯作者：
S. Evans
中科院分区：
文献类型：
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作者：
S. Evans
我们研究了$p$进数上$d$变量的随机族$d$多项式的同时零点。对于一组自然模型，我们获得了一个显式常数，用于表示$p$进数整数的$d$ -fold笛卡尔积中的期望零数。考虑到每个变量出现的最大程度为$N$的模型，对于最简单的模型，该期望值为$$ p^{d lfloor log_p N
floor} left(1 + p^{-1} + p^{-2} + cdots + p^{-d}
ight)^{-1} $$。
We study the simultaneous zeros of a random family of $d$ polynomials in $d$ variables over the $p$-adic numbers. For a family of natural models, we obtain an explicit constant for the expected number of zeros that lie in the $d$-fold Cartesian product of the $p$-adic integers. Considering models in which the maximum degree that each variable appears is $N$, this expected value is $$ p^{d lfloor log_p N
floor} left(1 + p^{-1} + p^{-2} + cdots + p^{-d}
ight)^{-1} $$ for the simplest such model.