随机系数代数方程平均实根个数的估计

The Estimation of the Average Number of Real Roots of a Random Algebraic Equation

考虑随机系数代数方程Ｆｎ（ｗ，ｔ）＝０（ｗ）＋１（ｗ）ｔ＋…＋ｎ－１（ｗ）ｔｎ－１＝０，其中ｉ（ｗ）（ｉ＝０，１，…，ｎ－１）为独立且服从标准正态分布的随机变量。令ＥＮＦ（ｗ）表示Ｆｎ（ｗ，ｔ）的平均实根个数。本文证明了ＥＮＦ（ｗ）＜２πｌｎｎ－２ｎπ＋１．２３７２７７１。

Let F n(w,t)= 0 (w)+ 1 (w)t+…+ n-1 (w)t n-1 =0 be a random algebraic equation where  i (w)(i=0, 1,…,n-1) is an indenpendent Gaussian random variable with mean o and deviation 1. Let EN F(w) be the average number of real roots of F n(w,t). The paper proves for all n>1, EN F(w)<2π ln n-2nπ+1.2372771.