随机系数代数方程平均实根个数的估计

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

考虑随机系数代数方程Fn(ω ,t) = 0 (ω) + 1(ω)t +… + n - 1(ω)tn- 1=0 ,其中 i(ω)服从标准正态分布且相互独立 (i=0 ,1,… ,n - 1) ,令ENF(ω)表示Fn(ω ,t)的平均实根个数 ,证明了ENF(ω) <2πlnn - 2nπ- 1806 31πn2 +1.2 372 771,且改进了骆振华。

Let F n(ω,t)= 0(ω)+ 1(ω)t+…+ n-1 (ω)t n-1 =0 be a random algebraic equation where i(ω)(i=0,1,…,n-1) is an independent Gaussian random variable with mean 0 and deviation 1.Let EN F(ω) be the number of real roots of F n(ω,t).This paper proves for all n>1,EN F(ω)<2πlnn-2nπ-180631πn 2+1.2372771.