| Luo Zhenhua.[J].数学年刊A辑,1980,1(3-4):541~544 |
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| THE AVERAGE NUMBER OF REAL ROOTS OF ARANDOM ALGEBRAIC EQUATION |
| Received:December 21, 1979 Revised:March 22, 1980 |
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| The average number of real roots of the random algebraio equation \[{F_n}(\omega ,t) = {a_0}(\omega ) + {a_2}(\omega )t + \cdots + {a_n}(\omega ){t^{n - 1}} = 0\] has been estimated by Kao, M.[5] for the case where
the \({a_i}(\omega ){\kern 1pt} {\kern 1pt} (i = 0,1, \cdots ,n - 1)\) are indenpendent Gaussian random variables with mean
0 and standard deviation 1. Let \(E{N_F}(\omega )\) be the average： aiumber of real roots of \({F_n}(\omega ,t)\) , Kao's main result is \[E{N_F}(\omega ) \le \frac{2}{\pi }{\rm{In}}n + \frac{{14}}{\pi }\]
Later in (8), Stevens obtained
\[\frac{2}{\pi }{\rm{In}}n - 0.6 < E{N_F}(\omega ) < \frac{2}{\pi }{\rm{In}}n + 1.4\]. The purpose of this paper is to prove the following theorem. Theorem. Let \[{F_n}(\omega ,t) = {a_0}(\omega ) + {a_2}(\omega )t + \cdots + {a_n}(\omega ){t^{n - 1}} = 0\] be a random algebraic equation where \({a_i}(\omega ){\kern 1pt} {\kern 1pt} (i = 0,1, \cdots ,n - 1)\) are indenpendent Gaussian random variables with mean 0 and standard deviation 1, Then for all \(n \ge 1\), \[\frac{2}{\pi }{\rm{In}}n \le E{N_F}(\omega ) \le \frac{2}{\pi }{\rm{In}}n + 1.2372771\]. |
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