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On the average number of real roots of a random algebraic equation. (English) Zbl 0609.60074
Let \(f(x)=a_ 0+a_ 1x+...+a_ nx^ n\) be a random algebraic polynomial with independent random coefficients where \(a_ i\in N(0,1)\), \(i=0,1,2,...,n\). In this paper, the author estimates the average number of real roots of the equation \(f(x)=k\), for the following cases:
(i) If \(k^ 2/n\) tends to zero as n tends to infinity then EN(-1,1)\(\sim (1/\pi)\log (n/k^ 2)\) and \(EN(-\infty,-1)=EN(1,\infty)\sim (2\pi)^{- 1}\log n.\)
(ii) If \(k^ 2/n\) tends to a nonzero positive constant then EN(- \(\infty,\infty)\sim (1/\pi)\log n.\)
This problem can also be reformulated as follows: F(x)\(\equiv f(x)- k\equiv (a_ 0-k)+a_ 1x+a_ 2x^ 2+...+a_ nx^ n=0\) where \(a_ 0-k,a_ 1,a_ 2,...,a_ n\) are independent normal random variables with \(a_ 0-k\in N(-k,1)\) and \(a_ i\in N(0,1)\), \(i=1,2,...,n\). Then the average number of real zeros of \(F(x)=0\) is (i) and (ii).
On this line, the reviewer feels that one can study the following problem also: let g(x)\(\equiv a_ 0+a_ 1x+...+a_ nx^ n\) where \(a_ i\), \(i=0,1,...,n\) are independent normal random variables and \(a_ i\in N(0,1)\), \(i=0,1,2,...,j-1,j+1,...,n\), \(a_ j\in N(-k,1)\). Then what is the average number of real zeros of \(g(x)=0\).
Reviewer: M.Sambandham

MSC:
60H99 Stochastic analysis
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