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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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