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How many zeros of a random polynomial are real? (English) Zbl 0820.34038
The authors provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. They show that the expected number of real zeros is simply the length of the moment curve $$(1,t,\dots, t^ n)$$ projected onto the surface of the unit sphere, devided by $$\pi$$. Also the authors consider random $$n$$th degree polynomials with independent normally distributed coefficients, each with mean zero, but with the variance of the $$i$$th coefficient equal to $$\left(\begin{smallmatrix} n\\ i\end{smallmatrix}\right)$$.
It has $$E_ n= \sqrt n$$ real zeros on average. Then the authors relax Kac’s assumptions by considering a variety of random sums, series and distributions.

##### MSC:
 34F05 Ordinary differential equations and systems with randomness 30B20 Random power series in one complex variable 60D05 Geometric probability and stochastic geometry
##### MathOverflow Questions:
Sign-oscillations for power series with random coefficients
testmatrix
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