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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
Software:
testmatrix
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