## How many zeros of a random polynomial are real?

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- by Alan Edelman and Eric Kostlan PDF
- Bull. Amer. Math. Soc.
**32**(1995), 1-37 Request permission

## Abstract:

We 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. We show that the expected number of real zeros is simply the length of the moment curve $(1, t, \ldots ,t^{n})$ projected onto the surface of the unit sphere, divided by $\pi$. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.## References

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## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Bull. Amer. Math. Soc.
**32**(1995), 1-37 - MSC: Primary 60G99; Secondary 30B20, 42A05
- DOI: https://doi.org/10.1090/S0273-0979-1995-00571-9
- MathSciNet review: 1290398