How many zeros of a random polynomial are real?

Authors:
Alan Edelman and Eric Kostlan

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 projected onto the surface of the unit sphere, divided by . 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0273-0979-1995-00571-9

Keywords:
Random polynomials,
Buffon needle problem,
integral geometry,
random power series,
random matrices

Article copyright:
© Copyright 1995
American Mathematical Society