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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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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.
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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