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How Many Zeros of a Random Polynomial are Real?

Year of Award: 1998

Award: Chauvenet

Publication Information: Bulletin of the AMS, vol. 32 (1995), pp.1-37


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About the Authors

Alan Edelman, a lifelong lover of linear algebra, recently celebrated his fortieth birthday with a cake bought by his wife with icing Ax = λx. He earned a B.S. and M.S. at Yale University and a Ph.D. at MIT, where he has been teaching since 1993.  He has also spent three years at UC Berkeley. Edelman's thesis on random matrices was supervised by Nick Trefethen (and a book is in progress). Random matrices, parallel computing, and numerical linear algebra remain passions. He has shared a Gordon Bell Prize for parallel computing, a Householder Prize for numerical linear algebra, and the MAA's Chauvenet Prize.

Eric Kostlan received his PhD at the University of California, Berkeley in 1985. At the time of the writing of the article, he was on the faculty at Kapiolani Community College. As of 2010, he is an  education specialist at Cisco Systems.

Author (old format): 
Alan Edelman, Eric Kostlan
Alan Edelman, Eric Kostlan
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Publication Date: 
Friday, October 10, 2008
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Edelman and Kostlan 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, ... , t") projected onto the surface of the unit sphere, divided by n . The probability density of the real zeros is proportional to how fast this curve is traced out.