The complex zeros of random polynomials

Larry Shepp, Robert J. Vanderbei

Research output: Contribution to journalArticlepeer-review

89 Scopus citations


Mark Kac gave an explicit formula for the expectation of the number, vn(Ω), of zeros of a random polynomial, in any measurable subset Ci of the reals. Here, ηo,., ηn-1are independent standard normal random variables. In fact, for each n > 1, he obtained an explicit intensity function g for which Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Ω of the complex plane ℂ. Namely, we show that where hnis an explicit intensity function. We also study the asymptotics of hnshowing that for large n its mass lies close to, and is uniformly distributed around, the unit circle.

Original languageEnglish (US)
Pages (from-to)4365-4384
Number of pages20
JournalTransactions of the American Mathematical Society
Issue number11
StatePublished - Nov 1995

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


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