Roots of random polynomials

  • Van Vu | Yale University

Consider a polynomial Pn = c0 + c1x + … + cn xn of degree n whose coefficients ci are iid real random variables with mean 0. In the late 1930s and eary 1940s, Littlewood and Offord surprised the mathematics community by showing that the number of real roots of Pn is quite small, somewhere between log n/log log n and log2 n. Subsequent deep works of Littlewood-Offord, Erdos-Offord, Turan, Kac, Stevens, Ibragimov-Maslova and others showed that the expectation of the number of real roots is 2/pi log n + o(log n).

In the case when the ci are gaussian, it has been showed by many researchers that the error term o(log n) is actually O(1), giving a very precise result. Nothing close to this has been known for other variables (such as +-1). The current best estimate from Ibragimov and Maslova’s paper is O(log1/2 n log log n).

We are going to present a new approach to the studies of roots of random polynomials (complex and real alike). As a consequence, we can show that the error term is O(1) (the same as is in the Gaussian case) for general non-Gaussian random polynomials.

The talk is based on joined works with T. Tao (UCLA) , and Hoi Nguyen (OSU) and Oanh Nguyen (Yale).

Speaker Details

Van H. Vu is the Percey F. Smith Professor of Mathematics at Yale University. He received his Ph.D. at Yale University in 1998 under the direction of László Lovász, worked as a postdoc at the Institute for Advanced Studies and Microsoft Research and prior to moving to Yale held professor positions at UCSD and Rutgers University. He is the recipient of an A. Sloan Fellowship (2002), an NSF CAREER Award (2003), the George Pólya Prize (2008) and the Fulkerson Prize (2012).

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      Jeff Running

Series: Microsoft Research Talks