Roots of random polynomials
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Consider a polynomial P n = c 0 + c1x + ... + c n x n of degree n whose coefficients c i 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 P n is quite small, somewhere between log n/log log n and log 2 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 c i 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(log 1/2 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).
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