How many real zeroes does a random polynomial have ? (Talk 2)

Sept 20, 1.45pm, LOM 215. (We start 15’ early due to a faculty meeting).

Consider a polynomial $P_n =c_0 + c_1x +…c+n x$ of degree n whose coefficients $c_i$ are real random variables. The problem of determining $N_n$, the number of real zeroesof $P_n$ goes back to Waring (1789), and has become popular since the work of Bloch-Polya and Littlewood-Offord in the 1940s. Deep works of Littlewood-Offord, Erdos-Offord, Kac, Stevens, Ibragimov-Maslova, Edelman-Kostlan, and many others give us a good understanding of $N_n$ in the case $c_i$ are iid random variables with mean $0$ and variance $1$ (see John Baez’s Lord of the Ring beautiful picture on http://math.ucr.edu/home/baez/). However, much less is known for the all other cases, when the $c_i$ have different variances and/or are dependent (a good example is the characteristic polynomial of a random matrix).

In the first talk, I give a brief survey on the state of the art of the problem, and introduce a new approach that will lead to the solution for a general set of random variables $c_i$. In the this talk, I will describe the main ideas behind the proof, most of which are combinatorial in nature.

Joint work with T. Tao.