Real roots of random polynomials: expectation and repulsion

Yen Do, Hoi Nguyen, Van Vu

Let $P_n(x)={\sum }_{i=0}^n{\xi }_i{x}^i$ be a Kac random polynomial where the coefficients ${\xi }_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitati…