Real zeroes of random polynomials. II: Descartes' rule of signs and anti-concentration on the symmetric group (Q2408032)

Based on Descartes' rule of signs and a new anti-concentration inequality for the symmetric group, the author contributes to the problem of finding bounds for the expected number of real roots of random polynomials \(P\). Specifically, assuming that \(P\) has exchangeable random coefficients, he proves that the upper bound does not grow faster than the logarithm of the degree \(P\).