Let pN be a degree N random polynomial in a complex variable. We obtain an explicit asymptotic formula for the covariance between the counting measures of its zeros and critical points, which we denote CovN(z,w). This formula shows that the correlation between a zero at z and a critical point at w is short range, decaying like e-N|z-w|2. With |z-w| on the order of N-1/2, however, CovN(z,w) is sharply peaked near z= w, causing zeros and critical points to appear in pairs. We prove bounds on the expected distance and angular dependence between a critical point and its paired zero.
All Science Journal Classification (ASJC) codes