  We study the conditional distribution of zeros of a Gaussian system of random
polynomials (and more generally, holomorphic sections), given that the
polynomials or sections vanish at a point p (or a fixed finite set of points).
The conditional distribution is analogous to the pair correlation function of
zeros, but we show that it has quite a different small distance behavior. In
particular, the conditional distribution does not exhibit repulsion of zeros in
dimension one. To prove this, we give universal scaling asymptotics for the
conditional zero distribution around p. The key tool is the conditional Szego
kernel and its scaling asymptotics.
