Estimated transversality and rational maps

In this paper, we address a question of Donaldson's on the best estimate that can be achieved for the transversality of an asymptotically holomorphic sequence of sections of increasing powers of a line bundle over an integral symplectic manifold. More specifically, we find an upper bound for the transversality of n such sequences of sections over a 2n-dimensional symplectic manifold. In the simplest case of S^2, we also relate the problem to a well known question in potential theory (namely, that of finding logarithmic equilibrium points), thus establishing an experimental lower bound for the transversality.