Counting decomposable multivariate polynomials

A polynomial f (multivariate over a field) is decomposable if f=g °h{f=g \circ h} with g univariate of degree at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and an approximation to their number over a finite field. The relative error in our approximations is exponentially decaying in the input size.