We present a finite algorithm for the computation of any moment of the solution (= the characteristic function of $\{p < 1\},$ with $p$ a polynomial, assumed to have isolated (complex) critical points) of the truncated extremal $n$-dimensional $L$-moment problem, linearly in terms of a finite set of generating moments, in the context of dynamic (i.e. time-dependent) moments. We find that a system of such generators is provided by the moments corresponding to a basis for ${\bf R}[x_1,...,x_n]/ I_{\nabla_p},$ where $I_{\nabla_p}$ is the gradient ideal of $p.$ From this, based on a well-known algebraic formalism for asymptotics of the Fourier transform, we obtain computations for the coefficients of the asymptotic expansions for the moments in terms of the generators and the monodromy of $p.$