  We study the decomposition of a multivariate Hankel matrix H\_$\sigma$ as a
sum of Hankel matrices of small rank in correlation with the decomposition of
its symbol $\sigma$ as a sum of polynomial-exponential series. We present a new
algorithm to compute the low rank decomposition of the Hankel operator and the
decomposition of its symbol exploiting the properties of the associated
Artinian Gorenstein quotient algebra A\_$\sigma$. A basis of A\_$\sigma$ is
computed from the Singular Value Decomposition of a sub-matrix of the Hankel
matrix H\_$\sigma$. The frequencies and the weights are deduced from the
generalized eigenvectors of pencils of shifted sub-matrices of H $\sigma$.
Explicit formula for the weights in terms of the eigenvectors avoid us to solve
a Vandermonde system. This new method is a multivariate generalization of the
so-called Pencil method for solving Prony-type decomposition problems. We
analyse its numerical behaviour in the presence of noisy input moments, and
describe a rescaling technique which improves the numerical quality of the
reconstruction for frequencies of high amplitudes. We also present a new Newton
iteration, which converges locally to the closest multivariate Hankel matrix of
low rank and show its impact for correcting errors on input moments.
