  We start a systematic study of the topology, geometry and singularities of
the Prony varieties $S_q(\mu)$, defined by the first $q+1$ equations of the
classical Prony system
  $$\sum_{j=1}^d a_j x_j^k = \mu_k, \ k= 0,1,\ldots \ .$$ Prony varieties,
being a generalization of the Vandermonde varieties, introduced in [5,21],
present a significant independent mathematical interest (compare [5,19,21]).
The importance of Prony varieties in the study of the error amplification
patterns in solving Prony system was shown in [1-4,19]. In [19] a survey of
these results was given, from the point of view of Singularity Theory.
  In the present paper we show that for $q\ge d$ the variety $S_q(\mu)$ is
diffeomerphic to an intersection of a certain affine subspace in the space
${\cal V}_d$ of polynomials of degree $d$, with the hyperbolic set $H_d$.
  On the Prony curves $S_{2d-2}$ we study the behavior of the amplitudes $a_j$
as the nodes $x_j$ collide, and the nodes escape to infinity. We discuss the
behavior of the Prony varieties as the right hand side $\mu$ varies, and
possible connections of this problem with J. Mather's result in [23] on
smoothness of solutions in families of linear systems.
