  Prony mapping provides the global solution of the Prony system of equations
\[ \Sigma_{i=1}^{n}A_{i}x_{i}^{k}=m_{k},\ k=0,1,...,2n-1. \] This system
appears in numerous theoretical and applied problems arising in Signal
Reconstruction. The simplest example is the problem of reconstruction of linear
combination of $\delta$-functions of the form
$g(x)=\sum_{i=1}^{n}a_{i}\delta(x-x_{i})$, with the unknown parameters $a_{i},\
x_{i},\ i=1,...,n,$ from the "moment measurements" $m_{k}=\int x^{k}g(x)dx.$
  Global solution of the Prony system, i.e. inversion of the Prony mapping,
encounters several types of singularities. One of the most important ones is a
collision of some of the points $x_{i}.$ The investigation of this type of
singularities has been started in \cite{yom2009Singularities} where the role of
finite differences was demonstrated.
  In the present paper we study this and other types of singularities of the
Prony mapping, and describe its global geometry. We show, in particular, close
connections of the Prony mapping with the "Vieta mapping" expressing the
coefficients of a polynomial through its roots, and with hyperbolic polynomials
and "Vandermonde mapping" studied by V. Arnold.
