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\section{Introduction}  Consider a signal system  $$  S =  \{ (a_j, x_j) \} $$  where a_j are signals amplitudes and x_j are signals positions. $\{a_j\}$ and $\{x_j\}$ may be represent by real or complex numbers.  \par Define the Prony map as a map $\{ (a_j, x_j) \} \mapsto \{m_k\}$ of the following form:  $$ 
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  $\{m_k\}$ are moments of the signal system.  There is a special case of the Prony system:  $$  S = \{ (a_j, \mu_j) \}; \quad m_k = \sum_{j=1}^N a_j e^{-2\pi i \mu_j k \delta}  $$  where $\delta \in \mathbb{R}_+$ is a parameter. We say that it is the trigonometric Prony map. The general Prony map can be reduced to the trigonometric Prony map by $x_j = e^{-2\pi i \mu_j \delta}$ if $|x_j| = 1$.  In the practical application, we usually know only moments of a source system and we want to reconstruct original amplitudes and positions of signals. This means that we should build the inverse Prony map. Unfortunately, measurements of moments frequently include some noise. A feature of the inverse Prony map is the following: even small noise can produce a significant reconstruction error. Therefore, it is very important to understand geometry of the Prony map and the inverse Prony map.  See also: \cite{2015arXiv150206932A}, \cite{azais_spike}, \cite{batenkov_numerical_2014}, \cite{batenkov_accurate_2014}, \cite{Bat.Sar.Yom}, \cite{Bat.Yom2}, \cite{Bat.Yom.Sampta13}, \cite{Bat.Yom1}, \cite{candes_towards_2014}, \cite{candes_super-resolution_2013}, \cite{demanet_super-resolution_2013}, \cite{donoho_superresolution_1992}, \cite{Don1}, \cite{duval_exact_2013}, \cite{fernandez-granda_support_2013}, \cite{heckel_super-resolution_2014}, \cite{Lev.Ful}, \cite{liao_music_2014}, \cite{McC}, \cite{Min.Kaw.Min}, \cite{moitra_threshold_2014}, \cite{Ode.Bar.Pis}, \cite{Sle}, \cite{Yom2}, \cite{Yom1}, \cite{Dem.Ngu}, \cite{Mor.Can}. 
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