Yehonatan Salman

Gil Goldman, Yehonatan Salman, Yosef Yomdin

We start a systematic study of the topology, geometry and singularities of the Prony varieties $S_q(μ)$, defined by the first $q+1$ equations of the classical Prony system $$\sum_{j=1}^d a_j x_j^k = μ_k, \ k= 0,1,\ldots \ .$$ 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 $S_q(μ)$ 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 $μ$ varies, and possible connections of this problem with J. Mather's result in [23] on smoothness of solutions in families of linear systems.

Gil Goldman, Yehonatan Salman, Yosef Yomdin

This is a survey paper discussing one specific (and classical) system of algebraic equations - the so called "Prony system". We provide a short overview of its unusually wide connections with many different fields of Mathematics, stressing the role of Singularity Theory. We reformulate Prony System as the problem of reconstruction of "Spike-train" signals of the form $F(x)=\sum_{j=1}^d a_jδ(x-x_j)$ from the noisy moment measurements. We provide an overview of some recent results of [1-3, 6, 8, 9, 11, 12, 5] on the "geometry of the error amplification" in the reconstruction process, in situations where the nodes $x_j$ near-collide. Some algebraic-geometric structures, underlying the error amplification, are described (Prony, Vieta, and Hankel mappings, Prony varieties), as well as their connection with Vandermonde mappings and varieties. Our main goal is to present some promising fields of possible applications of Singulary Theory.