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# Methods and Applications of Analysis

## Volume 25 (2018)

### Number 3

### In Memory of Professor John N. Mather: Part 1 of 3

Guest Editors: Sen Hu, University of Science and Technology, China; Stanisław Janeczko, Polish Academy of Sciences, Poland; Stephen S.-T. Yau, Tsinghua University, China; and Huaiqing Zuo, Tsinghua University, China.

### Geometry and singularities of Prony varieties

Pages: 257 – 276

DOI: https://dx.doi.org/10.4310/MAA.2018.v25.n3.a5

#### Authors

#### Abstract

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 $\sum^{d}_{j=1} a_j x^k_j = \mu_k , k = 0, 1, \dotsc$

Prony varieties, being a generalization of the Vandermonde varieties, introduced in [5,22], present a significant independent mathematical interest (compare [5, 20, 22]). The importance of Prony varieties in the study of the error amplification patterns in solving Prony system was shown in [1–4, 20]. In [20] a survey of these results was given, from the point of view of Singularity Theory.

In the present paper we show that for $q \geq d$ the variety $S_q (\mu)$ is diffeomerphic to the intersection of a certain affine subspace in the space $\mathcal{V}_d$ of polynomials of degree $d$, with the hyperbolic set $H_d$.

In case of 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 [24] on smoothness of solutions in families of linear systems.

#### Keywords

Prony system, spike-train signals, Prony and Vandermonde varieties, nodal collision singularities

#### 2010 Mathematics Subject Classification

32S05, 58K20, 65H10, 94A12

Received 4 June 2018

Accepted 12 October 2018

Published 1 November 2019