Geometry and Singularities of Prony varieties

TL;DR

This paper systematically studies the topology, geometry, and singularities of Prony varieties, revealing their structure, behavior under parameter variation, and implications for error analysis in solving Prony systems.

Contribution

It establishes the diffeomorphism of Prony varieties to intersections with hyperbolic sets and analyzes node collision and divergence behaviors.

Findings

Prony varieties are diffeomorphic to intersections with hyperbolic sets for q ≥ d.

Behavior of amplitudes and nodes analyzed as nodes collide or escape to infinity.

Insights into the variation of Prony varieties with respect to the data μ.

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 $$\sum_{j=1}^d a_j x_j^k = \mu_k, \ k= 0,1,\ldots \ .$$ 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.