Geometry and Singularities of Prony varieties
This work provides foundational geometric insights into Prony varieties, which are important for understanding error amplification in solving Prony systems, but the results are primarily theoretical and incremental.
The paper studies the geometry and singularities of Prony varieties, which arise from the Prony system of equations. It shows that for q ≥ d, the variety is diffeomorphic to an intersection of an affine subspace with the hyperbolic set, and analyzes amplitude behavior as nodes collide or escape to infinity.
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.