On Vandermonde Varieties

Fröberg, Ralf; Shapiro, Boris

doi:10.7146/math.scand.a-24185

Cited by 13 publications

(20 citation statements)

References 8 publications

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“…. , a d ) the signal Prony variety, intersected with A × P X d , coincides with the corresponding Vandermonde variety, as defined in [4,27,32]. We believe that the results of these papers may be important in study of Prony varieties, and we give more detail in [29].…”

Section: Prony Varietiesmentioning

confidence: 93%

Accuracy of noisy Spike-Train Reconstruction: a Singularity Theory point of view

Goldman¹,

Salman²,

Yomdin³

2018

jsing

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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) = d j=1 ajδ(x − xj) from the noisy moment measurements. We provide an overview of some recent results of [1][2][3]7,8,10,11,29,53] on the "geometry of the error amplification" in the reconstruction process, in situations where the nodes xj 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.

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Section: Prony Varietiesmentioning

confidence: 93%

Accuracy of noisy Spike-Train Reconstruction: a Singularity Theory point of view

Goldman¹,

Salman²,

Yomdin³

2018

jsing

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“…The ramification part R was studied by Fröberg and Shapiro [5], but they reached only partial results. Notably, it remains an open problem to find the dimension of R .…”

Section: Imagesmentioning

confidence: 99%

“…Assuming a 1 = g = 1, the first interesting case is n = 3, m = 5. This was studied in [5]. Proving the dimension of R to be the expected one is equivalent to showing that three complete homogeneous polynomials form a regular sequence.…”

Section: Imagesmentioning

confidence: 99%

Recovery from Power Sums

Melánová¹,

Sturmfels²,

Winter³

2021

Preprint

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We study the problem of recovering a collection of n numbers from the evaluation of m power sums. This yields a system of polynomial equations, which can be underconstrained (m < n), square (m = n), or overconstrained (m > n). Fibers and images of power sum maps are explored in all three regimes, and in settings that range from complex and projective to real and positive. This involves surprising deviations from the Bézout bound, and the recovery of vectors from length measurements by p-norms.

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“…, q k ) which are of codimension one. See [7] for some calculations regarding such varieties, also in the case of non quadratic Vandermonde-type matrices.…”

Section: Regularity and Relations To Representation Theorymentioning

confidence: 99%