Collision geometry and concentration in polynomial sign recovery
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Abstract
We study the degree needed for polynomial squares to recover negative directions of a signed measure. For a fixed number of negative atoms collapsing at arbitrary relative rates, we determine the Gaussian minimax degree of every fixed negative index from the corresponding Vandermonde eigenvalue, with leading coefficient 1/(8cd^2). A discrete monic minimax formulation identifies the separate geometric scales, and an exponential-tail counterpart determines their logarithmic recovery exponent. A Gaussian extension permits a shrinking negative probability measure with continuous or mixed support. Without a support gap, a local mass bound of order Rr^q yields sharp degree orders R^(p/[q(p-1)]) for tails e^(c|x|^p) with p≥2, and a sharp logarithmic degree order R^(1/q) at p=1, for every q>0. We give constructive upper bounds for 1<p<2. Classical orthogonal-polynomial and Christoffel estimates are separated from the signed-measure minimax deductions; exact rational finite controls accompany the proofs.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Theory · Classical and Real Analysis
- Keywords
- signed measures, polynomial certificates, negative index, Vandermonde matrices, Christoffel functions, local concentration
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Sharp Vandermonde-scale Gaussian minimax laws for fixed negative indices of collapsing atoms, with signed-measure and local-concentration extensions.
Disclosures
AI tools assisted with mathematical development, source preparation and verification. The author is responsible for the statements and proofs. The author declares no competing interests.
Reproduction
22-page manuscript and 10 source-package members. Compile main.tex and its included sections. Standard-library python verify_finite_controls.py, also under -O, checks the exact finite Vandermonde, Gram and concentration examples; the asymptotic laws are proved analytically.