Research Article · Preprint Mathematics
Collision geometry and concentration in polynomial sign recovery
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis · Linear Algebra and Matrix Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords signed measures · polynomial certificates · negative index · Vandermonde matrices · Christoffel functions · local concentration
Scope of the work
Sharp Vandermonde-scale Gaussian minimax laws for fixed negative indices of collapsing atoms, with signed-measure and local-concentration extensions.
Reproduction notes
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.
The manuscript
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The complete manuscript, as published in this edition.
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Author 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.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Collision geometry and concentration in polynomial sign recovery (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/geometry-concentration-polynomial-sign-recovery/v/2
BibTeX
@techreport{luminary_geometry_concentration_polynomial_sign_recovery_v2,
title = {Collision geometry and concentration in polynomial sign recovery},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/geometry-concentration-polynomial-sign-recovery/v/2}
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