Research Article · Preprint Mathematics
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
- 1 · 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/(8cd²). 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 exp(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, where a matching lower bound remains open in the present treatment. 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
Fixed-index minimax recovery for collapsing negative packets, and sharp gapless local-mass degree orders in the declared tail ranges. No growing-index uniformity, exact gapless leading constants, or theta positivity theorem is claimed.
Reproduction notes
The source package contains editable LaTeX, references and a standard-library exact rational Python verifier. Run python verify_finite_controls.py and python -O verify_finite_controls.py. Finite controls do not replace the analytic proofs.
The manuscript
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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). Geometry and Concentration in Polynomial Sign Recovery (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/geometry-concentration-polynomial-sign-recovery/v/1
BibTeX
@techreport{luminary_geometry_concentration_polynomial_sign_recovery_v1,
title = {Geometry and Concentration in Polynomial Sign Recovery},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/geometry-concentration-polynomial-sign-recovery/v/1}
}Loading discussion…