Geometry and Concentration in Polynomial Sign Recovery

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. Classical orthogonal-polynomial and Christoffel estimates are separated from the signed-measure minimax deductions; exact rational finite controls accompany the proofs.