A Finite-order Characterization of Entrywise Positivity Preservers
Abstract
Fix I = (0,ρ), where 0<ρ≤∞, and let ℙn(I) be the set of positive semidefinite n× n matrices with entries in I. A longstanding problem in matrix theory is to characterize the functions f: I → ℝ for which the entrywise calculus f[A] = [f(aij)]i,j = 1n preserves positive semidefiniteness for all A ∈ ℙn(I). We characterize these functions exactly: if f∈ C2n−2(I) and ℰ = xd/dx, then this holds if and only if f(k)(x)≥ 0 (0≤ k≤ n−1) and [ℰi+jf(x)]i,j = 0n−1⪰ 0 for every x∈ I. Regularization then removes all a priori smoothness: for n≥2, every preserver belongs to C2n−4(I) and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald--Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton--Guillot--Khare--Putinar and Khare--Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside I can detect positivity preservation on ℙn(I).
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BibTeX
@misc{bouthat2026finiteorder,
title = {{A Finite-order Characterization of Entrywise Positivity Preservers}},
author = {Ludovick Bouthat and Dominique Guillot},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.15904},
eprint = {2608.15904},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15904},
}
RIS
TY - UNPB
TI - A Finite-order Characterization of Entrywise Positivity Preservers
AU - Ludovick Bouthat
AU - Dominique Guillot
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.15904
UR - https://arxiv.org/abs/2608.15904
AB - Fix I = (0,ρ), where 0<ρ≤∞, and let ℙ_n(I) be the set of positive semidefinite n× n matrices with entries in I. A longstanding problem in matrix theory is to characterize the functions f: I → ℝ for which the entrywise calculus f[A] = [f(a_ij)]_i,j = 1^n preserves positive semidefiniteness for all A ∈ ℙ_n(I). We characterize these functions exactly: if f∈ C^2n−2(I) and ℰ = xd/dx, then this holds if and only if f^(k)(x)≥ 0 (0≤ k≤ n−1) and [ℰ^i+jf(x)]_i,j = 0^n−1⪰ 0 for every x∈ I. Regularization then removes all a priori smoothness: for n≥2, every preserver belongs to C^2n−4(I) and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald--Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton--Guillot--Khare--Putinar and Khare--Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside I can detect positivity preservation on ℙ_n(I).
ER -