Positivity preservers over finite fields II
Abstract
We say that a matrix over a finite field 𝔽q is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in 𝔽q. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on Mn(𝔽q) were classified in every case except when n=2, q≡1pmod4, and q is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension n≥2. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension n≥2, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.
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BibTeX
@misc{guillot2026positivity,
title = {{Positivity preservers over finite fields II}},
author = {Dominique Guillot and Himanshu Gupta and Prateek Kumar Vishwakarma and Chi Hoi Yip},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18978},
eprint = {2608.18978},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18978},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positivity preservers over finite fields II
AU - Dominique Guillot
AU - Himanshu Gupta
AU - Prateek Kumar Vishwakarma
AU - Chi Hoi Yip
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2608.18978
UR - https://arxiv.org/abs/2608.18978
AB - We say that a matrix over a finite field 𝔽_q is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in 𝔽_q. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on M_n(𝔽_q) were classified in every case except when n=2, q≡1pmod4, and q is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension n≥2. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension n≥2, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.
ER -