Spectral Equality for Tropical Totally Nonnegative Matrices: Dominant Cancellation and Newton Polygons
Abstract
Let A be a totally nonnegative matrix over a real closed non-Archimedean valued field, and suppose that its valuation A is tropically totally nonnegative. Although the valuations of the classical eigenvalues are weakly majorized by the tropical eigenvalues of A, equality can fail through cancellation. We give a necessary and sufficient condition for equality. The only obstruction is dominant cancellation in the principal minors that give the essential coefficients of the tropical characteristic polynomial. In other words, equality holds exactly when the classical and tropical Newton polygons agree at the breakpoints determined by the diagonal entries. We describe the exceptional lifts by explicit polynomial equations in the leading coefficients. We also study these equations on each irreducible component of the admissible parameter space. Finally, we give a complete 3× 3 criterion and a family that contains both ordinary lifts with spectral equality and exceptional totally nonnegative lifts without it.
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BibTeX
@misc{dahaki2026spectral,
title = {{Spectral Equality for Tropical Totally Nonnegative Matrices: Dominant Cancellation and Newton Polygons}},
author = {Mohammadrahim Saeidi Dahaki and Seyed Mahmoud Manjegani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06887},
eprint = {2610.06887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06887},
}
RIS
TY - UNPB
TI - Spectral Equality for Tropical Totally Nonnegative Matrices: Dominant Cancellation and Newton Polygons
AU - Mohammadrahim Saeidi Dahaki
AU - Seyed Mahmoud Manjegani
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.06887
UR - https://arxiv.org/abs/2610.06887
AB - Let A be a totally nonnegative matrix over a real closed non-Archimedean valued field, and suppose that its valuation A is tropically totally nonnegative. Although the valuations of the classical eigenvalues are weakly majorized by the tropical eigenvalues of A, equality can fail through cancellation. We give a necessary and sufficient condition for equality. The only obstruction is dominant cancellation in the principal minors that give the essential coefficients of the tropical characteristic polynomial. In other words, equality holds exactly when the classical and tropical Newton polygons agree at the breakpoints determined by the diagonal entries. We describe the exceptional lifts by explicit polynomial equations in the leading coefficients. We also study these equations on each irreducible component of the admissible parameter space. Finally, we give a complete 3× 3 criterion and a family that contains both ordinary lifts with spectral equality and exceptional totally nonnegative lifts without it.
ER -