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Rank and Border Rank Additivity of Complex Tensors with a Hankel Summand
Abstract
We prove that the CP rank of a complex Hankel tensor of order m≥3 is additive with that of every complex tensor of the same order. Together with the known equality of CP and symmetric ranks for complex Hankel tensors, this yields symmetric rank additivity whenever the other summand has equal CP and symmetric ranks. We also prove that ordinary and symmetric border ranks coincide and are additive on direct sums of complex Hankel tensors of any common order at least three. Their common value is the sum of the middle binary catalecticant ranks. For order three, we further prove border-rank additivity when the other summand has border rank at most three, or is symmetric with at most five essential variables.
Related
- Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces
- The Operator Systems of Hankel Matrices and Complex Polynomials
- Estimates for a Parameterized Hankel Functional of Starlike Mappings of Order α and Quasi-Convex Mappings of Type B in Complex Banach Spaces
- Bergman Kernels and Hankel Operators on Weighted Fock Spaces in Several Complex Variables
- The Riemann-Siegel remainder as a fractional summand
BibTeX
@misc{li2026rank,
title = {{Rank and Border Rank Additivity of Complex Tensors with a Hankel Summand}},
author = {Jianze Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24982},
eprint = {2610.24982},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24982},
}
RIS
TY - UNPB
TI - Rank and Border Rank Additivity of Complex Tensors with a Hankel Summand
AU - Jianze Li
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24982
UR - https://arxiv.org/abs/2610.24982
AB - We prove that the CP rank of a complex Hankel tensor of order m≥3 is additive with that of every complex tensor of the same order. Together with the known equality of CP and symmetric ranks for complex Hankel tensors, this yields symmetric rank additivity whenever the other summand has equal CP and symmetric ranks. We also prove that ordinary and symmetric border ranks coincide and are additive on direct sums of complex Hankel tensors of any common order at least three. Their common value is the sum of the middle binary catalecticant ranks. For order three, we further prove border-rank additivity when the other summand has border rank at most three, or is symmetric with at most five essential variables.
ER -