Oeding's symmetric border-rank conjecture for general monomials

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Let α=(0<α0≤⋯≤αd−1)∈Nd+1\alpha=(0<\alpha_0\leq\cdots\leq\alpha_{d-1})\in\mathbb{N}^{d+1}. For the monomial x0α0⋯xd−1αd−1x_0^{\alpha_0}\cdots x_{d-1}^{\alpha_{d-1}}, its symmetric border rank is the least number of powers of linear forms needed in a limiting symmetric decomposition. Oeding's conjecture. The symmetric border rank is

brk⁡(x0α0⋯xd−1αd−1)=∏i=0d−2(αi+1).\operatorname{brk}(x_0^{\alpha_0}\cdots x_{d-1}^{\alpha_{d-1}})=\prod_{i=0}^{d-2}(\alpha_i+1).

This gives the conjectured symmetric border rank of a general monomial and complements the known formula for its Waring rank. The supplied text does not state whether the conjecture has been resolved.

References

Primary source

Masoud Gharahi, “Classifying Entanglement by Algebraic Geometry”, arXiv:2408.12265 (2025).

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