Oeding's symmetric border-rank conjecture for general monomials

From papers

Let α=(0<α0αd1)\mathbbmNd+1\alpha=(0<\alpha_0\leq\cdots\leq\alpha_{d-1})\in\mathbbm{N}^{d+1}. For the monomial x0α0xd1αd1x_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(x0α0xd1αd1)=i=0d2(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.

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Sources & referencesView supporting material

Primary source

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

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