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arXiv · 2602.11309

Cactus barriers

Jarosław Buczyński
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Abstract

Determinantal methods for bounding the rank and border rank of tensors or polynomials are subject to a major barrier. For instance, it is known that using determinantal methods one cannot prove a lower bound for the border rank of a 3-way tensor of size m in each direction that exceeds 6m-4. We explain the precise geometric reason for this number (and analogous bounds in more general tensor spaces) using cactus varieties and, more generally, scheme theoretic methods in algebraic geometry.

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Jarosław Buczyński. 2026-02-11. Cactus barriers. https://arxiv.org/abs/2602.11309

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