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Mathematics > Algebraic Geometry

arXiv:2201.03615 (math)
[Submitted on 10 Jan 2022 (v1), last revised 28 Nov 2022 (this version, v2)]

Title:Geometric Rank and Linear Determinantal Varieties

Authors:Runshi Geng
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Abstract:There are close relations between tripartite tensors with bounded geometric ranks and linear determinantal varieties with bounded codimensions. We study linear determinantal varieties with bounded codimensions, and prove upper bounds of the dimensions of the ambient spaces. Using those results, we classify tensors with geometric rank 3, find upper bounds of multilinear ranks of primitive tensors with geometric rank 4, and prove the existence of such upper bounds in general. We extend results of tripartite tensors to n-part tensors, showing the equivalence between geometric rank 1 and partition rank 1.
Subjects: Algebraic Geometry (math.AG); Computational Complexity (cs.CC)
Cite as: arXiv:2201.03615 [math.AG]
  (or arXiv:2201.03615v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.2201.03615

Submission history

From: Runshi Geng [view email]
[v1] Mon, 10 Jan 2022 19:51:42 UTC (23 KB)
[v2] Mon, 28 Nov 2022 01:40:55 UTC (25 KB)
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