Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs. We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion. As a first application, we use geometric rank to prove a tight upper bound on the (border) subrank of the matrix multiplication tensors, matching Strassen's well-known lower bound from 1987.
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Towards a geometric approach to Strassen’s asymptotic rank conjecture
We make a first geometric study of three varieties inCm⊗Cm⊗Cm (for eachm), including the
Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our
motivation is to develop a geometric framework for Strassen’s asymptotic rank conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.
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- Award ID(s):
- 1814254
- NSF-PAR ID:
- 10226259
- Date Published:
- Journal Name:
- Collectanea mathematica
- Volume:
- 72
- Issue:
- 1
- ISSN:
- 2038-4815
- Page Range / eLocation ID:
- 63–86
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s13348-020-00280-8
