The next gap in the subrank of 3-tensors (2307.06115v2)
Abstract: Recent works of Costa-Dalai, Christandl-Gesmundo-Zuiddam, Blatter-Draisma-Rupniewski, and Bri\"et-Christandl-Leigh-Shpilka-Zuiddam have investigated notions of discreteness and gaps in the possible values that asymptotic tensor ranks can take. In particular, it was shown that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, or at least 2 (over any field), and that the set of possible values of these parameters is discrete (in several regimes). We determine exactly the next gap, showing that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, equal to 2, or at least 2.68.
- Large spaces of matrices of bounded rank. The Quarterly Journal of Mathematics, 31(3):253–262, 1980. doi:10.1093/qmath/31.3.253.
- Matthew D. Atkinson. Primitive spaces of matrices of bounded rank. II. J. Austral. Math. Soc. Ser. A, 34(3):306–315, 1983. doi:10.1017/S1446788700023740.
- Discreteness of asymptotic tensor ranks, 2023. doi:10.48550/arXiv.2306.01718.
- Algebraic complexity theory, volume 315 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1997. doi:10.1007/978-3-662-03338-8.
- Countably many asymptotic tensor ranks, 2022. doi:10.48550/arxiv.2212.12219.
- A tensor restriction theorem over finite fields, 2022. doi:10.48550/arxiv.2211.12319.
- On Degeneration of Tensors and Algebras. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58, page 19:1–19:11, 2016. doi:10.4230/LIPIcs.MFCS.2016.19.
- A gap in the slice rank of k𝑘kitalic_k-tensors. J. Comb. Theory, Ser. A, 177:105335, 2021. doi:10.1016/j.jcta.2020.105335.
- Rank and border rank of Kronecker powers of tensors and Strassen’s laser method. computational complexity, 31(1):1–40, 2022.
- A Gap in the Subrank of Tensors, 2022. doi:10.48550/arXiv.2212.01668.
- Universal points in the asymptotic spectrum of tensors. J. Amer. Math. Soc., 36(1):31–79, 2023. doi:10.1090/jams/996.
- Matrix multiplication via arithmetic progressions. J. Symb. Comput., 9(3):251–280, 1990. doi:10.1016/S0747-7171(08)80013-2.
- Vector spaces of matrices of low rank. Adv. in Math., 70(2):135–155, 1988. doi:10.1016/0001-8708(88)90054-0.
- On Linear spaces of matrices of bounded rank, 2023. doi:10.48550/arXiv.2306.14428.
- Volker Strassen. The asymptotic spectrum of tensors. J. Reine Angew. Math., 384:102–152, 1988. doi:10.1515/crll.1988.384.102.
- Volker Strassen. Degeneration and complexity of bilinear maps: some asymptotic spectra. J. Reine Angew. Math., 413:127–180, 1991. doi:10.1515/crll.1991.413.127.
- Asymptotic spectra: Theory, applications and extensions, 2022. URL: https://staff.fnwi.uva.nl/j.zuiddam/papers/convexity.pdf.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.