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Bounds on the real tensor rank of octonion multiplication — arXiv:2608.16649

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.16649
License (as archived): CC0-1.0
Archived: 2026-10-19T00:29:37.438Z
SHA-256 of the archived text: 8352bb80fafd741735888e467a15e200f4f8c9a033f9e89630d2dd6e7f3a2827
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Bounds on the real tensor rank of octonion multiplication — arXiv:2608.16649 — arXiv (en), https://arxiv.org/abs/2608.16649 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_7e441affcea2 (Metraipolis archive dc_7e441affcea2, sha256 8352bb80fafd741735888e467a15e200f4f8c9a033f9e89630d2dd6e7f3a2827)

BibTeX

@misc{dc7e441affcea2,
  title = {Bounds on the real tensor rank of octonion multiplication — arXiv:2608.16649},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.16649},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_7e441affcea2; sha256 8352bb80fafd741735888e467a15e200f4f8c9a033f9e89630d2dd6e7f3a2827},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.16649. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The tensor rank of a bilinear map is the least number of multiplications any bilinear algorithm needs to compute it; for the multiplication of an algebra it measures how cheaply the algebra can be multiplied at all. For the even-dimensional real normed division algebras it is $3$ for the complex numbers and $8$ for the quaternions, both classical, while for the octonions $\mathbb{O}$ only a range was known: at least $15$ (Fiduccia and Zalcstein, 1977) and at most $30$ (Cariow and Cariowa). We prove $$18 \le \operatorname{R}_{\mathbb{R}}(T_{\mathbb{O}}) \le 25.$$ The lower bound peels the eight slices of $T_{\mathbb{O}}$ down to two and bounds the rank of the surviving pencil through the octonion norm. Nothing in it is special to dimension $8$: the same steps give $\operatorname{R}_{\mathbb{R}}(T_A) \ge \frac{5}{2}n - 2$ for every real normed division algebra $A$ of even dimension $n$, sharp for $\mathbb{C}$ and $\mathbb{H}$ and the best bound we know for $\mathbb{O}$. The upper bound is a separate construction, an explicit rank-$25$ decomposition certified by a Krawczyk argument, in exact rational arithmetic, to sit within $10^{-6}$ of an exact one. The same two arguments pin down the rank of a smaller three-slice quaternion tensor $\tau$, giving $\operatorname{R}_{\mathbb{R}}(\tau) = 7$. The Lean 4 kernel checks the lower bounds and the Krawczyk existence principle; the accompanying scripts check the certificate's finitely many exact-rational inequalities.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-18 in cs.CC).
Categorie arXiv: Computational Complexity (cs.CC).
Pagina del lavoro: https://arxiv.org/abs/2608.16649 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-18T23:35:48.558Z dalla pagina https://arxiv.org/catchup/cs.CC/2026-08-18?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).