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On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings — arXiv:2611.00735

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.00735
License (as archived): CC0-1.0
Archived: 2026-11-03T11:46:25.080Z
SHA-256 of the archived text: 765cce57fb414fdcc2dffac966c2a149ae88e87b10eb263f94126ca171988842
Stable URL of this page: https://metraipolis.com/documento/dc_c81bc447e211

How to cite

On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings — arXiv:2611.00735 — arXiv (en), https://arxiv.org/abs/2611.00735 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_c81bc447e211 (Metraipolis archive dc_c81bc447e211, sha256 765cce57fb414fdcc2dffac966c2a149ae88e87b10eb263f94126ca171988842)

BibTeX

@misc{dcc81bc447e211,
  title = {On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings — arXiv:2611.00735},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.00735},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_c81bc447e211; sha256 765cce57fb414fdcc2dffac966c2a149ae88e87b10eb263f94126ca171988842},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.00735. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Tensors over commutative rings naturally appear in number theory, geometry, and group theory. For example, $2\times 2\times 2$ tensors over $\mathbb{Z}$ form the starting point of Bhargava's celebrated works generalising Gauss's composition law (Bhargava, Ann. Math., 2004). Symmetric tensors over $\mathbb{Z}$ are central to the classification of Calabi--Yau threefolds (Yau, Commun. Pure Appl. Math., 1978; Wall, Invent. Math., 1966), geometric objects of significance in string theory. Additionally, tensors over finite commutative rings closely correspond to finite nilpotent groups of class 2 (Baer, Trans. Am. Math. Soc., 1938). In this work, we study isomorphism testing of tensors over commutative rings from a complexity-theoretic viewpoint, and its applications. Some of our main results are: 1. Let $\mathrm{R}$ be a commutative ring. We introduce two complexity classes: $\mathsf{3TI}_\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three modules over $\mathrm{R}$, and $\mathsf{3FTI}_\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three free modules over $\mathrm{R}$. 2. We show that some classical problems considered by Grunewald and Segal (ibid.), and the problem of classifying Calabi--Yau threefolds, are $\mathsf{3FTI}_\mathbb{Z}$-complete. We also show that many natural problems are complete for $\mathsf{3TI}_{\mathbb{Z}/p^e\mathbb{Z}}$. 3. We show that testing isomorphism of tensors in $\mathbb{Z}^2\otimes\mathbb{Z}^2\otimes\mathbb{Z}^2$ is polynomial-time equivalent to the principal ideal problem in algorithmic number theory. 4. We present an $N^{O((\log N)^{8/9})}$-time algorithm for testing isomorphism of finite nilpotent groups of class $2$ and odd order $N$.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv cs.CC, nuovi lavori del giorno).
Categorie arXiv: Computational Complexity (cs.CC) ; Commutative Algebra (math.AC); Number Theory (math.NT).
Pagina del lavoro: https://arxiv.org/abs/2611.00735 — 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 2026-11-03T11:08:22.198Z dalla pagina https://arxiv.org/list/cs.CC/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).