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

@misc{grochow2026complexity,
  title = {{On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings}},
  author = {Joshua A. Grochow and Youming Qiao and Katherine E. Stange and Xiaorui Sun},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00735},
  eprint = {2611.00735},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00735},
}

TY  - UNPB
TI  - On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings
AU  - Joshua A. Grochow
AU  - Youming Qiao
AU  - Katherine E. Stange
AU  - Xiaorui Sun
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00735
UR  - https://arxiv.org/abs/2611.00735
AB  - Tensors over commutative rings naturally appear in number theory, geometry, and group theory. For example, 2× 2× 2 tensors over ℤ form the starting point of Bhargava's celebrated works generalising Gauss's composition law (Bhargava, Ann. Math., 2004). Symmetric tensors over 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 R be a commutative ring. We introduce two complexity classes: 3TI_R consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three modules over R, and 3FTI_R consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three free modules over R. 2. We show that some classical problems considered by Grunewald and Segal (ibid.), and the problem of classifying Calabi--Yau threefolds, are 3FTI_Z-complete. We also show that many natural problems are complete for 3TI_ℤ/p^eℤ. 3. We show that testing isomorphism of tensors in ℤ^2⊗ℤ^2⊗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.
ER  -