Generic and maximal rank of 3-tensors

### What is the conjecture

Let $T \in \mathbb{C}^{m_1} \otimes \mathbb{C}^{m_2} \otimes \mathbb{C}^{m_3}$. The *rank* of $T$
is the least $r$ such that $T = \sum_{i=1}^r a_i \otimes b_i \otimes c_i$. Two basic invariants of
the format $(m_1, m_2, m_3)$ are the *generic rank* $\operatorname{grank}(m_1, m_2, m_3)$, the rank
of a generic tensor, and the *maximal rank* $\operatorname{mrank}(m_1, m_2, m_3)$, the largest rank
attained.

Counting parameters, $r$ decomposable tensors carry $r(m_1 + m_2 + m_3 - 2)$ free parameters, so
one expects
$$\operatorname{grank}(m_1, m_2, m_3) = \left\lceil \frac{m_1m_2m_3}{m_1 + m_2 + m_3 - 2}
\right\rceil.$$
This fails once $m_3$ exceeds $(m_1-1)(m_2-1)$, where the generic rank is
$\min(m_3, m_1m_2)$, and it fails for the formats $(3, 2p+1, 2p+1)$, where the generic rank exceeds
it by one. No other exception is known.

**Conjecture (Friedland, Conjecture 5.1).** Let $3 \leq m_1 \leq m_2 \leq m_3 \leq (m_1-1)(m_2-1)$
with $(m_1, m_2, m_3) \neq (3, 2p+1, 2p+1)$ for every $p$. Then
$\operatorname{grank}(m_1, m_2, m_3) = \lceil m_1m_2m_3 / (m_1 + m_2 + m_3 - 2) \rceil$.

Friedland verified this numerically for $m_3 \leq 14$; the range has since been extended to $m_3 \leq 20$. It is the generic-rank shape of the
Abo-Ottaviani-Peterson conjecture that Segre varieties are never secant defective outside a known
list, which is still open in general.

Alongside it there are several known values worth recording: the unbalanced case
(Catalisano-Geramita-Gimigliano), $\operatorname{grank}(3, 2p, 2p)$ and
$\operatorname{grank}(3, 2p+1, 2p+1)$ (Strassen), and $\operatorname{grank}(n, n, n) =
\lceil n^3/(3n-2) \rceil$ for $n \neq 3$ (Lickteig), and $\operatorname{grank}(4, m, m)$
(Abo-Ottaviani-Peterson).

The maximal rank is much less well understood. It is known that
$\operatorname{mrank}(2, m, n) = m + \min(m, \lfloor n/2 \rfloor)$ for $2 \leq m \leq n$
(Kruskal, JaJa) and that $\operatorname{mrank}(3, 3, 3) = 5$: every $n \times n \times 3$ tensor has
rank at most $2n - 1$ (Atkinson-Stephens, proved by Sumi-Miyazaki-Sakata), and the generic rank is
already $5$. Blekherman and Teitler proved the general bound
$\operatorname{mrank} \leq 2\operatorname{grank}$. But no formula is known for
$\operatorname{mrank}(n, n, n)$, and even $\operatorname{mrank}(3, 3, 5)$ is undetermined: it is
known only to be $6$ or $7$. Both are open problems worth stating.

**Sources:**
- S. Friedland, *On the generic and typical ranks of 3-tensors*, Linear Algebra Appl. 436 (2012),
478-497, https://doi.org/10.1016/j.laa.2011.05.008, https://arxiv.org/abs/0805.3777.
Conjecture 5.1, and the survey of known values in Section 5.
- M. V. Catalisano, A. V. Geramita, A. Gimigliano, *Ranks of tensors, secant varieties of Segre
varieties and fat points*, Linear Algebra Appl. 355 (2002), 263-285,
https://doi.org/10.1016/S0024-3795(02)00352-X. The unbalanced case.
- V. Strassen, *Rank and optimal computation of generic tensors*, Linear Algebra Appl. 52/53
(1983), 645-685, https://doi.org/10.1016/0024-3795(83)80041-X. The families
$(3, 2p, 2p)$ and $(3, 2p+1, 2p+1)$.
- T. Lickteig, *Typical tensorial rank*, Linear Algebra Appl. 69 (1985), 95-120,
https://doi.org/10.1016/0024-3795(85)90070-9. Cubes.
- H. Abo, G. Ottaviani, C. Peterson, *Induction for secant varieties of Segre varieties*, Trans.
Amer. Math. Soc. 361 (2009), 767-792, https://doi.org/10.1090/S0002-9947-08-04725-9.
- J. B. Kruskal, *Rank, decomposition, and uniqueness for 3-way and N-way arrays*, in Multiway Data
Analysis, North-Holland (1989), 7-18, and J. JaJa, *Optimal evaluation of pairs of bilinear
forms*, SIAM J. Comput. 8 (1979), 443-462, https://doi.org/10.1137/0208037. Maximal rank for
$2 \times m \times n$.
- M. D. Atkinson, N. M. Stephens, *On the maximal multiplicative complexity of a family of bilinear
forms*, Linear Algebra Appl. 27 (1979), 1-8, https://doi.org/10.1016/0024-3795(79)90026-0, and
T. Sumi, M. Miyazaki, T. Sakata, *About the maximal rank of 3-tensors over the real and the
complex number field*, Ann. Inst. Statist. Math. 62 (2010), 807-822,
https://doi.org/10.1007/s10463-010-0294-5, https://arxiv.org/abs/0806.4048. Theorem 4.5 and
Proposition 4.9.
- W. Bruzda, S. Friedland, K. Zyczkowski, *Rank of a tensor and quantum entanglement*, Linear
Multilinear Algebra 72 (2024), 1796-1859, https://doi.org/10.1080/03081087.2023.2211717,
https://arxiv.org/abs/1912.06854. Survey; Conjecture 4.12, Sections 4.5 and 4.6.
- G. Blekherman, Z. Teitler, *On maximum, typical and generic ranks*, Math. Ann. 362 (2015),
1021-1031, https://doi.org/10.1007/s00208-014-1150-3, https://arxiv.org/abs/1402.2371.
Theorem 1.

### Prerequisites needed

None. Mathlib already has `Holor` and `Holor.cprank`, the CP rank of a holor, which is exactly
tensor rank. The generic rank can be stated as density of the locus of tensors of that rank in the
Euclidean topology, and the maximal rank as `IsGreatest` on the range of `Holor.cprank`.

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-14
* ams-15

### Choose either option

- [x] I plan on adding this conjecture to the repository
- [ ] This issue is up for grabs: I would like to see this conjecture added by somebody else

Read the existing Holor and Holor.cprank definitions first, then determine how the issue's generic-rank density and maximal-rank IsGreatest statements fit the available APIs. Done means adding formal statements for the generic and maximal rank conjectures, including the listed known cases and bounds, with suitable proofs or clearly stated conjectural results.

Written by the indexing model from the issue text.