Strassen's Asymptotic Rank Conjecture

• Strassen’s Asymptotic Rank Conjecture posits that for tight tensors the asymptotic tensor rank equals the maximum flattening rank, implying the polynomial-time computability of these invariants.
• Recent advances utilize algebraic and geometric tools such as Gröbner bases and Zariski-closed set analysis to formally bound and certify tensor ranks.
• If validated, the conjecture would lower the matrix multiplication exponent to 2 and enable sub-exponential algorithms for NP-hard problems like k-Set Cover and graph coloring.

Strassen’s Asymptotic Rank Conjecture is a central open question in algebraic complexity theory and tensor analysis that posits a sharp, algebraically tractable characterization of the asymptotic behavior of tensor rank. Resolving this conjecture would determine the optimal complexity of matrix multiplication and have deep consequences for computational complexity, combinatorics, and computational algebraic geometry. The conjecture asserts that, for highly structured tensors—concise and tight—the exponential growth rate of rank under Kronecker powers equals the dimension of their constituent spaces, generalizing the familiar invariance of matrix rank to the asymptotic regime. Recent advances have provided structural, geometric, and algorithmic frameworks for understanding asymptotic rank, clarified links to exponential-time algorithms, and characterized the relevant tensor varieties, but its truth remains open even for specific core examples.

1. Asymptotic Tensor Rank and Strassen’s Conjecture: Definition and Formal Statement

For a $k$-way tensor $T$ over a field $F$, the Kronecker (tensor) power is

$$T^{\boxtimes n} \in F^{d_1}\otimes\cdots\otimes F^{d_k} \boxtimes \cdots \boxtimes F^{d_k} = V^{\otimes n},$$

and the tensor rank $\mathrm{R}(T)$ is the minimal number of simple tensors summing to $T$. The asymptotic tensor rank is defined by

$$\mathrm{R}^{\otimes}(T) = \lim_{n \to \infty} \left( \mathrm{R}(T^{\boxtimes n}) \right)^{1/n}.$$

For order-$3$ tensors $T \in F^n \otimes F^n \otimes F^n$, one can take three matrix flattenings and define $\mathrm{R}_{\mathrm{f}}(T)$ as the maximum of their matrix ranks. Strassen’s Asymptotic Rank Conjecture asserts: $$\boxed{ \forall\, T \in F^n\otimes F^n\otimes F^n,\quad \mathrm{R}^{\otimes}(T) = \mathrm{R}_{\mathrm{f}}(T). }$$ This would imply that $\mathrm{R}^{\otimes}(T)$ is always integer, is computable in polynomial time, and, for the $2\times2$ matrix multiplication tensor, would set the matrix multiplication exponent $\omega$ to $2$ (Christandl et al., 2024).

2. Algebraic, Geometric, and Algorithmic Structure of Asymptotic Rank

A fundamental result is that the set of tensors whose asymptotic rank is at most $r$,

$$\{ T \in F^{d_1} \otimes \cdots \otimes F^{d_k} : \mathrm{R}^{\otimes}(T) \le r \}$$

is Zariski-closed. That is, there exist a finite set of polynomials $p_1(T),\ldots,p_\ell(T)$ such that

$$\mathrm{R}^{\otimes}(T) \le r\quad\Longleftrightarrow\quad p_1(T)=\cdots=p_\ell(T)=0.$$

This provides a semi-algorithm to certify upper bounds on asymptotic rank over any computable field and recasts the conjecture as a problem about polynomial ideal containment in the coordinate ring of the ambient tensor space. Notably, such sets being Zariski-closed brings tools from computational algebraic geometry (e.g., Gröbner bases, resultant computations, and symmetry reductions) directly to bear on the problem (Christandl et al., 2024).

Because these sublevel sets form descending chains of Zariski-closed sets in finite-dimensional spaces, the Noetherian property implies any such sequence stabilizes. Translated to rank values, this shows that possible values of asymptotic rank are well-ordered: no infinite strictly decreasing sequence of asymptotic ranks exists. In concrete terms, any tight upper bound on the matrix multiplication exponent must be exact; there is a “snap-to-grid” effect: sufficiently nearly optimal upper bounds force exactness (Christandl et al., 2024).

3. Combinatorial, Complexity-Theoretic, and Algorithmic Consequences

If Strassen’s Asymptotic Rank Conjecture holds, it would have sweeping implications in both algebraic and algorithmic domains. Concretely, it would set the exponent of matrix multiplication $\omega$ to $2$, yielding truly quadratic matrix multiplication algorithms. More unexpectedly, via reductions through tripartite partitioning and group-theoretic convolution tensors, it would also enable strictly sub-$2^n$ randomized (and, in some cases, deterministic) algorithms for classic exponential-time problems, including $k$-Set Cover and the Hamiltonian Cycle problem for bounded set sizes or graph degrees (Björklund et al., 2023, Pratt, 2023, Björklund et al., 2024). For example:

• Under the conjecture, one obtains a randomized $O((2-\epsilon)^n)$ algorithm for $k$-Set Cover for some $\epsilon > 0$ and for $k \le \kappa n$.
• Even a tiny improvement on known upper bounds for related tensor ranks (shaving a $1/k$ factor from a Fourier-analytic $8^k$ upper bound) would violate the exponential-time hypothesis on Set Cover (Pratt, 2023).
• Deterministic graph coloring (chromatic number computation) drops to $O(1.99982^n)$ time under ARC, strictly below the widely believed $2^n$ poly($n$) barrier unless the conjecture is false for explicit partitioning tensors (Björklund et al., 2024).

A summary of dependency relations:

Complexity Hypothesis	Consequence under ARC
Strassen’s conjecture (ARC)	Quadratic matrix mult. ($\omega=2$), sub-$2^n$ Set Cover and Hamiltonian
Set Cover Conjecture	Explicit tensor families have asymptotic rank $> N^{1.08}$
Sub-$2^n$ coloring alg. fails	Partitioning tensor with superlinear asymptotic rank exists

4. Geometric and Structural Approaches: Tight and Oblique Tensors

Tensors relevant to the conjecture are “tight” (admitting continuous regular symmetry) and concise (full-rank flattenings). These classes form large but codimension $m^2/4$ subvarieties in the space of $m \times m \times m$ tensors. The geometric structure of these varieties and their orbit closures under $GL(m)^3$ action facilitates dimension-theoretic and orbit-stabilizer arguments (Conner et al., 2018). Tight tensors are precisely those for which the conjecture is posited to hold, and these varieties admit large compressions, connecting to slice rank and related barrier arguments in cap-set and group-theoretic constructions.

Moreover, the exact characterization of Zariski-closed subvarieties of tight and oblique tensors, their dimensions matching, and the explicit incidence construction using Grassmannian correspondences, provide frameworks for comparing conjectured minimality of asymptotic rank in these classes and extend to combinatorial models for supports and symmetries.

5. Polynomial Barriers, Spectra, and Minimax Correspondence

Barrier approaches in the literature often rely on invariant-theoretic and entropy-based techniques. The introduction of the weighted slice rank and minimax correspondence to Strassen’s spectra program yields a framework in which asymptotic invariants for tensors (e.g., quantum functionals and slice-rank-type quantities) admit dual optimization problems over polytopes of flattening spectra (Christandl et al., 2020). For tight tensors, these invariants achieve their extremal values, and the conjecture coincides with the absence of further obstructions from spectral points.

Explicitly, if $\Sigma_r$ is the secant variety of tensors of rank $\le r$ and if there exists no nonzero polynomial of degree $\le p$ vanishing on it, then every tensor in the ambient space has asymptotic rank at most

$$\left( \binom{d^3-1+p}{p} \right)^{1/p} r^{(d^3-1)/p},$$

which, for specific cases, leads to nontrivial upper bounds $\sigma(7) \le 1.000\ldots$ (Kaski et al., 2024, Lee, 13 Jan 2026).

6. Explicit Universal Sequences and Numerical Methods

The construction of explicit universal sequences $\mathcal{U}_d$ and $\mathcal{T}_d$ of zero-one valued tensors provides families for which the asymptotic exponent captures the worst-case exponent over all $d \times d \times d$ tensors. The conjecture is equivalent to the assertion that, for all $d$, the worst-case exponent over tight tensors is $1$; i.e., the asymptotic rank never exceeds the dimension. The limit $\lim_{d \to \infty} \sigma(d)$ exists and is encoded by diagonalization constructions on these sequences (Kaski et al., 2024).

Recent systematic numerical work implements geometric frameworks based on secant variety implicitization to produce the first concrete improvements of asymptotic rank upper bounds below generic border rank for a range of formats (generic border rank $\leq 20$). These computations support the conjecture and clarify computational bottlenecks in high-dimensional interpolation and polynomial vanishing certification (Lee, 13 Jan 2026).

Format (r, a, b, c)	deg σ_r(X)	gen. bdr. rank	New $\underline{R}_\infty$ bound
(8, 3, 5, 7)	105	9	$< 8.366128$
(18, 7, 7, 7)	$1.87\times10^{5}$	19	$< 18.001169$

7. Symmetric Tensors, e-Computability, and Additivity

In the symmetric case, Strassen’s conjecture takes the form of additivity for the Waring rank of homogeneous polynomials: the rank of a sum of forms in disjoint variables equals the sum of their ranks. The notion of $e$-computability, whereby the rank is computed via Hilbert functions of colon ideals, gives infinitely many new cases where the symmetric additivity conjecture holds (Carlini et al., 2015). These methods connect apolarity, colon ideals, and exact combinatorial decompositions, and extend to tensor products via multiplicativity of asymptotic rank.

• (Christandl et al., 2024): Asymptotic tensor rank is characterized by polynomials
• (Björklund et al., 2023): The Asymptotic Rank Conjecture and the Set Cover Conjecture are not Both True
• (Kaski et al., 2024): A universal sequence of tensors for the asymptotic rank conjecture
• (Conner et al., 2018): Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture
• (Christandl et al., 2020): Weighted slice rank and a minimax correspondence to Strassen's spectra
• (Lee, 13 Jan 2026): Asymptotic rank bounds: a numerical census
• (Pratt, 2023): A stronger connection between the asymptotic rank conjecture and the set cover conjecture
• (Björklund et al., 2024): Fast Deterministic Chromatic Number under the Asymptotic Rank Conjecture
• (Carlini et al., 2015): Symmetric tensors: rank, Strassen's conjecture and e-computability