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Asymptotic tensor rank is characterized by polynomials

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Pith's one-line read Asymptotic tensor rank sublevel sets are cut out by finitely many polynomial equations.

desk verdict The structural result—Zariski-closed sublevel sets for asymptotic rank—is solid and important; the advertised algorithm does not follow from the proof. read the letter →

arxiv 2411.15789 v2 pith:FMZ6JRXA submitted 2024-11-24 cs.CC math.AGquant-ph

Matthias Christandl , Koen Hoeberechts , Harold Nieuwboer , Péter Vrana , Jeroen Zuiddam This is my paper · ORCID
classification cs.CCmath.AGquant-ph MSC 15A6968Q17
keywords asymptotictensorrankpolynomialequationsZariski-closedsublevelsetsmatrixmultiplicationexponentwell-orderedvaluesspectrumoftensorscomputabilityfromabovedegeneration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Asymptotic tensor rank $\tilde R(T)$ measures how quickly the ordinary tensor rank grows under Kronecker powers, and it is the quantity behind the matrix multiplication exponent $\omega$; deciding it is notoriously hard. This paper proves that, for any field, the sublevel set $\{T : \tilde R(T) \le r\}$ is Zariski-closed, meaning membership is equivalent to the vanishing of finitely many polynomials on the tensor entries, exactly as for matrix rank. Because of this, upper bounds on asymptotic rank become decidable from above: for computable fields and any real $r$, an algorithm can decide whether $\tilde R(T) \le r$ by evaluating polynomials. A further consequence, new for infinite fields such as $\mathbb{C}$, is that the set of values taken by asymptotic rank is well-ordered: every non-increasing sequence of asymptotic ranks stabilizes, so there is a positive gap above $2^\omega$ in the exponents of bilinear maps. These results hold not only for tensor rank but for every element of the asymptotic spectrum of tensors.

What carries the argument

The load-bearing object is an admissible functional: a family of functions $F_n$ on tensor powers $V^{\otimes n}$ satisfying subadditivity, submultiplicativity under tensor products, permutation invariance, $\mathbb{F}^\times$-homogeneity, and boundedness on $V$; tensor rank is the primary example, and the regularized limit $\tilde F(T)=\lim_{n\to\infty}F_n(T^{\otimes n})^{1/n}$ generalizes asymptotic rank. The key mechanism is the identity $\tilde F[A]=\tilde F[\bar A]$ between the supremum over a set and over its Zariski closure, proven by writing $T^{\otimes n}$ for $T\in \bar A$ as a linear combination of powers $S_i^{\otimes n}$ with $S_i\in A$ and then applying a double-blocking estimate (submultiplicativity and subadditivity) to the powers of those combinations. For the asymptotic-spectrum extension, the additional engine is a new lower bound on the max-rank quantities $Q_{i,j}(T)$: whenever $|F|>R_I(T)$, the product $\prod_{i\in I,j\notin I}Q_{i,j}(T)\ge R_I(T)$, which forces each spectrum element either to reduce to a lower-order tensor functional or to grow with the flattening rank.

What would settle it

Exhibit a tensor $T$ in the Zariski closure of a set $A$ with $\tilde R(T) > \sup_{S\in A}\tilde R(S)$; Theorem 2.2 forbids this. Over $\mathbb{C}$, it would be enough to find a converging sequence of tensors whose asymptotic ranks converge to a real number that is not the asymptotic rank of any tensor, or a strictly decreasing sequence of asymptotic ranks that does not stabilize.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for any field $F$, order $k \ge 3$, dimensions $d \in \mathbb{Z}_{\ge 1}^k$, and real $r$, the set $\{T \in F^{d_1}\otimes\cdots\otimes F^{d_k} : \tilde R(T) \le r\}$ is Zariski-closed. In concrete terms, for each format and threshold there is a finite list of polynomials $p_1,\dots,p_\ell$ such that $\tilde R(T)\le r$ iff all $p_i(T)=0$. The proof works at the level of a general admissible functional $F$: it shows that the supremum of the regularized function $\tilde F$ over a set equals the supremum over its Zariski closure, using a decomposition of $T^{\otimes n}$ into linear combinations of $S_i^{\otimes n}$ with $S_i$ in the original set. From this the paper derives lower-semicontinuity, well-orderedness of the value set, completeness over $\mathbb{C}$, and the analogous discreteness result for the asymptotic spectrum.

Load-bearing premise

The proof requires tensor rank to satisfy the axioms of an admissible functional, above all submultiplicativity under the Kronecker product; if tensor rank failed any of these inequalities, the regularized limit and the key closure estimate would no longer go through.

Editorial extensions

If this is right

  • Membership in every sublevel set of asymptotic tensor rank is decidable by evaluating a finite list of polynomials, giving an algorithm for "asymptotic rank at most $r$" over computable fields.
  • Any upper bound on $\tilde R$ proven for a Zariski-dense family of tensors automatically extends to all tensors in the closure, so degeneration arguments yield upper bounds without explicit sequences.
  • The value set $\{\tilde R(T)\}$ is well-ordered: any non-increasing sequence stabilizes, and in particular the matrix multiplication exponent cannot be approached from above by a strictly decreasing sequence of exponents of bilinear maps.
  • For every element of the asymptotic spectrum of tensors, the set of values is well-ordered; by duality, the set of tensors asymptotically restricted by a fixed tensor is Zariski-closed.
  • Over $\mathbb{C}$, the set of asymptotic ranks is complete: the limit of any converging sequence of asymptotic ranks is itself an asymptotic rank.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the sublevel sets are also irreducible, a dimension argument would give only finitely many asymptotic ranks per fixed format, a weak form of the asymptotic rank conjecture; the paper raises this as an open question, so this is an extrapolation rather than a claim.
  • The non-explicit polynomials may still be useful: once their degrees and sparsity are bounded, the decision procedure could be turned into a concrete algebraic witness for upper bounds, potentially connecting asymptotic rank to algebraic proof complexity.
  • The max-rank product inequality suggests a generic lower-bound strategy: to show a tensor has large asymptotic rank, it suffices to bound the individual $Q_{i,j}$ quantities, which are ordinary matrix-rank parameters and hence more accessible computationally.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proves that for any field F, the sublevel sets {T : eR(T) ≤ r} of asymptotic tensor rank are Zariski-closed, i.e., each is determined by the vanishing of finitely many polynomials. It derives that the set of all asymptotic ranks is well-ordered (discreteness from above), that over C it is Euclidean-closed, and it extends these properties to all elements of Strassen's asymptotic spectrum via a new lower bound on max-rank restrictions. It also gives an equivalent geometric condition for discreteness from below and leaves that as an open problem.

Significance. If the main theorem is correct, it establishes a fundamental algebraic structural property of asymptotic tensor rank that was previously unknown for infinite fields. The double-blocking proof of Theorem 2.2 is elegant and correct, and the paper is honest that the defining polynomials are not exhibited. The consequences for the matrix multiplication exponent (no upper bound arbitrarily close to omega without 'snapping' to it) and the generalization to the asymptotic spectrum are interesting. The main weakness is the unsubstantiated computational claim, which is not needed for the structural results but is prominently advertised.

major comments (2)
  1. [§1.1, Theorem 1.1; Abstract] The advertised algorithmic claim that asymptotic tensor rank is 'computable from above' is not established by the proof. Theorem 2.2 and Corollary 2.4 show that for each format and each r there exists a finite set of polynomials p1,...,pℓ whose vanishing decides eR(T) ≤ r, but the proof is purely existential: it uses Zariski closure and Noetherianity to assert existence without constructing the polynomials. Over a computable field one cannot simply enumerate candidate polynomial sets, because verifying a candidate requires an oracle for the predicate eR(T) ≤ r, which is precisely the decision problem the algorithm is meant to solve. The statement in footnote 2 that 'these polynomials are also computable' is therefore unsupported. Since Theorem 1.1 and the title of the paper rest on this computational claim, it must either be proven constructively or removed in favor of the (still substantial) structural statement of Theorem 1.2.
  2. [§3.1, Lemma 3.6] The proof of Lemma 3.6 is not correct as written. The function ϕ_j is defined on (k−1)-tensors, but the proof uses the expression ϕ_j(T) where T is a k-tensor; moreover, the inequalities 'S ⊠ ⟨m⟩_{i,j} ≥ ϕ_j(T)' and 'S ≤ ϕ_j(T) ⊠ ⟨m⟩_{i,j}' have a type mismatch and are not otherwise justified. Lemma 3.6 is the central reduction used in Theorem 3.1 to show that F coincides in value with an element F' ∈ ∆(F,k−1); without a clear proof of this lemma, the discreteness result for the asymptotic spectrum (Theorem 3.1/1.5) is incomplete. Please provide a full proof with correctly typed tensors.
minor comments (6)
  1. [§1, Introduction] The expression '2ω = eR(⟨2,2,2⟩)' should be typeset as 2^ω = eR(⟨2,2,2⟩) to avoid confusion with 2·ω.
  2. [§1.1, footnote 1] The non-uniformity in r is a significant caveat; it should appear in the statement of Theorem 1.1 and in the abstract, not only in a footnote.
  3. [§2.3] When defining 'well-ordered' for subsets of R, please make explicit that this is relative to the usual order and note the equivalence with stabilization of non-increasing sequences, since some readers may use the order-theoretic definition.
  4. [§3.1, Lemma 3.4] The block matrix notation [M1;...;Mt] is introduced with c blocks in Lemma 3.4 but with a different index variable in the surrounding text; unify the notation.
  5. [§3.1, Corollary 3.5] The expression 'T ⊠(k(k−1)/2)' is ambiguous; write (T ⊠ ... ) or define the Kronecker power explicitly.
  6. [§4.1, Theorem 4.1] Calling the set 'complete' may confuse readers; the proof shows Euclidean closedness, which is a different notion from completeness of a metric space.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.2 is derived from the admissible-functional axioms, Zariski closure, and Noetherianity, and is not assumed from the conclusion.

full rationale

The central claim (Theorem 1.2) is proved by a self-contained double-blocking argument: Theorem 2.2 establishes that eF[A]=eF[closure(A)] for any admissible functional using Definition 2.1, Lemma 2.3, and submultiplicativity, so Corollary 2.4 follows without any fitted parameter or appeal to the conclusion. The discreteness results follow from Noetherianity and from conciseness of tensors, not from any assumed structure of the value set. Section 3 uses a cited matrix lemma (Lemma 3.4, from BCL+24) whose authors overlap with the present paper, and it is load-bearing for the asymptotic-spectrum extension Theorem 3.1; however, this lemma is a parameter-free linear-algebra statement with stated assumptions, not an assumption of well-orderedness or Zariski-closedness, so it does not make the argument circular. Strassen's duality [Str88] is an external result. The authors explicitly state that they 'do not exhibit these polynomials explicitly,' which is an admitted limitation on the algorithmic claim in Theorem 1.1: the proof is existential via Zariski-closedness and Noetherianity. This is a potential proof gap or overclaim about computability from above, but it is not circularity, because the theorem does not rename its input or otherwise reduce to its own conclusion. No circular step was found.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. The central proof rests on standard facts about tensor rank, Zariski topology, Fekete's lemma, Baire category, and the Strassen asymptotic spectrum framework. The only non-standard cited input is the column-flushing lemma from BCL+24, which the authors cite rather than reprove.

assumptions (7)
  • domain assumption Tensor rank is an admissible functional (Definition 2.1): subadditive, submultiplicative under Kronecker products, permutation-invariant, homogeneous, bounded on one-tensors.
    This is the abstract framework in Section 2.1; it is standard for tensor rank and is required for Theorem 2.2.
  • standard math Fekete's lemma: the limit defining the regularized functional exists for submultiplicative sequences.
    Invoked in Section 2.1 to define the regularized functional \tilde F.
  • standard math The Zariski topology on a finite-dimensional vector space is Noetherian.
    Used in Corollary 2.5 to show descending chains of sublevel sets stabilize.
  • domain assumption Strassen's duality: asymptotic rank is the pointwise maximum over the asymptotic spectrum, and each spectral point is multiplicative, additive, monotone, and normalized.
    Used to extend well-orderedness to the asymptotic spectrum and to derive implications from Theorem 1.5 in Section 3.
  • domain assumption Field-extension invariance of the asymptotic spectrum (Strassen's Theorem 3.10).
    Used in the proof of Theorem 3.1 to assume the field is infinite.
  • domain assumption Lemma 3.4 from BCL+24, the column-flushing lemma, is correct.
    Cited without proof in the proof of Theorem 3.2; it is a prior result from the authors' group.
  • standard math Baire category theorem, and the facts that Zariski-open implies Euclidean-open and Euclidean-dense implies Zariski-dense.
    Used in Section 4 for completeness and for the geometric characterization of discreteness from below over the complex numbers.

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Pith. "Pith review of Asymptotic tensor rank is characterized by polynomials." pith.science (2026). https://pith.science/paper/FMZ6JRXA

@misc{pith2026241115789,
  author       = {Pith},
  title        = {Pith review of: Asymptotic tensor rank is characterized by polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMZ6JRXA}},
  note         = {Machine review of arXiv:2411.15789}
}
abstract

Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other hand, Strassen's asymptotic rank conjecture makes the bold claim that asymptotic tensor rank equals the largest dimension of the tensor and is thus as easy to compute as matrix rank. Despite tremendous interest, much is still unknown about the structural and computational properties of asymptotic rank; for instance whether it is computable. We prove that asymptotic tensor rank is "computable from above", that is, for any real number $r$ there is an (efficient) algorithm that determines, given a tensor $T$, if the asymptotic tensor rank of $T$ is at most $r$. The algorithm has a simple structure; it consists of evaluating a finite list of polynomials on the tensor. Indeed, we prove that the sublevel sets of asymptotic rank are Zariski-closed (just like matrix rank). While we do not exhibit these polynomials explicitly, their mere existence has strong implications on the structure of asymptotic rank. As one such implication, we find that the values that asymptotic tensor rank takes, on all tensors, is a well-ordered set. In other words, any non-increasing sequence of asymptotic ranks stabilizes ("discreteness from above"). In particular, for the matrix multiplication exponent (which is an asymptotic rank) there is no sequence of exponents of bilinear maps that approximates it arbitrarily closely from above without being eventually constant. In other words, any such upper bound on the matrix multiplication exponent that is close enough, will "snap" to it. Previously such discreteness results were only known for finite fields or for other tensor parameters (e.g., asymptotic slice rank). We obtain them for infinite fields like the complex numbers.

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