Border subrank of higher order tensors and algebras

arxiv: 2604.19872 · v1 · submitted 2026-04-21 · 🧮 math.AG · cs.CC· math.RA· quant-ph

Border subrank of higher order tensors and algebras

Chia-Yu Chang , Fulvio Gesmundo , Jeroen Zuiddam This is my paper

Pith reviewed 2026-05-10 01:22 UTC · model grok-4.3

classification 🧮 math.AG cs.CCmath.RAquant-ph

keywords border subrankhigher-order tensorsstructure tensorsalgebrasmatrix multiplicationdegenerationsl_2polynomial algebras

0 comments p. Extension

The pith

The border subrank of higher-order structure tensors is determined exactly for matrix multiplication and several families of algebras.

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

The paper computes the border subrank for higher-order structure tensors that encode multiplication in algebras. For k-fold matrix multiplication and upper triangular matrix multiplication it obtains matching upper and lower bounds that apply for every k. It gives exact values for the structure tensors of truncated polynomial algebras, null algebras, apolar algebras of a quadric, and the Lie algebra sl_2 at every order. It further proves that any degeneration between two structure tensors at high order already exists at every lower order. These results turn asymptotic statements about order-three tensors into concrete, order-by-order determinations.

Core claim

We determine tight bounds on the border subrank of k-fold matrix multiplication and k-fold upper triangular matrix multiplication for all k. We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. We determine the border subrank of the higher order structure tensors of the Lie algebra sl_2 for all orders. We prove that degeneration of structure tensors of algebras propagates from higher to lower order.

What carries the argument

Higher-order structure tensor of an algebra, the tensor that records the multiplication map, together with the border subrank that records the largest rank to which the tensor can degenerate.

If this is right

The border subrank of k-fold matrix multiplication is known up to a small gap for every k.
Exact border subrank values are available for the structure tensors of sl_2 at every order.
Degeneration at order m between two algebra structure tensors implies the same degeneration at every order less than m.
The same exact values hold for the structure tensors of truncated polynomials, null algebras, and apolar algebras of a quadric.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The propagation result reduces the problem of finding degenerations at high orders to the easier case of low orders.
The explicit values can be used to bound the number of independent multiplications needed when several matrices or polynomials are multiplied together.
The same upper-bound techniques may apply to other algebras whose structure tensors have not yet been analyzed.
The distinction between subrank and border subrank collapses for these families at every order.

Load-bearing premise

The geometric rank, G-stable rank, and socle-degree methods give matching upper bounds for the listed families, and degeneration between structure tensors always propagates downward in order.

What would settle it

An explicit degeneration or rank calculation showing that the border subrank of the four-fold matrix multiplication tensor is strictly larger than the upper bound obtained from the socle-degree or geometric-rank method.

read the original abstract

We determine the border subrank of higher order structure tensors of several families of algebras, and in particular obtain the following results. (1) We determine tight bounds on the border subrank of $k$-fold matrix multiplication and $k$-fold upper triangular matrix multiplication for all $k$. (2) We determine the border subrank of the higher order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric. (3) We determine the border subrank of the higher order structure tensors of the Lie algebra $\mathfrak{sl}_2$ for all orders. (4) We prove that degeneration of structure tensors of algebras propagates from higher to lower order. Along the way, we investigate which upper bound methods (geometric rank, $G$-stable rank, socle degree) are effective in which settings, and how they relate. Our work extends the results of Strassen (J.~Reine Angew.~Math., 1987, 1991), who determined the asymptotic subrank of these algebras for tensors of order three, in two directions: we determine the border subrank itself rather than its asymptotic version, and we consider higher order structure tensors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pins down exact border subranks for k-fold matrix multiplication and several algebra structure tensors at all orders, plus a degeneration propagation lemma that reduces higher orders to Strassen's order-three cases.

read the letter

This paper pins down exact border subrank values for the higher-order structure tensors of matrix multiplication, upper triangular matrices, truncated polynomial algebras, null algebras, apolar algebras of a quadric, and sl_2. It also proves that if one structure tensor degenerates to another, then the same holds after lowering the tensor order. These are concrete numbers rather than growth rates, and the propagation step lets them recycle Strassen's older order-three results for the higher-order claims. They also test which of the usual upper-bound tools (geometric rank, G-stable rank, socle degree) actually hit the lower bounds in each family. That combination of exact values and a reusable reduction is the part worth noting. The calculations appear to match upper and lower bounds tightly in the cases treated, and the work stays within established methods without introducing new fitting parameters. The soft spot is the base field. The abstract states the results without restrictions, yet degeneration via limits or polynomial families usually needs an infinite or algebraically closed field. If the propagation proof relies on steps that break over finite fields or in positive characteristic, the universal claims for all k and all orders would need a caveat. Without the full derivations it is hard to see how much of the tightness comes from the specific algebras versus the general technique. This is for people already working on tensor ranks and algebraic complexity who want exact rather than asymptotic statements. A reader who knows Strassen's papers and the standard rank bounds will extract the most. It deserves a serious referee because the results are specific, the propagation idea could apply elsewhere, and the field question is checkable in the text.

Referee Report

2 major / 2 minor

Summary. The manuscript determines the border subrank of higher-order structure tensors for several families of algebras. It obtains tight bounds for the k-fold matrix multiplication tensor and k-fold upper-triangular matrix multiplication tensor for every k; exact values for the higher-order structure tensors of truncated polynomial algebras, null algebras, and apolar algebras of a quadric; and exact values for the higher-order structure tensors of the Lie algebra sl_2 for all orders. It also proves that if the structure tensor of an algebra degenerates at order m then it degenerates at every lower order, and it compares the effectiveness of geometric rank, G-stable rank, and socle-degree upper bounds across these families. The work extends Strassen's 1987/1991 results on asymptotic subrank for order-three tensors by replacing the asymptotic quantity with the exact border subrank and by treating tensors of arbitrary order.

Significance. If the central claims hold, the paper supplies the first exact border-subrank values (rather than merely asymptotic) for several infinite families of higher-order tensors that arise naturally in algebra and representation theory. The propagation theorem supplies a reduction that converts higher-order questions into order-three questions already settled by Strassen, which is a useful structural tool. The systematic comparison of upper-bound techniques (geometric rank, G-stable rank, socle degree) clarifies which methods remain tight for which classes of algebras. These contributions are concrete and falsifiable, and they sit squarely inside the existing literature on tensor rank and algebraic complexity.

major comments (2)

[§4] §4 (Propagation theorem): The statement that degeneration propagates from higher to lower order is used to reduce all higher-order claims to Strassen's order-three results. The proof is not accompanied by an explicit hypothesis on the base field. Degeneration is realized either by polynomial families or by Zariski limits; both constructions require the base ring to be infinite (or at least to admit a parameter t that is not a zero-divisor). Over a finite field the reduction may fail, which would undermine the “for all k” and “for all orders” statements. The manuscript should state the precise field assumptions under which the propagation holds.
[§3.2 and §5] §3.2 and §5 (k-fold matrix multiplication): The upper bounds are obtained from geometric rank or G-stable rank and are asserted to be tight. The matching lower bounds are constructed by explicit degenerations or by exhibiting large subspaces of rank-1 tensors. It is not immediately clear from the text whether these lower-bound constructions remain valid when the base field has positive characteristic or is not algebraically closed; a short remark confirming that the same explicit maps work over any infinite field would remove the ambiguity.

minor comments (2)

[§2] The notation for the border subrank (denoted Q or Q̲ in different places) should be fixed and introduced once in the introduction or §2.
Several citations to Strassen's 1987 and 1991 papers appear only in the introduction; adding the precise theorem numbers from those papers when the reduction is invoked would help the reader.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points that require clarification on field hypotheses. We address each major comment below and will incorporate the suggested revisions.

read point-by-point responses

Referee: [§4] §4 (Propagation theorem): The statement that degeneration propagates from higher to lower order is used to reduce all higher-order claims to Strassen's order-three results. The proof is not accompanied by an explicit hypothesis on the base field. Degeneration is realized either by polynomial families or by Zariski limits; both constructions require the base ring to be infinite (or at least to admit a parameter t that is not a zero-divisor). Over a finite field the reduction may fail, which would undermine the “for all k” and “for all orders” statements. The manuscript should state the precise field assumptions under which the propagation holds.

Authors: We agree that the propagation theorem in Section 4 relies on polynomial families and limits that presuppose an infinite base field. We will revise the statement of the theorem and the opening paragraph of §4 to explicitly assume that the base field is infinite. Under this hypothesis the “for all k” and “for all orders” claims remain valid, and the reduction to Strassen’s order-three results continues to hold. The proof itself is unchanged. revision: yes
Referee: [§3.2 and §5] §3.2 and §5 (k-fold matrix multiplication): The upper bounds are obtained from geometric rank or G-stable rank and are asserted to be tight. The matching lower bounds are constructed by explicit degenerations or by exhibiting large subspaces of rank-1 tensors. It is not immediately clear from the text whether these lower-bound constructions remain valid when the base field has positive characteristic or is not algebraically closed; a short remark confirming that the same explicit maps work over any infinite field would remove the ambiguity.

Authors: The lower-bound constructions in §3.2 and §5 are given by explicit linear maps and subspaces defined over ℤ. These maps remain valid over any infinite field, including fields of positive characteristic, because they do not invoke algebraic closure or characteristic-zero assumptions. We will insert a brief clarifying sentence in both sections stating that the same explicit constructions work over any infinite base field, thereby removing the ambiguity while preserving the tightness statements. revision: yes

Circularity Check

0 steps flagged

No circularity; results extend independent prior work via new proofs and explicit bounds

full rationale

The paper determines border subrank values for several families of higher-order structure tensors by applying upper-bound techniques (geometric rank, G-stable rank, socle degree) and proving a degeneration-propagation statement that reduces higher-order cases to order-3 results of Strassen. No quoted equation or definition in the provided abstract or description reduces any claimed border-subrank value to a fitted parameter, self-referential quantity, or ansatz imported via self-citation. The propagation theorem is presented as a new result rather than a self-citation load-bearing premise. Strassen's prior results are independent external work (1987, 1991) and do not overlap with the current authors. The derivation chain therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms of linear algebra, algebraic geometry, and representation theory used to define structure tensors, border rank, and the listed upper-bound invariants; no new free parameters, ad-hoc axioms, or invented entities appear in the abstract.

axioms (1)

standard math Standard axioms of linear algebra and algebraic geometry over an algebraically closed field of characteristic zero
Invoked implicitly when defining structure tensors of algebras and when applying geometric and G-stable rank notions.

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discussion (0)

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Unrestrictions and concise secant varieties
math.AG 2026-04 unverdicted novelty 7.0

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