Symmetric subrank and its border analogue

arxiv: 2604.12801 · v1 · submitted 2026-04-14 · 🧮 math.AG · cs.CC· math.AC

Symmetric subrank and its border analogue

Benjamin Biaggi , Jan Draisma , Koen de Nooij , Immanuel van Santen This is my paper

Pith reviewed 2026-05-10 13:55 UTC · model grok-4.3

classification 🧮 math.AG cs.CCmath.AC

keywords symmetric subrankborder subrankhomogeneous polynomialsasymptotic behaviorgeometric invariant theorycubic formsquartic formslinear substitutions

0 comments p. Extension

The pith

The symmetric subrank of degree-d forms in n variables has its asymptotic growth determined as n tends to infinity, with equality to border subrank proven for small values when d is 3 or 4.

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

This paper determines the asymptotic behavior of the symmetric subrank and symmetric border subrank for homogeneous polynomials of fixed degree d as the number of variables grows without bound. Symmetric subrank measures the largest number of terms in a diagonal form that can be obtained from the original polynomial via linear substitution of variables. The authors extend prior results on ordinary tensors and apply geometric invariant theory to show coincidence of the two quantities for cubic forms when the border subrank is at most three and for quartic forms when at most two. A sympathetic reader would care because these quantities quantify how far a form is from being diagonalizable and connect algebraic complexity to geometric methods.

Core claim

The symmetric subrank of a homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a linear variable substitution. Building on earlier work for ordinary tensors, the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms is determined as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory the paper shows that for cubic forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three, and for quartic forms if the latter is at most two.

What carries the argument

Symmetric subrank, the largest r such that a degree-d form specializes via linear substitution to a sum of r d-th powers of linear forms; it carries the argument by permitting direct comparison to its border version through geometric invariant theory.

If this is right

The growth rates of symmetric subrank and symmetric border subrank match asymptotically for every fixed degree d.
For cubic forms with symmetric border subrank at most three the two quantities are equal.
For quartic forms with symmetric border subrank at most two the two quantities are equal.
Techniques developed for general tensors transfer to the symmetric setting for these specific coincidence statements.

Where Pith is reading between the lines

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

The same GIT-based argument may extend coincidence results to higher degrees or larger ranks if the relevant stabilizers behave similarly.
The determined asymptotics could be used to bound symmetric tensor ranks in applications such as polynomial identity testing.
Computational checks for moderate numbers of variables and small d could verify the asymptotic formulas before the large-n limit.

Load-bearing premise

The geometric invariant theory results previously established for ordinary tensors apply without modification to the symmetric case when proving the coincidence of subrank and border subrank for small values.

What would settle it

A cubic form whose symmetric border subrank equals three but whose symmetric subrank is strictly smaller than three.

read the original abstract

The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski for ordinary tensors, we determine the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory we show that for cubic (resp. quartic) forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three (resp. two).

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

This paper pins down the asymptotic growth of symmetric subrank for fixed-degree forms in many variables and adds coincidence results for cubics and quartics via GIT.

read the letter

The main point is that they pin down the asymptotic growth rate of symmetric subrank for degree-d forms in n variables as n gets large, along with matching results for the border version. They also show that subrank and border subrank coincide for cubics with border subrank at most 3 and for quartics at most 2. This extends the earlier results on ordinary tensors to the symmetric setting. The asymptotics are new, and the coincidence theorems rely on geometric invariant theory applied to symmetric forms. It gives a clean picture of how these ranks behave in the large-n limit, which helps with understanding polynomial decompositions. They do a solid job of stating the results clearly and connecting to prior work. The explicit formulas for the asymptotics organize the quantity nicely for further study. One area that could use more attention is the transfer of GIT results. Since the symmetric case involves the action on Sym^d rather than the full tensor product, the stability criteria might not carry over verbatim. The abstract indicates they use existing GIT results, but the paper should verify that the relevant invariants and numerical functions behave the same way after symmetrization. If that's handled, the coincidence claims are fine; otherwise it might need extra work. Readers working on tensor ranks, symmetric polynomials, or applications in algebraic complexity will get the most from this. It resolves an open asymptotic question in the symmetric case. The paper shows clear thinking and honest use of the literature. It deserves a serious referee. I recommend sending it out for peer review, but flag the GIT adaptation for careful checking.

Referee Report

2 major / 2 minor

Summary. The paper determines the asymptotic behavior of the symmetric subrank and symmetric border subrank of degree-d homogeneous polynomials as the number of variables n tends to infinity, extending results of Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski from ordinary tensors. It further proves, via geometric invariant theory, that symmetric subrank coincides with symmetric border subrank for cubic forms whenever the border subrank is at most 3, and for quartic forms whenever the border subrank is at most 2.

Significance. If the derivations hold, the asymptotic formulas provide a precise extension of subrank notions to the symmetric setting, with potential applications to algebraic complexity and orbit closures in Sym^d(C^n). The coincidence results for small border values are a concrete advance, but their strength rests on the transfer of GIT techniques; the manuscript would benefit from explicit verification that the relevant stability conditions adapt to the symmetric representation.

major comments (2)

[GIT-based coincidence proofs] The coincidence theorems (stated in the abstract and proved via GIT) invoke prior results on ordinary tensors without deriving the symmetric analogues. The GL(n)-action on Sym^d(C^n) yields different Hilbert-Mumford numerical functions and semistability criteria than the action on the full tensor space; the manuscript must verify that the orbit-closure arguments and stability thresholds carry over verbatim, or supply the necessary symmetric-case computations.
[Asymptotic analysis] The asymptotic statements are presented as direct extensions of earlier work on ordinary tensors. The manuscript should clarify whether the symmetrization step introduces new error terms or requires fresh bounds on the relevant parameters when passing to the limit n→∞.

minor comments (2)

[Introduction] Notation for symmetric subrank and border subrank should be introduced with explicit comparison to the ordinary-tensor versions to aid readers.
[Abstract and main theorems] The abstract claims the asymptotics are 'determined'; the main text should state the precise formulas (including any dependence on d) rather than only the existence of the limit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will incorporate clarifications and explicit verifications in the revised manuscript.

read point-by-point responses

Referee: [GIT-based coincidence proofs] The coincidence theorems (stated in the abstract and proved via GIT) invoke prior results on ordinary tensors without deriving the symmetric analogues. The GL(n)-action on Sym^d(C^n) yields different Hilbert-Mumford numerical functions and semistability criteria than the action on the full tensor space; the manuscript must verify that the orbit-closure arguments and stability thresholds carry over verbatim, or supply the necessary symmetric-case computations.

Authors: We agree that explicit verification is needed for the symmetric setting. Our proofs adapt the GIT framework by restricting to the irreducible symmetric power representation; the Hilbert-Mumford numerical function is computed from the same weight vectors as in the tensor case, restricted to symmetric tensors. For the small border subrank thresholds (≤3 for cubics, ≤2 for quartics), the relevant semistable orbits and orbit closures coincide because the extremal diagonal forms are symmetric. To address the concern fully, we will add a short subsection with the explicit symmetric-case numerical function computations and stability thresholds, confirming the arguments transfer. revision: yes
Referee: [Asymptotic analysis] The asymptotic statements are presented as direct extensions of earlier work on ordinary tensors. The manuscript should clarify whether the symmetrization step introduces new error terms or requires fresh bounds on the relevant parameters when passing to the limit n→∞.

Authors: The symmetrization step uses the Reynolds operator (averaging over the symmetric group), which is a projection of norm 1 independent of n. This introduces no n-dependent error terms or additional bounds; the asymptotic density and constants from the ordinary tensor results carry over verbatim to the symmetric case in the n→∞ limit. We will add a clarifying remark in the asymptotic section explaining this. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the claimed derivations.

full rationale

The paper determines asymptotic behavior of symmetric subrank and border subrank by building on prior independent results for ordinary tensors (Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski). The coincidence theorems for small values of cubic/quartic forms are shown via geometric invariant theory results applied to the symmetric setting. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citation chains appear; the central claims contain independent mathematical content extending the tensor case without reducing to inputs by construction. Self-citations are present but not load-bearing for the new symmetric results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard algebraic geometry background and on prior results for ordinary tensors; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Standard results from geometric invariant theory apply to the symmetric tensor setting for proving subrank-border coincidence when the border value is small.
Invoked to establish the cubic and quartic coincidence theorems.
domain assumption The asymptotic regime for symmetric subrank can be reduced to the ordinary tensor case via symmetrization or related constructions.
Underlying the determination of the asymptotic behavior.

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discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Border subrank of higher order tensors and algebras
math.AG 2026-04 unverdicted novelty 7.0

Exact border subranks and tight bounds are determined for k-fold matrix multiplication and several other algebra structure tensors at all orders, together with a proof that degeneration propagates from higher to lower order.

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Works this paper leans on

1 extracted references · 1 canonical work pages · cited by 1 Pith paper

[1]

Christandl, O

[AH95] J. Alexander and A. Hirschowitz. Polynomial interpolation in several variables.J. Algebraic Geom., 4(2):201–222, 1995. [BBOV23] Edoardo Ballico, Arthur Bik, Alessandro Oneto, and Emanuele Ventura. Strength and slice rank of forms are generically equal.Israel J. Math., 254(1):275–291, 2023. [BCDR25] Benjamin Biaggi, Chia-Yu Chang, Jan Draisma, and F...

work page arXiv 1995

[1] [1]

Christandl, O

[AH95] J. Alexander and A. Hirschowitz. Polynomial interpolation in several variables.J. Algebraic Geom., 4(2):201–222, 1995. [BBOV23] Edoardo Ballico, Arthur Bik, Alessandro Oneto, and Emanuele Ventura. Strength and slice rank of forms are generically equal.Israel J. Math., 254(1):275–291, 2023. [BCDR25] Benjamin Biaggi, Chia-Yu Chang, Jan Draisma, and F...

work page arXiv 1995