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Bounds for Geometric rank in Terms of Subrank

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Pith's one-line read Subrank bounds geometric rank for fixed-order tensors — and quadratically in order three.

desk verdict Genuinely new bounds for geometric rank in terms of subrank; the order-three theorem is solid, but the proof of the general large-field theorem has a repairable but real gap in Lemma 5.1. read the letter →

arxiv 2506.16132 v1 pith:GGSMWPS4 submitted 2025-06-19 math.CO cs.CCmath.ACmath.AG

Qiyuan Chen , Ke Ye This is my paper · ORCID
classification math.COcs.CCmath.ACmath.AG MSC 15A6905D0513P10
keywords geometricranksubrankborderpartitionanalyticfieldextensionR1-sequencestrengthofforms
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

The paper proves that geometric rank, a tensor invariant measuring the codimension of the set of directions where a tensor's multilinear form vanishes, is always dominated by subrank, the size of the largest identity tensor that can be embedded into the tensor. For order-three tensors over any field the domination is quadratic: an explicit bound GR(T) ≤ 2Q(T)^2 + 3Q(T) holds over infinite fields, with an explicit quadratic bound over finite fields as well. For tensors of arbitrary fixed order, geometric rank is bounded by a function of subrank over algebraically closed fields, and over fields of characteristic zero or characteristic larger than the order the same conclusion holds after passing to a field extension of bounded degree. These results close an open problem on whether subrank controls geometric rank, and they imply new bounds on partition rank, analytic rank, and border subrank.

What carries the argument

The machinery is a two-layer reduction. First, Proposition 4.2 expresses GR(T) as min_c(codim X_c + c), where X_c is the set of d-th slices of T having geometric rank c, so the entire tensor's geometric rank is controlled once the slices' ranks are controlled. Second, the slices are controlled by commutative algebra: repeatedly applying Proposition 3.14 (bounded-length R1-sequences of forms, i.e. sequences whose successive quotient rings stay regular in codimension one) and Proposition 3.16 (rational points in bounded-degree extensions) produces vectors that avoid the derivative ideal of the slice span while retaining one prescribed nonzero value, forcing the Kronecker-delta contractions that build an identity tensor of size Q(T). The quantitative strength bounds underlying these propositions, and their transfer under field extensions, are what turn an existence argument into an explicit GR(T) ≤ A(d, Q(T)) bound. In the order-three case, Lemma 2.2 replaces the heavier algebra: c linearly independent slices whose nonzero linear combinations all have rank at least 2c(c−1) already force Q(T) ≥ c, so the proof reduces to a Grassmannian counting estimate.

What would settle it

Enumerate order-three tensors of small format (for example, 4×4×4 over Q or F_2), compute Q(T) by exhaustive restriction search and GR(T) by the codimension of the zero set of the associated multilinear form, and check the claimed quadratic inequality; a single tensor with GR(T) > 2Q(T)^2+3Q(T) over an infinite field would refute Theorem 1.4.

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Extended reading notes

Core claim

The central claim is that for tensors of fixed order, subrank is a genuine measure of complexity: it controls geometric rank quantitatively. Theorem 1.4 gives GR(T) ≤ 2Q(T)^2 + 3Q(T) for every order-three tensor over an infinite field, and an explicit quadratic polynomial for finite fields, so GR(T) = O(Q(T)^2). The quadratic growth is optimal, since for a generic n×n×n tensor one has Q(T) = Θ($n^{{1/2}}$) and GR(T) = n. For algebraically closed fields, Theorem 1.2 supplies a function C(d, Q(T)) with GR(T) ≤ C(d, Q(T)) for every order d, answering the open problem raised in [35, Section 9]; Theorem 1.1 is the analogue for large characteristic and bounded-degree extensions. The proof reworks the geometry of slices: GR(T) is read off from the geometric ranks of its d-slices, and high geometric rank of the span of the slices is forced to produce the contractions that realize a large identity tensor.

Load-bearing premise

The argument depends on a quantitative trade-off: the geometric rank of the space spanned by the tensor's slices must be large enough relative to the number of low-degree forms needed to generate the ideal they avoid, and this transfer must survive passing to a field extension; if that gap closes at any induction step, the construction that forces a large identity restriction collapses.

Editorial extensions

If this is right

  • For order-three tensors, subrank is stable under field extensions: Q_K(T) = O(Q(T)^2), generalizing known real-field stability to arbitrary fields (Corollary 1.6).
  • Subrank of a direct sum is controlled by the sum of the bounding functions: Q(S⊕T) ≤ C(d,Q(S)) + C(d,Q(T)), and in order three the maximal quadratic gap between Q(S⊕T) and Q(S)+Q(T) is attained (Corollary 1.8).
  • Border subrank is de-bordered: Q(T) ≤ Q̄(T) ≤ C(d,Q(T)) over algebraically closed fields, with a quadratic bound in order three (Corollary 1.10).
  • For order-three tensors, partition rank and analytic rank are each O(Q(T)^2), the first upper bounds of this kind (Corollaries 1.11 and 1.12).
  • A gap theorem follows for the subrank of Kronecker powers over algebraically closed fields: either every power has subrank 1, or subrank grows at least exponentially in k/2 (Corollary 1.13).

Reading between the lines

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

  • The same slice-avoidance technique may yield a polynomial bound of degree d−1 for all orders if the base-field transfer in Propositions 4.5 and 4.6 can be made effective over arbitrary fields; the paper itself proposes this as Conjecture 8.1, and the order-three case is its first nontrivial confirmation.
  • Because geometric, partition, and analytic rank are linearly equivalent over large fields, the quadratic-in-subrank bound effectively promotes a combinatorial lower bound on subrank into a geometric lower bound on all three ranks; cap-set-style constructions that bound subrank from below may now yield new extremal results.
  • The explicit constants in Theorem 1.4 are small enough that the proof is close to algorithmic: for a tensor with subrank r, a certificate of geometric rank could in principle be exhibited as a slice of the form built by the induction.
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Editorial analysis

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

Referee Report

3 major / 4 minor

Summary. The paper studies the relation between geometric rank GR(T) and subrank Q(T) for tensors of fixed order d over a field K. Its three main theorems are: (Theorem 1.1) under a characteristic condition, GR(T) is bounded by a function of the subrank over a field extension of bounded degree; (Theorem 1.2) over algebraically closed fields, GR(T) ≤ C(d, Q(T)) for a function C, resolving an open problem of Kopparty–Moshkovitz–Zuiddam; and (Theorem 1.4) for order-three tensors over arbitrary fields, GR(T) = O(Q(T)^2), with an explicit quadratic bound for infinite fields. The paper derives several corollaries concerning stability of subrank, direct sums, de-bordering of border subrank, bounds on partition and analytic rank, and a gap theorem. The proofs are built on polynomial-ideal results in the style of Ananyan–Hochster, rational-point lemmas in bounded-degree extensions, and structural lemmas about geometric rank.

Significance. If the proofs are correct, this is a substantial contribution. The order-three quadratic bound is optimal for generic tensors, and the algebraic-closure result answers a named open problem. The applications to border subrank, partition rank, and analytic rank are natural and have not, to my knowledge, appeared before. The paper is generally careful and self-contained, and it gives explicit functions in several places, including the clean bound GR(T) ≤ 2Q(T)^2 + 3Q(T) for infinite fields.

major comments (3)
  1. [§2.2, definition of R1-sequence] The sentence "Clearly, an R1-sequence of forms in K[x_1,...,x_n] is a prime sequence" is false. For the form xy in K[x,y], the quotient K[x,y]/(xy) satisfies Serre's condition (R1) but is not a domain. This implication is used in Proposition 3.16 to conclude that (f_1,...,f_m) is prime and that f_1,...,f_m,g form a regular sequence, and in Lemma 3.15 to treat the sequence produced by Proposition 3.14 as regular. Since Proposition 3.16 is invoked in Step 1.2 of Lemma 5.1 and is similarly reused in Lemma 6.1, Theorems 1.1 and 1.2 are not supported as written. The authors should either prove that the specific R1-sequences produced by Propositions 3.13 and 3.14 and Theorem 2.5 are prime sequences, or replace this construction by a theorem that produces prime sequences.
  2. [§5, Lemma 5.1, Step 1.1] The displayed contradiction "I(d,N+(d−1)m) ≥ GR(span_K{T_1,...,T_c})/2 ≥ M_1(d,c)/2" is unjustified. From g ∈ (g_1,...,g_t) one obtains str_{L_m}(g) ≤ t ≤ I, hence GR_{L_m}(T'_{m+1}) ≤ 2I by Proposition 4.5; however T'_{m+1} lies in span_{L_m}{T_i}, not necessarily in span_K{T_i}. To contradict assumption (iii) one must bound GR_K(span_K{T_i}), and Proposition 4.6 gives GR_K(span_K{T_i}) ≤ J(d,c,2I), not ≤ 2I. The condition M_1(d,c) > 2I(...) does not imply M_1(d,c) > J(d,c,2I(...)). The argument can likely be repaired by defining M_1 through J ∘ I, but as written the induction in Lemma 5.1 is incomplete.
  3. [§5, proof of Theorem 1.1] The definition of the integer s is off by one. The contrapositive of Lemma 5.1 with c = s requires Q_F(T) ≤ s−1 for every extension F/K with [F:K] ≤ M_2(d,s), but the text says "degree M_2(d,s−1)" and then uses the same index in the subsequent minimality argument. Replacing M_2(d,s−1) with M_2(d,s) in the definition appears to fix the step, but as written the conclusions [L:K] = M_2(d,s−1) ≤ B(d,GR(T)) and GR(T) ≤ A(d,Q_L(T)) do not follow from the stated condition.
minor comments (4)
  1. [§1.3, Corollary 1.6] The notation Q_K(T) is not defined; if it denotes subrank over the algebraic closure, the equality GR_K(T) = GR(T) needs a justification or a reference, and the notation should be introduced explicitly.
  2. [§4, proof of Proposition 4.2] The expression "∪_{m}^{c=0}" should read "∪_{c=0}^{m}".
  3. [§6, proof of Theorem 1.2] The set X is defined as a subset of (K^{n_3})^* although the tensor has order d; it should be (K^{n_d})^*, and the condition should refer to GR(T_u), not rank(T_u).
  4. [References] In the reference list, "Disrete Analysis" should be "Discrete Analysis", and "Theroem B" should be "Theorem B".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main bounds are derived from independent ideal-theoretic and analytic-rank results, not from their own conclusions.

full rationale

The paper's derivation chain is self-contained rather than circular. The central bounds GR(T) <= A(d,Q_L(T)), GR(T) <= C(d,Q(T)), and GR(T) <= 2Q(T)^2+3Q(T) are obtained by reducing the geometric rank to the height of a polynomial ideal via Proposition 4.2, then combining height/strength bounds from Ananyan–Hochster ([4], Theorems 2.5 and 2.7), strength-stability results ([10], Theorem 2.8), and a geometric-rank stability estimate (Proposition 4.6) that derives from external theorems. Lemma 5.1's contradiction uses only the lower bound GR(span_K{T_1,...,T_c}) >= M_1(d,c) together with the independent inequality 2 str(T) >= GR(T); it does not assume the theorem being proved. The finite-field branches invoke the analytic-rank/geometric-rank equivalence from [5,14,46] (Theorem 2.3), which is jointly attributed and does not mention subrank, so the use of the authors' own [14] is not load-bearing and not circular. Theorem 1.4 is proved directly from the external Lemma 2.2 and Proposition 4.2, with no fitted parameter renamed as a prediction and no ansatz smuggled in through self-citation. The proof-level gap noted in Lemma 5.1, concerning the field-extension transfer between GR_{L_m} and GR_K, is a correctness concern rather than a circularity concern, because nothing in that step reduces the theorem's conclusion to its hypothesis. Therefore no step in the claimed derivation is equivalent by construction to its inputs, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard commutative algebra and algebraic geometry, on non-explicit functions from Ananyan-Hochster and related work, and on previously established inequalities such as Q(T) <= GR(T) and AR(T) approx GR(T). There are no fitted parameters and no new postulated objects.

assumptions (5)
  • standard math Ananyan-Hochster-type bounded generation of subalgebras by R1-sequences and strength bounds.
    Imported from [4] and [10] as Theorems 2.5, 2.7 and 2.8; these non-explicit functions control the main bounds in Theorems 1.1, 1.2 and 1.4.
  • domain assumption Linear equivalence between analytic rank and geometric rank over finite fields.
    Imported from [5,14,46] as Theorem 2.3; used in Proposition 4.4 and in the finite-field case of Theorem 1.1. The present authors' own [14] is one of the three cited sources.
  • domain assumption Subrank is bounded above by geometric rank.
    Imported from [35, Theorem 5]; used to compare Q_L(T) with GR(T) in the proof of Theorem 1.1 and in several corollaries.
  • domain assumption Characteristic condition char(K)=0 or char(K)>d is sufficient for the strength and derivative arguments.
    Structural condition in Theorem 1.1 and Propositions 3.13, 3.14, 3.15 and 3.16; it is removed for algebraically closed fields and for d=3.
  • standard math Hilbert Nullstellensatz, dimension theory and height arguments for polynomial rings.
    Used throughout Sections 3 and 4, for example in Lemma 3.1 and Proposition 3.16, without proof.

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Pith. "Pith review of Bounds for Geometric rank in Terms of Subrank." pith.science (2026). https://pith.science/paper/GGSMWPS4

@misc{pith2026250616132,
  author       = {Pith},
  title        = {Pith review of: Bounds for Geometric rank in Terms of Subrank},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGSMWPS4}},
  note         = {Machine review of arXiv:2506.16132}
}
read the original abstract

For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is bounded by a function in its subrank in some field extension of bounded degree. Secondly, we show that, over any algebraically closed field, the geometric rank of a tensor is bounded by a function in its subrank. Lastly, we prove that, for any order three tensor over an arbitrary field, its geometric rank is bounded by a quadratic polynomial in its subrank. Our results have several immediate but interesting implications: (1) We answer an open question posed by Kopparty, Moshkovitz and Zuiddam concerning the relation between the subrank and the geometric rank; (2) For order three tensors, we generalize the Biaggi-Chang- Draisma-Rupniewski (resp. Derksen-Makam-Zuiddam) theorem on the growth rate of the border subrank (resp. subrank), in an optimal way; (3) For order three tensors, we generalize the Biaggi- Draisma-Eggleston theorem on the stability of the subrank, from the real field to an arbitrary field; (4) We confirm the open problem raised by Derksen, Makam and Zuiddam on the maximality of the gap between the subrank of the direct sum and the sum of subranks; (5) We derive, for the first time, a de-bordering result for the border subrank and upper bounds for the partition rank and analytic rank in terms of the subrank; (6) We reprove a gap result for the subrank.

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