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Strength and partition rank under limits and field extensions

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Pith's one-line read Border rank polynomially bounds strength and partition rank.

desk verdict Strong paper with real new results on de-bordering for strength and partition rank, but the finite-field branch rests on a degree estimate whose written proof doesn't deliver the claimed log bound. read the letter →

arxiv 2502.10007 v2 pith:QX2GOWMV submitted 2025-02-14 math.AG cs.CCmath.RT

Arthur Bik , Jan Draisma , Amichai Lampert , Tamar Ziegler This is my paper · ORCID
classification math.AGcs.CCmath.RT MSC 15A6914L30
keywords strengthpartitionrankborderde-borderingfieldextensionspolynomialfunctorshomogeneouspolynomialstensors
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

Strength is the minimal number of products of lower-degree homogeneous polynomials needed to express a form; partition rank is the analogous measure for multilinear tensors. For fixed degree $d$, the paper proves that the border versions of these ranks, defined via Zariski closures, control the actual ranks over the ground field by a polynomial bound: if the border strength is $r$, then the strength over $K$ is at most $O_d(r^{d-1})$ for infinite $K$ and $O_d(r^{d-1}\log r)$ for finite $K$, and the analogous statements hold for partition rank (with no characteristic restriction) and for tuples. This makes the previously non-explicit or field-dependent control of rank drops under field extensions and rank jumps in limits explicit and uniform. A by-product shows that a low-border-strength form lies in a subalgebra generated by few elements of its derivative space.

What carries the argument

The load-bearing mechanism is a reconstruction theorem (Theorem 3.0.1), imported from [BDE19, Section 4] and generalized to $m$-tuples of tensors. It says that if a closed subvariety $X$ of $m$-tuples of tensors is defined by polynomials with coefficients in the prime field, and a nonzero partial derivative $h=\partial f/\partial x_1$ of a defining polynomial does not vanish at a tuple, then the $m$-th tensor can be reconstructed from the other $m-1$ tensors and from smaller tensor factors, so the collective partition rank is bounded by the number of small-tensor summands in a shift of $T_{[d]}^m$. The proof then runs an induction on the minimal degree of a vanishing polynomial for such subvarieties: either the tuple lands in a smaller subvariety where the induction hypothesis applies, or the reconstruction theorem bounds its partition rank. For finite fields, an additional degree estimate (Proposition 4.0.1) shows the minimal degree of a vanishing polynomial is $O_d(\log(r+m))$, and a finite-field-extension trick (Proposition 2.2.1) multiplies the partition rank bound by the extension degree, producing the $\log$-factor.

What would settle it

Exhibit, for some fixed degree $d\ge 3$ and some field $K$ with $\mathrm{char}(K)=0$ or $>d$, a sequence of forms $f_N$ with border strength $s(f_N)\le r_N$ but honest strength $s_K(f_N)$ growing faster than $C_d r_N^{d-1}$ (over infinite $K$) or faster than $C_d r_N^{d-1}\log r_N$ (over finite $K$), for every constant $C_d$; an analogous counterexample for tensors would disprove Theorem 1.6.3.

Watch

Extended reading notes

Core claim

The central discovery is a de-bordering theorem: border rank is not just a topological relaxation but a true upper bound on actual rank, up to a polynomial in fixed degree. For strength, Theorem 1.2.3 establishes that if a degree-$d$ form $f$ has border strength $s(f)=r$ and $\mathrm{char}(K)=0$ or $>d$, then $s_K(f)\ll_d r^{d-1}$ for infinite $K$ and $s_K(f)\ll_d r^{d-1}\log r$ for finite $K$. For partition rank, Theorem 1.6.3 gives the same polynomial control with no characteristic condition, and the collective versions (Theorems 1.5.3 and 1.7.2) extend the bounds to $m$-tuples with an extra factor $m^3$ and, in the finite-field case, a factor $\log(r+m)$. The paper also derives Theorem 1.3.1: such a form lies in a subalgebra generated by $\ll_d r^d$ (or $\ll_d r^d\log r$ over finite fields) elements of the space $D(f)$ of its partial derivatives. The paper thus establishes polynomial bounds that are explicit in $d$ and uniform across all fields satisfying the stated characteristic assumptions.

Load-bearing premise

The proof depends on the reconstruction theorem (Theorem 3.0.1), imported from earlier work, that a single nonzero partial derivative of a defining polynomial lets one rebuild the last tensor from the others; if that theorem or its $m$-tuple generalization fails, the polynomial bounds collapse.

Editorial extensions

If this is right

  • For any field $K$ with $\mathrm{char}(K)=0$ or $>d$, a form of border strength $r$ has honest strength at most $C_d r^{d-1}$; hence ranks cannot jump by more than a fixed polynomial in a limit.
  • Partition rank of tensors obeys the same polynomial control with no characteristic restriction, giving a field-independent de-bordering result for tensors.
  • The collective versions imply that for $m$-tuples of forms or tensors, bounded border collective rank bounds the actual collective rank by $O_d(m^3 r^{d-1})$ (with an extra $\log(r+m)$ over finite fields).
  • Because strength over a field extension can only drop, the bounds control the drop: passing to the algebraic closure can reduce strength by at most the same polynomial factor.
  • Theorem 1.3.1 yields a structural statement: low-border-strength forms are generated as a polynomial algebra by at most $O_d(r^d)$ (or $O_d(r^d\log r)$ over finite fields) elements of their derivative space.

Reading between the lines

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

  • The proof's constants depend only on $d$, but the paper does not compute them; tracking them through the induction and the degree estimates could yield explicit (not just existential) bounds, which would make the de-bordering algorithmic.
  • The log factor for finite fields comes from the degree estimate in Proposition 4.0.1; if that estimate could be improved to $O_d(1)$, the finite-field bounds would become polynomial without a log term, matching the infinite-field rate.
  • The same reconstruction-from-derivative scheme could apply to other rank models (for instance, Waring rank or analytic rank) whenever an analogous defining-polynomial and derivative pair exists, potentially extending de-bordering beyond strength and partition rank.
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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

3 major / 4 minor

Summary. The paper studies how the strength and partition rank of homogeneous polynomials and tensors behave under field extensions and under passage to limits (border rank). The main theorems (Theorems 1.2.3, 1.6.3, 1.7.2) assert that for fixed degree d, the ordinary strength or partition rank is bounded by a polynomial in the corresponding border rank, uniformly in the ground field under mild characteristic assumptions, with an additional logarithmic factor over finite fields. The proofs use polynomial functor machinery, import a reconstruction theorem from the authors' earlier work [BDE19], and proceed by dimension counts and, in the finite-field case, by a degree bound for equations vanishing on the relevant parameterized varieties. The paper also derives consequences for the drop of ranks under field extensions and for jumps under limits.

Significance. If the theorems are correct, they constitute a significant advance: they give the first quantitative, field-uniform de-bordering results for strength and partition rank in fixed degree, subsuming and strengthening earlier work of Lampert–Ziegler and others. The infinite-field part of the proof is coherent and appears sound, and the paper has useful corollaries (for example, control of rank drops under field extensions and control of jumps in limits). The manuscript is clearly written and the framework of polynomial functors is well suited to the problem. However, the finite-field branch rests on a delicate degree estimate (Proposition 4.0.1) that is not established by the written argument; this gap affects several of the main stated results.

major comments (3)
  1. [Section 4] The argument that log(D) ≪_d log(r+m) does not follow from the displayed binomial counting. After dividing the inequality (1/4)n^d log D ≥ m n^d log(m n^d) + (m-1/4)n^d log(4e/((m-1/4)n^d)) by (1/4)n^d, one obtains log D ≥ 4m log(m n^d) + 4(m-1/4) log(4e/((m-1/4)n^d)). With n = 4(r + m^2/d), the first term is of order m log(r+m), which is not O_d(log(r+m)). The subsequent "sufficient condition" displayed in the paper, log D ≥ 4 log(m n^d) + 4(m-1/4) log(3e/((m-1/3)n^d)), is not an algebraic consequence of the prior inequality; it appears to drop the factor m from the first term. Thus the claimed bound is not proven, and because Proposition 4.0.1 is the basis for the extension-degree estimate [L:K] ≪_d log(r+m) in the finite-field proof of Theorem 1.7.2, the finite-field cases of Theorems 1.7.2, 1.6.3, 1.5.3, 1.2.3, and 1.1.2 are not established.
  2. [Section 3] The theorem is introduced as a summary of results from [BDE19, §4] "but generalised from tensors to m-tuples of tensors". No proof or precise reference for the m-tuple generalization is given. This theorem is load-bearing: Corollary 3.0.2 and the inductive argument in the proof of Theorem 1.7.2 depend on it. The authors should either prove the m-tuple version or indicate explicitly where in [BDE19] the m-tuple case is proved.
  3. [Section 3] The statement that the minimal n satisfying (4) is "linear in r + m2/d" is ambiguous and, if read as r + m^2/d, is false (for d=3 the minimal n is O(r + m^{2/3})). The proof of the infinite-field case of Theorem 1.7.2 uses this to conclude that the expression in (6) is ≪_d m^3 r^{d-1}. The authors should clarify the exponent and confirm that (6) is indeed bounded as claimed for their choice of n. If the intended bound is n = O(r + m^{2/d}), the argument goes through; if n = 4(r + m^2/d) is used, the bound (6) becomes O(m r^{d-1} + m^{2d-1}), which is not the claimed m^3 r^{d-1} in all regimes.
minor comments (4)
  1. [Section 4] In Proposition 4.0.1, the notation "n = 4(r +m2/d)" is ambiguous; please write either 4(r + m^2/d) or 4(r + m^{2/d}) consistently throughout the paper.
  2. [Section 4] In the derivation within Proposition 4.0.1, the transition from the logarithmic inequality to the displayed "sufficient condition" is algebraically incorrect; please correct the equations and re-derive the bound.
  3. [Section 3] The reference for Theorem 3.0.1 cites [BDE19, §4] but does not give a theorem number or page; adding one would help the reader verify the m-tuple generalization.
  4. [Title and Abstract] There are several typographical errors (e.g., "P ARTITION" in the title, "th e" and "anal ogue" in the abstract) that should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the de-bordering theorems are derived from a prior, parameter-free reconstruction theorem in BDE19; the self-citation is load-bearing but independent, and neither the strength nor the partition-rank conclusion is assumed in its hypotheses.

full rationale

The central claim (Theorem 1.7.2) is obtained by applying Theorem 3.0.1, quoted as 'a summary of results proved in [BDE19, §4]', to the subvariety X_r of m-tuples of border partition rank at most r. That theorem is a published, parameter-free statement about reconstructing the m-th tensor from the remaining ones when a partial derivative of a defining polynomial is nonzero; its assumptions do not contain the partition-rank bound being proven, so citing it is real evidence rather than circularity. The m-tuple generalization is asserted rather than proved in this text; if it were nontrivial that would be a missing proof, not a circular reduction. The finite-field branch depends on Proposition 4.0.1's bound log D ≪_d log(r+m); the proof's binomial estimates contain the line 'these monomials have more structure, but we ignore this', and the displayed sufficient condition as written appears to contain a term linear in m, so the written derivation of that proposition may be incomplete. That is a potential correctness gap, not an instance of the paper fitting a parameter and renaming it a prediction or defining the target into the input. The strength results are then deduced from the partition-rank results through the standard linear maps π and ι (Proposition 2.1.1), not by assuming the strength bound. Hence no step in the claimed derivation reduces by construction to its own input; the self-citations are load-bearing but external and independent.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central proof is an argument over polynomial functors. It imports one heavy theorem from BDE19 by two of the present authors; that theorem is not the target result and is treated as an independent published tool. No data are fitted and no entities are invented. The only hand-chosen quantity in the proof is the auxiliary dimension n, which is used to force a nontrivial vanishing polynomial and disappears from the final bound.

free parameters (1)
  • Auxiliary vector-space dimension n = 4(r + m^2/d)
    Chosen in Section 4 so that inequality (4) holds and a nonzero vanishing polynomial exists on X_r(U). It is an internal proof parameter, not a physical or final-model parameter.
assumptions (3)
  • standard math Reconstruction theorem for closed subvarieties of tensor tuples (Theorem 3.0.1, from BDE19 Section 4, generalized to m-tuples).
    Load-bearing tool: from a nonzero partial derivative of a defining polynomial it reconstructs the m-th tensor coordinate and bounds collective partition rank. It is cited, not proved here.
  • standard math Constructibility: images of algebraic morphisms are constructible, so border rank is the Zariski closure of the union of parameterisation images (Chevalley's theorem).
    Used in Section 1.2 and Proposition 3.0.5 to define border strength and partition rank, and to produce a vanishing polynomial.
  • domain assumption Polynomial functor formalism and covariance under GL(V_i) used to identify the M-component of the reconstruction map as a combination of tensor-product morphisms.
    Invoked in Corollary 3.0.2 via BDE19 Section 4.8; the current paper relies on this structural decomposition.

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Cite this review

Pith. "Pith review of Strength and partition rank under limits and field extensions." pith.science (2026). https://pith.science/paper/QX2GOWMV

@misc{pith2026250210007,
  author       = {Pith},
  title        = {Pith review of: Strength and partition rank under limits and field extensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QX2GOWMV}},
  note         = {Machine review of arXiv:2502.10007}
}
read the original abstract

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop under field extensions, and both can jump in a limit. We show that, for fixed degree and under mild conditions on the characteristic of the ground field, the strength is at most a polynomial in the border strength. We also establish an analogous result for partition rank. Our results control both the jump under limits and the drop under field extensions.

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Forward citations

Cited by 1 Pith paper

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    Over Brauer fields, systems of degree-d forms with Birch rank above s^{2^{d-1}} C_d have Zariski dense solution sets, with C_d an explicit product of diagonal-form thresholds.

Reference graph

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