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Sharp bounds for minimal dependencies of linear-form powers

T0 review · 0 major / 3 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read If t linear forms have m-th powers forming a minimal dependence, their span has dimension at most (t+m-2)/m.

desk verdict The paper proves the sharp bound dim L ≤ (t+m-2)/m for spans of linear forms with circuit m-th powers, via a direct Schur-power reduction to the Mirandola-Zémor Kneser theorem. read the letter →

arxiv 2606.24349 v1 pith:FJDFMUD3 submitted 2026-06-23 math.CO

Heng Li , Xizhi Liu This is my paper · ORCID
classification math.CO
keywords VeronesecircuitslinearformsSchurpowersKnesertheoremdimensionboundsminimaldependenciesrationalnormalcurvescircuitdependence
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 when the m-th powers of t nonzero homogeneous linear forms are minimally linearly dependent over a field of characteristic zero, the dimension of the span of those forms is bounded above by (t + m - 2)/m. This bound is attained by configurations arising from rational normal curves for infinitely many pairs of t and dimension. The result settles the dimension part of a question of Bukh and shows that the optimal leading constant there is 1/m. The argument translates the setup into coding theory, where the coefficient space of the powers becomes the m-th Schur power of a code that is forced to be a full-support hyperplane, allowing application of a Schur-product version of Kneser's theorem.

What carries the argument

The m-th Schur power of the coefficient row space of the linear forms, which the circuit assumption turns into a full-support hyperplane to which the Schur-product Kneser theorem applies.

What would settle it

An explicit set of t linear forms over a characteristic-zero field whose m-th powers form a circuit but whose span has dimension strictly larger than (t + m - 2)/m would disprove the bound.

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Extended reading notes

Core claim

If ℓ1,…,ℓt are nonzero homogeneous linear forms over a field of characteristic zero such that the powers ℓ1^m,…,ℓt^m form a circuit, then dim L ≤ (t+m-2)/m where L = Span{ℓ1,…,ℓt}. Rational-normal-curve configurations attain equality for infinitely many pairs (t, dim L). The same method produces flat-concentration and interpolation criteria, a Cayley-Bacharach lower bound, Segre-Veronese and positive-characteristic variants, and Hilbert-function constraints for equality cases.

Load-bearing premise

The dependence among the m-th powers must be minimal, i.e., they must form a circuit, and the base field must have characteristic zero.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

Summary. The manuscript proves the sharp bound dim L ≤ (t + m - 2)/m, where ℓ1, …, ℓt are nonzero homogeneous linear forms over a characteristic-zero field whose m-th powers form a circuit and L = Span{ℓ1, …, ℓt}. Equality is attained by rational-normal-curve configurations for infinitely many pairs (t, dim L). The argument translates the circuit hypothesis into the statement that the m-th Schur power of the coefficient row space is a full-support hyperplane, then invokes the Mirandola–Zémor Schur-product Kneser theorem; the same reduction produces flat-concentration criteria, an interpolation criterion, a Cayley–Bacharach lower bound, Segre–Veronese and positive-characteristic variants, and Hilbert-function constraints for equality cases.

Significance. If the central claim holds, the paper supplies the sharp finite-dimensional bound for the dimension part of Bukh’s problem and identifies the optimal leading constant 1/m. The coding-theoretic translation is direct and parameter-free, and the explicit equality constructions via rational normal curves confirm sharpness for infinitely many parameters. The method simultaneously yields several auxiliary results on Hilbert functions and interpolation, increasing the manuscript’s utility beyond the main bound.

minor comments (3)
  1. [Introduction / coding-theoretic translation paragraph] The abstract states that the m-th Schur power is a full-support hyperplane, but the precise definition of the coefficient code and the row-space construction should be given explicitly in the first section that introduces the coding-theoretic translation (currently only sketched in the abstract).
  2. [Equality cases] The statement that rational-normal-curve configurations attain equality for infinitely many pairs (t, dim L) is asserted without a reference or self-contained verification; a short appendix or subsection exhibiting the explicit linear forms and verifying the circuit property for at least one infinite family would strengthen the sharpness claim.
  3. [Abstract] The positive-characteristic variant is mentioned only in the abstract; if the manuscript contains a full statement or counter-example, it should be cross-referenced from the main theorem so readers can locate the precise range of validity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the detailed summary of its contributions, and the recommendation of minor revision. No major comments appear in the report, so we have no specific points requiring response or revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; external theorem application

full rationale

The central bound is obtained by a coding-theoretic reduction that maps the circuit hypothesis on m-th powers to a full-support hyperplane in the m-th Schur power, then invokes the external Mirandola-Zémor Schur-product Kneser theorem (distinct authors). No self-citation is load-bearing, no parameter is fitted and renamed as a prediction, and equality cases are exhibited by explicit rational-normal-curve constructions rather than by redefinition. The derivation therefore remains independent of its own inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on two domain assumptions and an external theorem.

assumptions (2)
  • domain assumption The base field has characteristic zero.
    Required for the main theorem and for the applicability of the Schur-product Kneser theorem.
  • domain assumption The powers ℓ1^m,…,ℓt^m form a circuit (minimal linear dependence).
    Central hypothesis that makes the Schur power a full-support hyperplane to which the Kneser theorem applies.

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Cite this review

Pith. "Pith review of Sharp bounds for minimal dependencies of linear-form powers." pith.science (2026). https://pith.science/paper/FJDFMUD3

@misc{pith2026260624349,
  author       = {Pith},
  title        = {Pith review of: Sharp bounds for minimal dependencies of linear-form powers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FJDFMUD3}},
  note         = {Machine review of arXiv:2606.24349}
}
abstract

Motivated by the dimension-bound part of a problem of Bukh, we study Veronese circuits: how large can the span of $t$ linear forms be if their $m$-th powers are minimally linearly dependent? We prove the sharp finite dimension bound \[ \dim L\leq \frac{t+m-2}{m}. \] Here $\ell_1,\ldots,\ell_t$ are nonzero homogeneous linear forms over a field of characteristic zero, the powers $\ell_1^m,\ldots,\ell_t^m$ form a circuit, and $L=\Span\{\ell_1,\ldots,\ell_t\}$. Rational-normal-curve configurations attain equality for infinitely many pairs $(t,\dim L)$; in particular, the affine bound itself is sharp and the optimal leading constant in Bukh's question is $1/m$. The proof uses a coding-theoretic translation: the coefficient row space of the powers is the $m$-th Schur power of the coefficient code, and the minimality hypothesis makes this Schur power a full-support hyperplane to which the Schur-product Kneser theorem of Mirandola and Z\'emor applies. The same method yields flat-concentration and interpolation criteria, a Cayley--Bacharach lower bound, Segre--Veronese and positive-characteristic variants, and Hilbert-function constraints for equality and near-equality in Veronese circuits.

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Reference graph

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