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arxiv: 2606.23814 · v1 · pith:PEJNTGF2new · submitted 2026-06-22 · 🧮 math.AG · math.AC

Polynomials of minimal border rank

Cosimo Flavi , Weronika Obcowska , Tim Seynnaeve This is my paper

Pith reviewed 2026-06-26 06:09 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords border rankGorenstein algebrashomogeneous polynomialssmoothable rankmultiplication tensorstensor classificationalgebraic geometry
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The pith

A correspondence with iterated multiplication tensors of Gorenstein algebras classifies homogeneous polynomials of minimal border rank for high degree in up to seven variables.

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

The paper establishes a classification of homogeneous polynomials that achieve minimal border rank when the number of variables is at most seven and the degree is large enough. It does so by invoking a known link that identifies such polynomials with iterated multiplication tensors coming from Gorenstein algebras of minimal smoothable rank. A reader would care because border rank measures the complexity of a polynomial or tensor, and the minimal cases often determine the geometry of the entire rank stratification. The method converts the border-rank question into an algebraic classification problem that becomes tractable once the degree exceeds a threshold depending on the number of variables.

Core claim

Using the correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank, the authors classify all homogeneous polynomials of minimal border rank of sufficiently high degree in up to seven variables.

What carries the argument

The correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank, which equates the two minimal-rank problems.

If this is right

  • All minimal-border-rank polynomials in the stated range arise from Gorenstein algebras whose multiplication tensors achieve minimal smoothable rank.
  • Explicit lists or parametrizations of the polynomials become available once the Gorenstein algebras are enumerated.
  • Geometric properties of the minimal-border-rank locus can be read off from the corresponding algebra structures.
  • The classification is exhaustive for each fixed number of variables up to seven once the degree is high enough.

Where Pith is reading between the lines

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

  • The same correspondence could be used to test whether the minimal-border-rank condition stabilizes at lower degrees than the paper requires.
  • The classification supplies candidate examples that might be checked for uniqueness or for behavior under specialization to fewer variables.
  • If the correspondence extends beyond seven variables, the same method would produce classifications in higher dimensions once the relevant Gorenstein algebras are known.

Load-bearing premise

The correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank remains valid and supplies a complete list for high degree.

What would settle it

Exhibit one homogeneous polynomial in at most seven variables, of degree above the paper's threshold, that has minimal border rank yet does not arise from any iterated multiplication tensor of a Gorenstein algebra of minimal smoothable rank.

read the original abstract

We use the correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank to classify polynomials of minimal border rank of sufficiently high degree in up to 7 variables.

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.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to classify homogeneous polynomials of minimal border rank of sufficiently high degree in up to 7 variables by applying the correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank.

Significance. If the correspondence is valid and the classification is complete and correct, the result would supply explicit, concrete information on border rank in low numbers of variables, a regime where such classifications remain rare in the literature on tensors and algebraic geometry.

minor comments (1)
  1. The abstract states the classification result but provides no indication of the degree threshold or the explicit forms obtained; the manuscript should include a clear statement of the degree bound and at least one low-variable example in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their report and summary of the manuscript. The recommendation is listed as uncertain, but the report contains no specific major comments to address. We therefore provide no point-by-point responses. If the uncertainty concerns the validity of the Gorenstein algebra correspondence or the completeness of the classification, we note that these are established in the literature cited in the paper and are applied directly here.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper invokes an external correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank as the basis for its classification of minimal border rank polynomials in high degree. No step in the claimed derivation chain reduces a result to its own inputs by construction, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose validity is established only inside the present work. The classification is presented as an application of the cited correspondence rather than a self-referential or tautological construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5543 in / 922 out tokens · 33070 ms · 2026-06-26T06:09:19.206678+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Hankel and Multiplication Tensor Completions for Cactus Rank

    math.AC 2026-06 unverdicted novelty 6.0

    Establishes equivalence between Hankel flat extension and multiplication tensor completion for cactus rank in Artinian Gorenstein algebras, plus reduction of basis shapes via Borel-fixed staircases.

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