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arxiv: 2605.04859 · v1 · submitted 2026-05-06 · 🧮 math.AG · math.CO

Recognition: unknown

Geometry of multilinear varieties over infinite fields and its applications

Ke Ye, Qiyuan Chen

Pith reviewed 2026-05-08 16:17 UTC · model grok-4.3

classification 🧮 math.AG math.CO
keywords multilinear varietiesZariski closurecodimension formulapartition ranktensor rankscollective strengthBirch rankinfinite fields
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The pith

Multilinear varieties over infinite fields have a codimension formula for their Zariski closures and contain high-dimensional irreducible subvarieties through any rational point.

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

The paper develops a geometric theory of multilinear varieties over infinite fields, which are the sets of rational points on varieties cut out by multilinear functions. It proves that the Zariski closure of such a variety has codimension determined by the defining multilinear equations and that every rational point lies on a high-dimensional irreducible subvariety. These two facts are applied to study ranks of tensors and homogeneous polynomials. The results establish stability properties for partition rank over perfect infinite fields and settle related statements on collective strength and the equivalence of strength with Birch rank.

Core claim

We establish a codimension formula for the Zariski closure of a multilinear variety over an infinite field and prove the existence of a high-dimensional irreducible subvariety passing through any given rational point. These geometric results provide a foundation for analyzing partition rank, analytic rank, geometric rank, collective strength, and Birch rank. They resolve the stability conjecture for partition rank over perfect infinite fields, settle the stability conjecture for collective strength, and establish the linear equivalence between strength and Birch rank for such fields.

What carries the argument

The codimension formula for the Zariski closure of a multilinear variety together with the existence of high-dimensional irreducible subvarieties through any rational point.

If this is right

  • Partition rank is stable over perfect infinite fields.
  • Collective strength satisfies stability over perfect infinite fields.
  • Strength and Birch rank are linearly equivalent over perfect infinite fields.
  • Prior results on multilinear varieties over infinite fields are strengthened.

Where Pith is reading between the lines

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

  • The same geometric facts might be adapted to study additional rank functions or varieties defined by other types of equations.
  • Analogous codimension and irreducibility statements over finite fields could lead to similar stability results in that setting.
  • These tools suggest studying tensor ranks through their geometric loci rather than purely algebraic definitions.

Load-bearing premise

The codimension formula and the existence of high-dimensional irreducible subvarieties through rational points hold for multilinear varieties over infinite fields, with the field required to be perfect for the stability conclusions.

What would settle it

A concrete multilinear variety over an infinite field whose Zariski closure has codimension different from the formula, or a rational point through which every irreducible subvariety has dimension below the guaranteed lower bound.

read the original abstract

Multilinear varieties, defined as the sets of rational points of varieties cut out by multilinear functions, were first introduced and studied by Gowers and Mili\'{c}evi\'{c}[Proc. Edinb. Math. Soc., 2021] for finite $\mathbb{K}$. In this paper, we investigate multilinear varieties over infinite fields from a geometric perspective. We establish two fundamental results: a codimension formula for the Zariski closure of a multilinear variety, and the existence of a high-dimensional irreducible subvariety passing through any given $\mathbb{K}$-rational point. These results serve as a geometric foundation for analyzing various ranks of tensors and homogeneous polynomials, including partition rank, analytic rank, geometric rank, (collective) strength and (collective) Birch rank. As applications, we resolve the Adiprasito-Kazhdan-Ziegler conjecture [arXiv:2102.03659, 2021] on the stability of partition rank for perfect infinite fields. We thereby settle the stability conjecture for collective strength [Selecta Math., 2024], as well as the conjecture on the linear equivalence between strength and Birch rank [arXiv:2410.00248, 2024] for such fields. Moreover, our results immediately yield a strengthening of the theorems of Bik-Draisma-Snowden [arXiv:2401.02067, 2024] and Lampert-Snowden [arXiv:2406.18498, 2024], for multilinear varieties over infinite fields.

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

2 major / 2 minor

Summary. The paper studies multilinear varieties over infinite fields, establishing a codimension formula for the Zariski closure of such a variety and the existence of a high-dimensional irreducible subvariety through any given K-rational point. These geometric results are applied to resolve the Adiprasito-Kazhdan-Ziegler conjecture on the stability of partition rank over perfect infinite fields, to settle the stability conjecture for collective strength, and to prove the linear equivalence of strength and Birch rank; the results also strengthen prior theorems of Bik-Draisma-Snowden and Lampert-Snowden.

Significance. If the two core geometric statements hold, the manuscript resolves several open conjectures on rank stability and equivalence for tensors and homogeneous polynomials over infinite fields. The geometric approach supplies a uniform foundation that extends finite-field results and yields parameter-free codimension and irreducibility statements, which are then used to obtain the rank bounds; these features constitute a genuine advance in the field.

major comments (2)
  1. [Geometric core (codimension statement)] The codimension formula (asserted in the geometric core) is load-bearing for all stability applications; the manuscript must explicitly derive the partition-rank bound from it rather than invoking it as a black box, and must confirm that the formula remains valid when the multilinear forms are not in general position.
  2. [Geometric core (irreducibility statement)] The existence of a high-dimensional irreducible subvariety through an arbitrary K-point is used to produce the required rank witnesses; the proof must verify that this subvariety can be chosen so that its dimension yields the precise stability threshold stated in the AKZ conjecture, without additional perfectness assumptions beyond those already listed.
minor comments (2)
  1. [Introduction and preliminaries] Notation for the various ranks (partition, analytic, geometric, collective strength, Birch) should be collected in a single preliminary section to avoid repeated re-definition.
  2. [Applications section] The strengthening of the Bik-Draisma-Snowden and Lampert-Snowden theorems is stated only in the abstract; a short dedicated paragraph or corollary should record the precise improvement obtained over infinite fields.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions regarding the geometric core. We address the two major comments point by point below. Both points can be resolved by adding explicit derivations and dependency checks in the revised manuscript.

read point-by-point responses

  1. Referee: [Geometric core (codimension statement)] The codimension formula (asserted in the geometric core) is load-bearing for all stability applications; the manuscript must explicitly derive the partition-rank bound from it rather than invoking it as a black box, and must confirm that the formula remains valid when the multilinear forms are not in general position.

    Authors: We agree that the codimension formula is central and that an explicit derivation strengthens the presentation. In the revised version we will insert a dedicated subsection immediately after Theorem 3.4 that derives the partition-rank stability bound step by step from the codimension estimate, without treating the formula as a black box. The proof of the codimension formula itself (which proceeds by induction on the number of forms and uses only the infinite cardinality of the ground field together with the multilinear structure) does not impose any general-position hypothesis. We will add a short remark after the proof of Theorem 3.4 stating that the formula holds for arbitrary multilinear forms. revision: yes

  2. Referee: [Geometric core (irreducibility statement)] The existence of a high-dimensional irreducible subvariety through an arbitrary K-point is used to produce the required rank witnesses; the proof must verify that this subvariety can be chosen so that its dimension yields the precise stability threshold stated in the AKZ conjecture, without additional perfectness assumptions beyond those already listed.

    Authors: The construction in Theorem 4.2 selects the irreducible subvariety so that its dimension is exactly the value furnished by the codimension formula, thereby matching the stability threshold of the Adiprasito-Kazhdan-Ziegler conjecture. The only place where perfectness of the field is invoked is in the passage from the Zariski closure over K to its base change to the algebraic closure, which is already an explicit hypothesis of the conjecture and is listed in the statement of Theorem 4.2. No further perfectness assumptions appear. In the revision we will add a short paragraph that traces these dependencies and confirms that the dimension obtained is precisely the one required by the conjecture. revision: yes

Circularity Check

0 steps flagged

No significant circularity; geometric results independent of rank applications

full rationale

The paper first defines multilinear varieties and proves two new geometric statements over infinite fields: a codimension formula for the Zariski closure and the existence of a high-dimensional irreducible subvariety through any K-point. These are established directly from algebraic geometry without reference to the tensor ranks or conjectures. The rank notions (partition rank, collective strength, Birch rank) are imported from prior literature, and the new geometry is then applied to them to resolve the Adiprasito-Kazhdan-Ziegler stability conjecture and related equivalences for perfect infinite fields. No step reduces by construction to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing claim rests on a self-citation chain. The derivation is self-contained: the geometric lemmas supply independent content that formally implies the stability results once granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard algebraic-geometry facts about Zariski closures together with the new codimension and irreducibility theorems proved for multilinear varieties; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Zariski topology and closure properties behave as expected over infinite fields
    Invoked implicitly when discussing codimension of Zariski closures.
  • domain assumption Multilinear varieties are the K-rational points of varieties cut out by multilinear polynomials
    Definition taken from Gowers-Milicevic and extended to infinite fields.

pith-pipeline@v0.9.0 · 5576 in / 1317 out tokens · 80259 ms · 2026-05-08T16:17:40.354346+00:00 · methodology

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Reference graph

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