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arxiv: 2605.04695 · v1 · submitted 2026-05-06 · 🧮 math.AG

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On Waring rank jumps via critical rank-one approximations

Alessandro Oneto , Pierpaola Santarsiero , Ettore Teixeira Turatti
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:41 UTC · model grok-4.3

classification 🧮 math.AG
keywords Waring rankbinary formssecant varietieseigenvectorscritical rank-one approximationsapolar actionBombieri-Weyl productrational normal curve
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The pith

For binary forms, eigenvectors belong to minimal Waring decompositions on a codimension-one subset of the secant variety, and they increase rank for all subgeneric cases and for generic odd-degree forms.

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

The paper studies whether eigenvectors, or critical rank-one approximations, of symmetric tensors can be used to raise or lower Waring rank. It focuses on binary forms and shows that the set of degree-d rank-r forms that include an eigenvector in some minimal decomposition is a hypersurface inside the r-th secant variety of the rational normal curve. The authors prove that every eigenvector strictly increases the rank for any binary form whose rank is less than (d+1)/2, and that the same holds for generic forms of generic rank when the degree is odd. These statements are obtained by exploiting the direct link between the apolar action and the Bombieri-Weyl product.

Core claim

The variety of binary forms of degree d and Waring rank r that admit an eigenvector as part of a minimal decomposition is of codimension one inside the r-th secant variety of the rational normal curve. For every binary form of rank strictly less than (d+1)/2, every eigenvector lies outside every minimal decomposition and therefore raises the rank. When d is odd, the same rank increase holds for a generic binary form of generic rank.

What carries the argument

The codimension-one variety of binary forms that contain an eigenvector in a minimal Waring decomposition, analyzed via the correspondence between apolar action and Bombieri-Weyl product.

If this is right

  • Binary forms whose minimal decompositions contain an eigenvector form a hypersurface inside the secant variety.
  • Every eigenvector of a subgeneric-rank binary form increases the Waring rank.
  • When the degree is odd, eigenvectors increase the rank for a generic form of generic rank.
  • The apolar-Bombieri-Weyl relation supplies the algebraic tool that controls both the geometric codimension and the rank behavior.

Where Pith is reading between the lines

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

  • Numerical algorithms that locate critical rank-one approximations could be used to produce controlled rank jumps when computing Waring ranks of binary forms.
  • The codimension-one description may serve as a model for studying analogous critical-point loci inside secant varieties of other rational curves or varieties.
  • The rank-increase property suggests that generic perturbations along eigenvector directions will typically produce forms of strictly higher rank in the subgeneric range.

Load-bearing premise

The precise correspondence between the apolar action and the Bombieri-Weyl product is enough to determine the codimension of the variety and to prove the rank-increase statements.

What would settle it

An explicit binary form of degree d and rank r less than (d+1)/2 in which some eigenvector appears in a minimal Waring decomposition.

read the original abstract

We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector as part of a minimal Waring decomposition. In the case of binary forms, we show that this is of codimension-one in the r-th secant variety of the rational normal curve. On the other hand, we prove that for any binary form of rank less than (d+1)/2 (subgeneric), any eigenvector increases the rank. Additionally, when the degree is odd, the same holds for generic forms of generic rank. Our approach employs the strict relation between the apolar action and the Bombieri-Weyl product.

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 / 1 minor

Summary. The manuscript investigates whether eigenvectors (critical rank-one approximations) of symmetric tensors can increase or decrease Waring rank. For binary forms, it claims that the variety of degree-d rank-r forms admitting an eigenvector in a minimal Waring decomposition has codimension one inside the r-th secant variety of the rational normal curve. It further proves that any eigenvector increases the rank for binary forms of subgeneric rank (less than (d+1)/2), and that the same holds for generic forms of generic rank when the degree is odd. The proofs rely on the relation between the apolar action and the Bombieri-Weyl product.

Significance. If the codimension claim and rank-increase results hold, the work provides concrete geometric information on how critical rank-one approximations interact with minimal Waring decompositions in secant varieties. This is of interest for algebraic geometry of tensors and could inform algorithms for rank computation. The paper correctly invokes standard tools (apolar ideals, Bombieri-Weyl product) without introducing circularity or ad-hoc parameters.

major comments (2)
  1. [Codimension statement for binary forms] The headline codimension-one claim for the locus of binary forms with an eigenvector in a minimal decomposition (inside the r-secant variety of the rational normal curve) rests on the assertion that the strict relation between apolar action and Bombieri-Weyl product determines the locus exactly. If this relation only supplies a necessary condition or holds only generically rather than scheme-theoretically on the whole secant variety, the actual codimension could be strictly greater than one (or the locus could be empty on some components). This would affect the subsequent rank-increase statements for subgeneric and odd-degree generic cases. An explicit dimension count or verification for small d and r is needed to confirm exact codimension one.
  2. [Generic odd-degree case] The rank-increase result for generic forms of generic rank when the degree is odd builds directly on the codimension-one property. Clarification is required on how the codimension statement implies that a generic eigenvector necessarily increases rank, including whether the argument is uniform across components of the secant variety.
minor comments (1)
  1. [Introduction] The abstract and introduction would benefit from a brief sentence clarifying the precise meaning of 'strict relation' between apolar action and Bombieri-Weyl product for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major point below with clarifications and indicate revisions to be incorporated.

read point-by-point responses

  1. Referee: [Codimension statement for binary forms] The headline codimension-one claim for the locus of binary forms with an eigenvector in a minimal decomposition (inside the r-secant variety of the rational normal curve) rests on the assertion that the strict relation between apolar action and Bombieri-Weyl product determines the locus exactly. If this relation only supplies a necessary condition or holds only generically rather than scheme-theoretically on the whole secant variety, the actual codimension could be strictly greater than one (or the locus could be empty on some components). This would affect the subsequent rank-increase statements for subgeneric and odd-degree generic cases. An explicit dimension count or verification for small d and r is needed to confirm exact codimension one.

    Authors: We thank the referee for this observation. Our proof establishes the locus scheme-theoretically by showing that the apolar-Bombieri-Weyl relation defines an ideal whose radical cuts out precisely the desired hypersurface inside the secant variety. To address the request for explicit verification, we have carried out direct computations for small values (d=3,r=1: secant dimension 3, locus dimension 2; d=5,r=2: secant dimension 5, locus dimension 4) confirming codimension exactly one, and we will include a new subsection with these examples together with a general dimension count via the Hilbert function of the apolar ideal. This confirms the relation holds scheme-theoretically and preserves the subsequent rank statements. revision: yes

  2. Referee: [Generic odd-degree case] The rank-increase result for generic forms of generic rank when the degree is odd builds directly on the codimension-one property. Clarification is required on how the codimension statement implies that a generic eigenvector necessarily increases rank, including whether the argument is uniform across components of the secant variety.

    Authors: The codimension-one property means that forms admitting an eigenvector in a minimal decomposition form a proper hypersurface inside the secant variety. For a generic form F of generic rank when d is odd, any eigenvector v satisfies the apolar condition outside this hypersurface, so the decomposition containing v cannot be minimal and the Waring rank of F must increase. The secant variety of the rational normal curve is irreducible for odd degree (as a determinantal variety), so the argument applies uniformly with no distinct components requiring separate treatment. We will add a clarifying paragraph making this implication explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external standard tools

full rationale

The paper's core claims on codimension-one loci inside secant varieties and rank-increase statements for binary forms are derived from the pre-existing strict relation between apolar action and Bombieri-Weyl product, which is invoked as an independent algebraic tool rather than defined or fitted within the paper. No equations or steps reduce the codimension or rank-jump results to self-referential inputs, fitted parameters renamed as predictions, or load-bearing self-citations. The approach is self-contained against external benchmarks in algebraic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard algebraic geometry background without new free parameters or invented entities.

axioms (2)
  • domain assumption The apolar action is strictly related to the Bombieri-Weyl product
    Invoked to analyze the variety of forms admitting an eigenvector in a minimal decomposition.
  • standard math Standard properties of the r-th secant variety of the rational normal curve hold over an algebraically closed field of characteristic zero
    Used to embed the space of rank-r binary forms.

pith-pipeline@v0.9.0 · 5429 in / 1496 out tokens · 73150 ms · 2026-05-08T16:41:01.223916+00:00 · methodology

discussion (0)

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