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This paper strengthens two conjectures on tensor eigenvalue multiplicities, proving them for all 2×2×…×2 tensors and for eigenschemes whose top-dimensional components form a hypersurface, and establishing a rank bound for the zero eigenvalu

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2026-08-01 07:31 UTC pith:GPQHZDZR

load-bearing objection Solid progress: the n=2 and codimension-1 multiplicity conjectures are proven, and the rank-to-zero-eigenvalue bounds are new; the Lemma 3.6 complaint in the stress-test is a misreading, though Proposition 4.3's WLOG step needs work. the 4 major comments →

arxiv 2607.21422 v1 pith:GPQHZDZR submitted 2026-07-23 math.AG math.AC

New conjectures on multiplicities of tensor eigenvalues

Stefano Canino , Cosimo Flavi , Francesco Galuppi , Yuze Luan This is my paper
classification math.AG math.AC MSC 15A69
keywords tensor eigenvaluesalgebraic multiplicitygeometric multiplicityeigenschemeresultanttensor rankalgebraic geometryeigenvalue conjectures
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper works on two open conjectures about how many times an eigenvalue of a tensor should count. It proposes stronger versions that take into account the whole eigenscheme — the geometric locus of eigenvectors, with multiplicities — rather than just its dimension: the algebraic multiplicity of an eigenvalue should be at least a degree-weighted sum over the irreducible pieces of the eigenscheme, and at least the dimension of the smallest linear space containing that scheme. The authors prove these refined conjectures for every 2×2×…×2 tensor, for eigenschemes whose largest pieces form a hypersurface, and in several other geometric situations. They also prove that a tensor of rank r≤n has zero-eigenvalue algebraic multiplicity at least (n−r)d^{n−1}, connecting multiplicity to the tensor's rank. If the conjectures hold in full, multiplicity becomes computable from the geometry of the eigenscheme alone.

Core claim

The central claim is Conjecture 1.5: for a non-singular reference tensor t and any t-eigenvalue λ0 of T, the t-algebraic multiplicity of λ0 is at least Σ_i deg(E_i) dim(E_i) d^{dim(E_i)−1}, where the E_i are the irreducible components of the eigenscheme E_{t,T}(λ0). This refines the earlier Conjecture 1.3, and the companion Conjecture 1.4 replaces the span multiplicity by the dimension of the scheme-theoretic linear span. The paper proves Conjecture 1.5 in full when n=2, when the eigenscheme is all of C^n or has all components of dimension n−1, and when the union of the maximal components is a hypersurface of degree b, in which case the exact algebraic multiplicity is a recursively defined n

What carries the argument

The t-characteristic polynomial φ_{t,T}(λ)=Res(T−λt) is the engine: its roots are the t-eigenvalues and its root multiplicity is the algebraic multiplicity. The t-eigenscheme E_{t,T}(λ0), defined by the n equations F_i−λ0G_i=0, carries the geometric side. The conjectures compare these two. The proofs manipulate resultants (notably Lemma 3.6 and Proposition 3.8), use the recursive function A_{d,b,n,t} that gives the exact resultant multiplicity in the codimension-one case, and, for n=3 with geometric multiplicity 1, use the Hilbert-Burch theorem to present the eigenscheme ideal by 2×2 minors and extract a lower bound d^2/2+1. Semi-continuity transfers the computed multiplicities from general

Load-bearing premise

The argument assumes that a property that holds for most tensors also holds for every tensor; the paper proves this 'genericity transfer' only in some cases, and its own Example 4.4 shows the exact bound can fail outside those cases.

What would settle it

Compute, with a computer algebra system, the t-characteristic polynomial and the degrees and dimensions of all irreducible components of the eigenscheme for a non-singular t and a tensor T with n=3, d=2 that is not general among tensors with geometric multiplicity 1 — for instance T=(y^2,z^2,yz), t=(x^2,y^2,z^2). If the algebraic multiplicity is strictly below Σ deg(E_i) dim(E_i) d^{dim(E_i)−1}, Conjecture 1.5 is false; the paper's Example 4.4 already shows the general-tensor lower bound am≥3 can fail (am=2), so this is a natural test case.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every 2×2×…×2 tensor satisfies both refined conjectures for all eigenvalues, so the algebraic multiplicity is always at least the number of lines in the eigenscheme, counted with multiplicity.
  • When the top-dimensional components of the eigenscheme form a hypersurface of degree b, the algebraic multiplicity is at least (n−1)bd^{n−2}, and exactly A_{d,b,n,n} for general tensors in that family.
  • A tensor of rank r≤n has zero-eigenvalue algebraic multiplicity at least (n−r)d^{n−1}; for general rank-r tensors the geometric multiplicity of 0 is exactly n−r.
  • Non-zero eigenvalues of a rank-r tensor have geometric multiplicity at most r, so high rank forces large eigenschemes for non-zero eigenvalues.
  • The refined Conjecture 1.4 holds in every case covered, including low-rank tensors.

Where Pith is reading between the lines

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

  • If Conjecture 1.5 is true in full, algebraic multiplicity could be read off directly from the eigenscheme's component degrees and dimensions, giving applied tensor analysts a way to bound eigenvalue multiplicity without expanding the characteristic polynomial.
  • The rank-to-multiplicity bound suggests a concrete testable diagnostic: a tensor whose zero eigenvalue has unusually high algebraic multiplicity must have small rank, so multiplicity could serve as a lower-bound certificate for tensor rank.
  • Example 4.4 shows the n=3, gm=1 lower bound d^2/2+1 is not universal for non-general tensors (am=2 there), so any complete proof of the conjectures must handle non-generic eigenschemes by a mechanism other than the general-tensor resultant computation.
  • The refined conjectures effectively identify algebraic multiplicity with a weighted degree of the eigenscheme; if true, they connect tensor spectral theory to standard enumerative geometry, where such weighted degrees are computable by intersection theory.

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

4 major / 6 minor

Summary. The paper proposes two new conjectures refining prior conjectures of Qi and Hu–Ke on tensor eigenvalue multiplicities. Conjecture 1.4 asserts that the t-algebraic multiplicity of an eigenvalue is at least the scheme-theoretic dimension of the linear span of the eigenscheme, and Conjecture 1.5 asserts a lower bound in terms of the sum over irreducible components of degree times dimension times d^{dim-1}. The main proving tools are resultants, Hilbert–Burch, and semicontinuity. The paper establishes the conjectures for all n=2 tensors (Corollary 4.1), for eigenschemes whose maximal components form a hypersurface (Theorem 3.10 and Corollary 3.11), and for tensors of rank at most n it gives lower bounds on the geometric and algebraic multiplicity of the zero eigenvalue (Theorem 1.7, Propositions 5.2 and 5.3). It also proves a lower bound for general ternary tensors with one-dimensional eigenscheme (Proposition 4.3). The abstract emphasizes the novelty of the refined statements and the algebraic-geometric methods.

Significance. If the main results are correct, this is a meaningful step beyond the known cases of tensor eigenvalue multiplicity conjectures. The new Conjectures 1.4 and 1.5 are natural strengthenings that take into account the scheme structure, and the proofs use standard but nontrivial algebraic-geometric techniques (resultant theory, Hilbert–Burch, semicontinuity). The complete result for n=2 and the low-rank bounds are concrete, potentially useful advances. The paper also includes explicit examples and careful use of resultant manipulations. However, the strongest universal claims rest on genericity hypotheses and on a displayed identity whose statement is at least ambiguous, so the conclusion currently calls for substantial revision.

major comments (4)
  1. [§3, Lemma 3.6] The displayed identity in Lemma 3.6 is internally inconsistent. The proof derives a quotient identity with denominator Res(G2,M2,F3,...,Fn), and Example 4.2 uses the quotient form. As printed, the equation appears to state a product formula with no denominator. This is load-bearing: Proposition 3.8's recursion obtains its subtractive term precisely from this denominator, and Theorem 3.10 relies on Proposition 3.8. If the product version is intended, the proof and recursion are wrong; if the quotient version is intended, the display must be corrected. Please state the identity unambiguously and adjust the proof or statement accordingly.
  2. [§3, Theorem 3.10 and Lemma 3.4] Theorem 3.10 is phrased confusingly: it says both that X is a general algebraic hypersurface and that X is the union of all maximal-dimensional components of E_{t,T}(λ0). More importantly, the proof of the lower bound for arbitrary T uses semicontinuity via Lemma 3.4, but Lemma 3.4 is stated for the set Y of tensors with deg(E_{t,T}(λ0))=b exactly. If E has additional lower-dimensional components, deg(E) > b, so the tensors in Theorem 3.10 with such components are not in Y and the semicontinuity passage is not justified as written. The argument is likely salvageable by a limiting argument inside the family Ψ_{n,d,λ0}(R_d ⊕ R_{d-b}^{⊕n}), but this needs to be spelled out. This affects the proof of Theorem 1.6(3).
  3. [§4, Proposition 4.3] The step 'Without loss of generality, assume that 1≤deg(A3)≤d/2' is not justified. For a Hilbert–Burch matrix whose three 2×2 minors are all homogeneous of the same degree d, the degrees of the entries must satisfy a system of equations; the text should explain why a row/column change of basis can be chosen so that one row has common degree ≤ d/2 and why the index 3 is no restriction. Generically this is plausible, but the manuscript does not supply the argument. Since the claimed lower bound am ≥ d^2/2 + 1 depends on this degree assumption, it is a load-bearing gap in the proof. Example 4.4 shows the genericity assumptions are essential, so the precise open conditions used in the proof should also be stated.
  4. [§5, Corollary 5.4] In the proof of part (2), the chain contains the equality '≥ gm_{t,T}(0)d^{n-r-1} = gm_{t,T}(0)d^{gm_{t,T}(0)}d^{n-r-1}', which is false as written. For a general rank-r tensor one has gm_{t,T}(0)=n-r, so the intended equality should be d^{n-r-1}=d^{gm_{t,T}(0)-1}, not the displayed expression. This is likely a typo, but as written the proof of the general-tensor case of Conjecture 1.3(2) is not spelled out correctly. Please correct the displayed chain.
minor comments (6)
  1. [Abstract and Introduction] The phrase 'all 2×2×...×2 tensors' should be made precise: it means n=2 with arbitrary order d. The notation D(n,d)=nd^{n-1} is introduced only later; a forward reference would help.
  2. [§3, Remark 3.1] There is a typo: 'lm_{t,T}(λ0 =n' should be 'lm_{t,T}(λ0)=n'.
  3. [§3, Lemma 3.6] Even in the quotient form, the identity is valid only up to a sign arising from the permutation of G2 and H, as the proof itself signals with 'up to a sign' in Proposition 3.8. The lemma should either state the sign or explicitly say 'up to sign'.
  4. [Appendix A] The appendix title uses the notation A_{d,b,t,n}, while the body uses A_{d,b,n,t}. Please standardize.
  5. [§3, Corollary 3.11] In the proof, the line 'am_{t,T}(λ0) ≥ A_{d,b,2,2} ≥ b ≥= lm_{t,T}(λ0)' contains a typographical error: '≥=' should be '=' or '≥'. The intended inequality is clear but the text should be corrected.
  6. [Throughout] There are many OCR-like spacing issues and broken parentheses, e.g., 'lm_{t,T}(λ0' in Remark 3.1 and several displayed equations. A careful copyediting pass is needed.

Circularity Check

0 steps flagged

No significant circularity: the conjectures are treated as targets, not assumptions; the proofs are self-contained and no prediction reduces to a fitted input or to a self-citation.

full rationale

The derivation chain is not circular. Conjectures 1.4 and 1.5 are introduced as open statements and are not used as hypotheses in any proof; each theorem is established by independent algebraic-geometric arguments. The main computational engine is the resultant factorizations of Lemma 2.3 and Lemma 3.6, the Hilbert–Burch determinantal description in Proposition 4.3, and semi-continuity in Lemma 3.4. None of these inputs contains the target multiplicity inequalities. The rank bound in Proposition 5.2 follows from Lemma 2.3(1) applied to the system F1−λG1,...,Fr−λGr, λG_{r+1},...,λG_n; it is a direct resultant computation, not a dressed-up definition of rank. Theorem 3.10 computes am_{t,T}(λ0) as the multiplicity of a resultant and then compares it with the conjectural bound via the explicit formula for A_{d,b,n,n}; the conjectural quantity is not fed into the computation. The only self-citation is [GGTV23] for the observation that eigenvector/eigenvalue definitions depend on the algebraic maps rather than on the tensor representation (Section 2), and this is also attributed to [HK16, Section 5.2] and is elementary; it is not load-bearing. The manuscript itself flags limitations: Proposition 4.3 depends on an unquantified genericity condition and the unexplained assumption 1≤deg(A3)≤d/2, and Example 4.4 shows that the generality assumptions cannot simply be dropped. The skeptic's Lemma 3.6/Example 4.2 discrepancy is a possible algebraic error in the displayed resultant formula, i.e. a correctness risk, not a circularity: even if the formula is mis-stated, the bound is not assumed in order to prove itself. Accordingly no circular step is identified and the score is 1, reflecting only the presence of a minor, non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No numerical fitting or invented physical entities. The paper's mathematical assumptions are the non-singularity of t, standard theorems (resultants, Hilbert–Burch, semi-continuity), and genericity conditions in Propositions 3.8 and 4.3. The new conjectures themselves are proposed statements, not axioms used to prove the results.

axioms (5)
  • domain assumption Non-singularity of t: Res(t) ≠ 0 (Remark 2.5).
    All definitions of t-eigenvalues, eigenschemes and the characteristic polynomial require t(w)≠0 for every nonzero w. The conjectures are stated only under this assumption.
  • standard math Standard properties of resultants (Lemma 2.3 from [Jou91], [GKZ94]).
    The proofs of Proposition 3.8 and Theorem 3.10 rely on the multiplicative and invariance properties of resultants.
  • standard math Hilbert–Burch theorem for codimension-2 ideals (used in Proposition 4.3).
    The proof that a 1-dimensional eigenscheme ideal is generated by the 2×2 minors of a 2×3 matrix is an application of Hilbert–Burch.
  • standard math Semi-continuity theorem (used in Lemma 3.4).
    The proof that a lower bound on algebraic multiplicity for general tensors extends to all tensors in the same irreducible family uses the semi-continuity of cohomological quantities.
  • domain assumption Genericity conditions: 'general polynomials' H,M_i and 'T general among tensors with gm=1'.
    The exact multiplicity formulas in Propositions 3.8 and 4.3 are only proven for general choices; Example 4.4 shows the n=3 bound can fail outside this locus.

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read the original abstract

We work on two conjectures on tensor eigenvalue multiplicities. By using the language of algebraic geometry we give stronger, more refined versions of the conjectures and we prove them in many new cases, notably for all $2\times 2\times\dots\times 2$ tensors. We also establish a connection between the rank of a tensor and the multiplicities of the zero eigenvalue.

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