Matrix tuples with linearly dependent invariant subspaces

Tam\'as Bencze

arxiv: 2604.22733 · v1 · submitted 2026-04-24 · 🧮 math.AG

Matrix tuples with linearly dependent invariant subspaces

Tam\'as Bencze This is my paper

Pith reviewed 2026-05-08 10:17 UTC · model grok-4.3

classification 🧮 math.AG

keywords matrix tuplesinvariant subspacesalgebraic hypersurfacelinear dependencepolynomial equationvector spacealgebraic geometry

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The pith

Matrix tuples with linearly dependent invariant subspaces lie on an algebraic hypersurface defined by a single polynomial.

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

The paper establishes that the set of tuples of matrices on an n-dimensional space that admit invariant subspaces whose dimensions add to exactly n, yet whose sum fails to be the full space, coincides with the zero locus of one polynomial in the entries of the tuple. This turns a geometric spanning condition on collections of invariant subspaces into an algebraic hypersurface in the parameter space of all matrix tuples. A reader would care because it supplies an explicit polynomial test for when a tuple of linear operators has a deficient but dimensionally balanced set of invariant subspaces. The authors derive the explicit form of this polynomial, showing the set is always codimension one.

Core claim

The set of matrix tuples possessing invariant subspaces whose dimensions sum to the ambient dimension n but do not span the whole space forms an algebraic hypersurface, and the paper determines the explicit polynomial equation that cuts out this hypersurface.

What carries the argument

The explicit polynomial in the matrix entries that vanishes precisely when the tuple admits invariant subspaces of total dimension n that fail to span the ambient space.

If this is right

Membership in the set can be decided by direct evaluation of the single polynomial on the matrix entries.
The hypersurface is the closure of all tuples whose invariant subspaces satisfy the dimension-sum but non-spanning condition.
The result supplies an algebraic equation whose degree can be read off once the polynomial is written down.
The description holds for any number of matrices and any ambient dimension n.

Where Pith is reading between the lines

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

The explicit polynomial may allow computation of the degree or singular locus of the hypersurface by standard algebraic methods.
This equation could be used to test small cases computationally and confirm the codimension claim by direct calculation.
Similar polynomial conditions might exist for other linear-algebraic dependence relations among subspaces.

Load-bearing premise

The condition on the dimensions and spanning of the invariant subspaces defines an algebraic set of codimension exactly one rather than higher codimension or a non-algebraic set.

What would settle it

Take a concrete low-dimensional matrix tuple with invariant subspaces summing in dimension to n but not spanning the space; check whether the proposed polynomial vanishes at that tuple.

read the original abstract

The set of matrix tuples with invariant subspaces whose dimensions sum up to the dimension of the space, but which do not span the whole space form an algebraic hypersurface. We found the equation of this hypersurface. This generalizes previous joint work.

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.

Desk Editor's Note private letter to a colleague

read the letter

The paper supplies an explicit equation for the hypersurface of matrix tuples that admit invariant subspaces with dimension sum equal to n but not spanning the space, as a direct generalization of prior joint work. The concrete polynomial is the main addition; earlier results apparently handled existence or other properties, and this version writes down the defining equation in the parameter space of tuples. That output can be useful for explicit calculations in low dimensions or for checking membership in the locus. The setup stays within the usual affine space over an algebraically closed field, where the invariant-subspace condition is algebraic, so the framing is standard and appropriate. The soft spot is the missing support for the claim that the locus has codimension exactly one. The abstract asserts the set forms a hypersurface without a dimension calculation, an irreducibility argument, or even a small example. The stress-test concern holds up on the given information: fixing the dimension sum and adding the non-spanning condition can produce components of higher codimension or fill the whole space for some choices of subspace counts, and nothing visible rules that out. If the full manuscript contains a clear codimension computation or shows the ideal is principal, that would fix it; otherwise the central statement rests on unshown steps. This is aimed at specialists working on varieties of matrices or tuples with prescribed invariants. Readers seeking new general methods or wider applications in linear algebra will find little here. It deserves a serious referee if the full proof of codimension one is rigorous and checkable, because an explicit equation is the sort of thing that can be verified and used downstream. I would send it to review rather than desk reject.

Referee Report

1 major / 1 minor

Summary. The manuscript claims that the set of tuples of matrices admitting invariant subspaces V_i with ∑ dim V_i equal to the ambient dimension n, but with the V_i not spanning the whole space, forms an algebraic hypersurface in the parameter space, and provides an explicit equation for this hypersurface as a generalization of prior joint work.

Significance. If the central claim is established with a correct codimension calculation, the result would supply an explicit polynomial equation for a geometrically natural locus of matrix tuples defined by linearly dependent invariant subspaces. This could be useful in the algebraic geometry of representations and invariant subspace problems, particularly as an explicit generalization of earlier results.

major comments (1)

The central claim that the locus is a hypersurface (i.e., the zero set of a single non-zero polynomial, hence pure codimension 1) is load-bearing but unsupported by any dimension calculation, irreducibility argument, or explicit verification that the invariant-subspace condition with the given dimension sum and non-spanning requirement yields codimension exactly 1 rather than higher codimension or the full space. This must be addressed in the section stating the main theorem and deriving the equation.

minor comments (1)

The abstract and introduction should specify the base field (assumed algebraically closed), the number of matrices in the tuple, and the precise ambient dimension n to make the setup unambiguous.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on our manuscript. We address the major concern below and will revise the paper to incorporate an explicit codimension argument.

read point-by-point responses

Referee: The central claim that the locus is a hypersurface (i.e., the zero set of a single non-zero polynomial, hence pure codimension 1) is load-bearing but unsupported by any dimension calculation, irreducibility argument, or explicit verification that the invariant-subspace condition with the given dimension sum and non-spanning requirement yields codimension exactly 1 rather than higher codimension or the full space. This must be addressed in the section stating the main theorem and deriving the equation.

Authors: We agree that the manuscript would benefit from an explicit codimension verification to rigorously establish that the locus is a hypersurface of pure codimension 1. In the revised version, we will expand the section containing the main theorem and the derivation of the equation by adding a dedicated paragraph (or short subsection) that performs the following: (i) we count the dimension of the incidence variety consisting of matrix tuples together with choices of subspaces V_i of the prescribed dimensions that are invariant and whose sum of dimensions equals n; (ii) we show that the additional non-spanning condition imposes precisely one independent algebraic constraint, which is captured by the polynomial we derive; (iii) we verify that this polynomial is not identically zero by exhibiting an explicit matrix tuple inside the locus (where the polynomial vanishes) and another outside the locus (where it does not vanish). These additions will confirm that the zero set has codimension exactly 1 and coincides with the geometrically defined locus, without changing the statement or the explicit form of the equation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained algebraic computation.

full rationale

The paper asserts that the locus of matrix tuples admitting invariant subspaces V_i with sum of dimensions equal to n but not spanning the space is an algebraic hypersurface and provides its explicit equation as a generalization of prior joint work. No step reduces the central claim to a fitted parameter, self-definition, or unverified self-citation chain; the algebraicity follows from the standard closedness of the invariant-subspace condition in the affine space of tuples, and the hypersurface property is presented as the result of the derivation rather than an input assumption. The generalization is framed as an extension, not as the sole justification for codimension 1. The derivation chain remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard concepts from linear algebra (invariant subspaces) and algebraic geometry (hypersurfaces in affine space); no free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math The set of matrix tuples satisfying the invariant-subspace dimension condition is an algebraic set in the ambient matrix space.
Invoked implicitly when asserting that the set forms a hypersurface.

pith-pipeline@v0.9.0 · 5311 in / 1286 out tokens · 37455 ms · 2026-05-08T10:17:52.845883+00:00 · methodology

discussion (0)

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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Frenkel:The triangulant,Linear Algebra and its Applications 709 (2025), 92–110

[1] Tamás Bencze, Péter E. Frenkel:The triangulant,Linear Algebra and its Applications 709 (2025), 92–110. Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, 1117 Hungary Email address:benczetamas11@gmail.com

work page 2025

[1] [1]

Frenkel:The triangulant,Linear Algebra and its Applications 709 (2025), 92–110

[1] Tamás Bencze, Péter E. Frenkel:The triangulant,Linear Algebra and its Applications 709 (2025), 92–110. Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, 1117 Hungary Email address:benczetamas11@gmail.com

work page 2025