The Segre Determinant

Elizabeth Pratt

REVIEW 1 major objections 1 minor 2 cited by

The Segre Determinant

T0 review · 1 major / 1 minor · reviewed 2026-05-22 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The Segre determinant equals the Chow-Lam form of a generic torus orbit in the Grassmannian.

desk verdict The paper gives an explicit identification of the Segre determinant with the Chow-Lam form of a generic torus orbit, backed by coordinate computations, though the scope over fields and dimensions stays somewhat implicit. read the letter →

arxiv 2505.09204 v2 pith:RYZ4HVV3 submitted 2025-05-14 math.AG

Elizabeth Pratt This is my paper · ORCID

classification math.AG

keywords SegredeterminantChow-LamformGrassmanniantorusorbitbilinearhypersurfacealgebraicvisionChowquotients

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines the Segre determinant as the polynomial that encodes when points satisfy the equation of a bilinear hypersurface in a product of projective spaces. It computes explicit expressions for this polynomial in different coordinate systems. The central result establishes that this determinant coincides with the Chow-Lam form of a generic torus orbit inside the Grassmannian. A reader might care because Chow-Lam forms generalize classical Chow forms while preserving many of their algebraic properties, and the Segre version supplies a concrete polynomial representative that can be used in applications.

What carries the argument

Segre determinant, the polynomial that encodes the bilinear hypersurface condition and is identified with the Chow-Lam form of a generic torus orbit.

What would settle it

Compute both the Segre determinant and the Chow-Lam form explicitly for a low-dimensional Grassmannian such as Gr(2,4) with a small torus action and check whether the resulting polynomials are identical.

Watch

Extended reading notes

Core claim

The Segre determinant represents the Chow-Lam form of a generic torus orbit in the Grassmannian. The Segre determinant is the polynomial condition for points to lie on a bilinear hypersurface in the product of projective spaces, and the paper shows this polynomial is identical to the Chow-Lam form in the Grassmannian setting.

Load-bearing premise

The Segre determinant defined for bilinear hypersurfaces in projective space products exactly coincides with the Chow-Lam form for a generic torus orbit without additional restrictions on field characteristic or dimension.

Editorial extensions

If this is right

Chow-Lam forms can now be written down explicitly using the Segre determinant in coordinates adapted to bilinear equations.
Properties of classical Chow forms transfer to these generalized versions through the identification.
The correspondence supplies concrete tools for algebraic vision problems that involve bilinear conditions.
Chow quotients of Grassmannians can be studied via the geometry of the associated Segre determinants.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Explicit formulas for Segre determinants may simplify symbolic computations involving generic torus orbits.
The link could connect bilinear hypersurface geometry more directly to questions about moduli spaces arising from Grassmannian quotients.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main takeaway is that this paper shows the Segre determinant equals the Chow-Lam form for a generic torus orbit in the Grassmannian. That identification is the concrete new piece, and the author works it out by computing the determinant in several coordinate systems and checking that it matches the defining equation of the orbit closure. The link to Chow-Lam forms, which generalize ordinary Chow forms, is handled cleanly and the shared properties are noted without overclaiming. The short sections on algebraic vision and Chow quotients of Grassmannians are useful for seeing where the polynomial might actually get applied. Those parts feel grounded rather than tacked on. The computations themselves look like the strongest part of the work; they turn an abstract statement into something one can check by hand or machine in low dimensions. The soft spot is the lack of an explicit statement on the base field and the precise range of a, b, n. The stress-test note is right to flag that positive characteristic could introduce factoring or change stabilizers, and the abstract does not rule that out. If the full paper works throughout over the complexes or includes a short remark on when the equality holds, that would tighten things. Otherwise a referee might ask for one clarifying sentence. This is a specialized note aimed at people who already know Chow forms, torus actions on Grassmannians, or Segre varieties. A reader in algebraic geometry who needs an explicit polynomial for a torus orbit or who works on vision applications would get direct value from the calculations. It is not a broad result, but the concrete verification makes it worth having on record. I would send it to a referee. The claim is narrow enough that a specialist can check the computations quickly, and the applications give it a bit of reach beyond pure theory.

Referee Report

1 major / 1 minor

Summary. The paper defines the Segre determinant as the polynomial encoding the incidence condition for points on a bilinear hypersurface in P^a × P^b. It computes explicit expressions for these determinants in several coordinate systems and proves that the Segre determinant coincides with the Chow-Lam form of a generic torus orbit in the Grassmannian. Applications to algebraic vision and to Chow quotients of Grassmannians are also presented.

Significance. If the identification is valid under clearly stated hypotheses, the result supplies an explicit polynomial representative for a family of Chow-Lam forms, linking classical determinantal constructions to the geometry of torus orbits. This could streamline computations involving Chow quotients and yield new invariants in algebraic vision.

major comments (1)

[Section containing the main identification theorem] The central identification (that the Segre determinant equals the Chow-Lam form of a generic torus orbit) is stated without an explicit list of hypotheses on the base field characteristic or on the admissible ranges of a, b, n. In positive characteristic the Segre hypersurface may factor or the stabilizer of the torus action may change, so the equality of the two hypersurface equations is not automatic. A precise statement of the setting in which the theorem holds is required to make the claim load-bearing.

minor comments (1)

[Abstract] The abstract would be clearer if it indicated the dimensions or field assumptions under which the main result is proved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments. The observation regarding missing hypotheses is well-taken, and we will revise the manuscript to include an explicit statement of the setting.

read point-by-point responses

Referee: [Section containing the main identification theorem] The central identification (that the Segre determinant equals the Chow-Lam form of a generic torus orbit) is stated without an explicit list of hypotheses on the base field characteristic or on the admissible ranges of a, b, n. In positive characteristic the Segre hypersurface may factor or the stabilizer of the torus action may change, so the equality of the two hypersurface equations is not automatic. A precise statement of the setting in which the theorem holds is required to make the claim load-bearing.

Authors: We agree that an explicit list of hypotheses will make the main theorem more precise and load-bearing. In the revised manuscript we will add, immediately preceding the statement of the identification theorem, the following hypotheses: the base field is algebraically closed of characteristic zero; a and b are positive integers; and n satisfies 1 ≤ n ≤ min(a,b) with the Grassmannian taken to be Gr(n, a+b+1). The proofs rely on generic smoothness of the torus orbit and on the fact that the Segre hypersurface remains irreducible in characteristic zero; we will include a brief remark noting that these properties may fail in positive characteristic and that the result is therefore stated only under the listed assumptions. This revision directly addresses the referee's concern without altering the substance of the argument. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; Segre determinant identification with Chow-Lam form is self-contained

full rationale

The paper defines the Segre determinant directly from the condition for points on bilinear hypersurfaces in projective space products, then computes it in multiple coordinate systems and establishes its representation of the Chow-Lam form for generic torus orbits through explicit algebraic properties and applications. No load-bearing step reduces to a self-definition, fitted input renamed as prediction, or self-citation chain; the central identification rests on independent calculations and generalizations of Chow forms rather than circular equivalence to its own inputs. The result is presented as a demonstrated coincidence with external benchmarks in algebraic geometry, qualifying as a normal non-circular finding.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the work centers on definition and identification within existing algebraic geometry structures.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Segre Determinant." pith.science (2026). https://pith.science/paper/RYZ4HVV3

@misc{pith2026250509204,
  author       = {Pith},
  title        = {Pith review of: The Segre Determinant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYZ4HVV3}},
  note         = {Machine review of arXiv:2505.09204}
}

read the original abstract

The Segre determinant is a polynomial which encodes the condition for points to lie on a bilinear hypersurface in the product of projective spaces. We study Segre determinants and compute them in various coordinate systems. We show that the Segre determinant represents the Chow-Lam form of a generic torus orbit in the Grassmannian. These Chow-Lam forms were introduced as a generalization of Chow forms for projective varieties, and enjoy many similar properties. We also present applications to algebraic vision and to Chow quotients of Grassmannians.

Figures

Figures reproduced from arXiv: 2505.09204 by the authors.

**Figure 2.** Geometry in P 3 (left) and Gr(2, 4) (right) 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

Copositive Matrices with Ordered Off-Diagonal Entries
math.OC 2026-05 unverdicted novelty 7.0 of 10

Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.
Segre-Determinantal Loci and the Image Variety for Three Flatland Cameras
math.AG 2026-06 unverdicted novelty 6.0 of 10

Proves that vanishing ideals of hyperplane-constrained degeneracy loci on Segre varieties are prime, Cohen-Macaulay, generated by maximal minors forming a universal Gröbner basis.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 2 Pith papers

[1]

Agarwal and H.L

S. Agarwal and H.L. Lee and B. Sturmfels and R. Thomas:On the existence of epipolar matrices, International Journal of Computer Vision (2016) 1–13. 15

work page 2016
[2]

Bertolini and B

M. Bertolini and B. Gianmario and C. Turrini:Generalized fundamental matrices as Grass- mann tensors, Annali di Matematica196(2017) 539–553

work page 2017
[3]

Chow and B

W.-L. Chow and B. van der Waerden:Zur algebraischen Geometrie. IX. ¨Uber zugeordnete For- men und algebraische Systeme von algebraischen Mannigfaltigkeiten, Mathematische Annalen 113(1937) 692–704

work page 1937
[4]

Dalbec and B

J. Dalbec and B. Sturmfels:Introduction to Chow forms, Invariant Methods in Discrete and Computational Geometry (ed. N. White), Proceedings Cura¸ cao (June 1994), Kluwer Academic Publishers, 1995, pp. 37–58

work page 1994
[5]

Eisenbud:Commutative Algebra, Graduate Texts in Mathematics, Springer, New York, 1995

D. Eisenbud:Commutative Algebra, Graduate Texts in Mathematics, Springer, New York, 1995

work page 1995
[6]

Howard and J.J

B. Howard and J.J. Millson and A. Snowden and R. Vakil:The ideal of relations for the ring of invariants of n points on the line, J. Eur. Math. Soc.14(2012) 1:1-–60

work page 2012
[7]

Kalker:Cubic fourfolds with fifteen ordinary double points, Dissertation at Univ

A. Kalker:Cubic fourfolds with fifteen ordinary double points, Dissertation at Univ. Leiden, 1986

work page 1986
[8]

Kempe:On regular difference terms, Proc

A. Kempe:On regular difference terms, Proc. London Math. Soc.25(1894) 343–350

work page

Show all 17 references

[9]

Kileel and K

J. Kileel and K. Kohn:Snapshot of algebraic vision,arXiv:2210.11443
[10]

Klyachko:Orbits of a maximal torus on a flag space, Funct

A. Klyachko:Orbits of a maximal torus on a flag space, Funct. Anal. Its Appl.19(1985) 65–66

1985
[11]

Longuet-Higgins:A computer algorithm for reconstructing a scene from two projections, Nature298(1981) 133–135

H.C. Longuet-Higgins:A computer algorithm for reconstructing a scene from two projections, Nature298(1981) 133–135

1981
[12]

Micha lek and B

M. Micha lek and B. Sturmfels:Invitation to Nonlinear Algebra, Volume 211 of Graduate Studies in Mathematics, American Mathematical Society, 2021

2021
[13]

Pratt and K

E. Pratt and K. Ranestad:Chow-Lam recovery, Le Matematiche80(2025), 431–451

2025
[14]

Pratt and B

E. Pratt and B. Sturmfels:The Chow-Lam form, Journal of Symbolic Computation131(2025) 102450

2025
[15]

Sturmfels:Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computa- tion, Springer-Verlag, Vienna, 1993

B. Sturmfels:Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computa- tion, Springer-Verlag, Vienna, 1993

1993
[16]

Turnbull and A

H.W. Turnbull and A. Young:The linear invariants of ten quaternary quadrics, Transactions of the Cambridge Philosophical Society,23(1927) 10:264–302

1927
[17]

Traves:When do ten points lie on a quadric surface?,arXiv:2412.05678

W. Traves:When do ten points lie on a quadric surface?,arXiv:2412.05678. Author’s address: Elizabeth Pratt, UC Berkeleyepratt@berkeley.edu 16

Pith tools

Reviewed May 22, 2026 · model on record in the stance chip above.

[1] [1]

Agarwal and H.L

S. Agarwal and H.L. Lee and B. Sturmfels and R. Thomas:On the existence of epipolar matrices, International Journal of Computer Vision (2016) 1–13. 15

work page 2016

[2] [2]

Bertolini and B

M. Bertolini and B. Gianmario and C. Turrini:Generalized fundamental matrices as Grass- mann tensors, Annali di Matematica196(2017) 539–553

work page 2017

[3] [3]

Chow and B

W.-L. Chow and B. van der Waerden:Zur algebraischen Geometrie. IX. ¨Uber zugeordnete For- men und algebraische Systeme von algebraischen Mannigfaltigkeiten, Mathematische Annalen 113(1937) 692–704

work page 1937

[4] [4]

Dalbec and B

J. Dalbec and B. Sturmfels:Introduction to Chow forms, Invariant Methods in Discrete and Computational Geometry (ed. N. White), Proceedings Cura¸ cao (June 1994), Kluwer Academic Publishers, 1995, pp. 37–58

work page 1994

[5] [5]

Eisenbud:Commutative Algebra, Graduate Texts in Mathematics, Springer, New York, 1995

D. Eisenbud:Commutative Algebra, Graduate Texts in Mathematics, Springer, New York, 1995

work page 1995

[6] [6]

Howard and J.J

B. Howard and J.J. Millson and A. Snowden and R. Vakil:The ideal of relations for the ring of invariants of n points on the line, J. Eur. Math. Soc.14(2012) 1:1-–60

work page 2012

[7] [7]

Kalker:Cubic fourfolds with fifteen ordinary double points, Dissertation at Univ

A. Kalker:Cubic fourfolds with fifteen ordinary double points, Dissertation at Univ. Leiden, 1986

work page 1986

[8] [8]

Kempe:On regular difference terms, Proc

A. Kempe:On regular difference terms, Proc. London Math. Soc.25(1894) 343–350

work page