arxiv: 2605.14010 · v1 · pith:MWBFHCDLnew · submitted 2026-05-13 · 🧮 math.CO

The Cullis' determinant as Pfaffian

Alexander Guterman , Andrey Yurkov This is my paper

Pith reviewed 2026-05-15 02:49 UTC · model grok-4.3

classification 🧮 math.CO

keywords Cullis determinantPfaffianrectangular matricesmaximal minorsalternating sumskew-symmetric matrixpolynomial-time algorithm

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The pith

The Cullis determinant of a rectangular matrix equals the Pfaffian of a skew-symmetric matrix obtained from it by multiplication and transposition.

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

The Cullis determinant extends the usual determinant to rectangular matrices by taking the alternating sum of all maximal minors. The paper proves this sum equals the Pfaffian of a skew-symmetric matrix built from the input matrix through multiplication and transposition. The resulting identity supplies a polynomial-time algorithm for the Cullis determinant because Pfaffians admit the same efficient evaluation methods as ordinary determinants.

Core claim

The Cullis determinant of a matrix X is equal to the Pfaffian of the matrix obtained from X by matrix multiplication and transposition.

What carries the argument

The skew-symmetric matrix constructed from rectangular matrix X by multiplication and transposition, whose Pfaffian recovers the alternating sum of maximal minors.

If this is right

Cullis determinants of rectangular matrices can be evaluated in polynomial time by reduction to existing Pfaffian routines.
Any algorithm or identity known for Pfaffians immediately transfers to the Cullis determinant.
The construction supplies an explicit bridge between the theory of maximal minors and the linear-algebraic properties of skew-symmetric matrices.

Where Pith is reading between the lines

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

The same reduction technique might apply to other alternating sums over rectangular arrays, such as certain hyperdeterminants.
Numerical linear-algebra libraries could implement the Cullis determinant by calling their Pfaffian routines after the matrix construction step.
The identity may shorten proofs that previously handled rectangular minors by direct expansion, by moving the argument to the skew-symmetric setting.

Load-bearing premise

The matrix produced by the multiplication-and-transposition construction is skew-symmetric and its Pfaffian exactly equals the alternating sum of maximal minors for every rectangular input.

What would settle it

Take any small rectangular matrix such as the 2-by-3 matrix with entries 1,2,3 in the first row and 4,5,6 in the second; compute the alternating sum of its 2-by-2 minors and the Pfaffian of the constructed matrix; the two values must match.

read the original abstract

The Cullis' determinant is a generalization of the ordinary determinant for rectangular matrices. It is defined as the alternating sum of maximal minors of given matrix. In this paper we express the Cullis' determinant of a matrix $X$ as the Pfaffian of the matrix obtained from $X$ by matrix multiplication and transposition. Relying on this result, we present an efficient polynomial-time algorithm for calculating the Cullis' determinant of given matrix.

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

The paper reduces the Cullis determinant to a Pfaffian via an explicit matrix construction from multiplication and transposition, with a direct proof that supports a polynomial-time algorithm.

read the letter

The main thing here is that Guterman and Yurkov show how to write the Cullis determinant of a rectangular matrix X as the Pfaffian of a skew-symmetric matrix built from X by a specific multiplication and transposition step. The construction produces the right size and the proof equates the Pfaffian directly to the alternating sum of maximal minors. This also yields the polynomial-time algorithm they describe. The stress-test note confirms the manuscript gives an explicit construction and handles the general rectangular case without leftover assumptions on field characteristic or dimension parity, which is a clear plus for the argument staying grounded in the definitions. What the paper does well is keep the reduction straightforward and avoid circular steps or fitted parameters. The identity follows from the two quantities' definitions, and the algorithmic consequence is immediate once the Pfaffian link is in place. On the softer side, the result stays inside combinatorial linear algebra and does not appear to open connections to wider questions or supply new proof techniques beyond this identity. The algorithm claim is polynomial in theory, but the paper might still benefit from a short note on matrix size growth or a comparison to direct minor computation for moderate dimensions. This is the sort of work that would interest specialists tracking determinant generalizations and Pfaffian identities. A reader focused on combinatorial matrix theory would get value from the concrete construction and the proof details. The thinking looks clear and the engagement with the basic definitions is honest. I would send it to peer review so a referee can check the full steps of the construction and the algorithm's correctness.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that the Cullis determinant of a rectangular matrix X—defined as the alternating sum of its maximal minors—equals the Pfaffian of a skew-symmetric matrix constructed from X by a specific multiplication-and-transposition operation. It then derives a polynomial-time algorithm for evaluating the Cullis determinant from this identity.

Significance. The explicit construction and direct proof supply a parameter-free identity that converts an alternating-sum quantity into a Pfaffian, immediately yielding an efficient algorithm via standard Pfaffian methods. This strengthens the link between generalized determinants and Pfaffian theory in combinatorial matrix algebra and provides a concrete computational payoff.

minor comments (2)

[§3] §3, construction of the auxiliary matrix: the proof that the resulting block matrix is skew-symmetric for arbitrary rectangular dimensions is stated but would benefit from an explicit low-dimensional example (e.g., 2×3 case) to verify the sign pattern.
[§5] §5, algorithm statement: the claimed polynomial-time bound relies on the O(n³) Pfaffian algorithm; a single sentence citing the reference for this complexity would make the overall runtime transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for highlighting the significance of the explicit identity linking the Cullis determinant to a Pfaffian. The recommendation for minor revision is noted; we will incorporate any editorial improvements to enhance clarity and presentation in the revised version.

Circularity Check

0 steps flagged

Derivation is self-contained from definitions with no circular steps

full rationale

The paper presents an explicit matrix construction (via multiplication and transposition of X) that yields a skew-symmetric matrix, followed by a direct proof that its Pfaffian equals the Cullis determinant (alternating sum of maximal minors). No load-bearing self-citations, no fitted parameters renamed as predictions, no self-definitional loops, and no ansatz smuggled via prior work. The identity is derived from the definitions of both quantities for arbitrary rectangular dimensions, making the derivation independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard algebraic definitions of the Cullis' determinant and the Pfaffian together with the algebraic properties of matrix multiplication and transposition; no free parameters or new entities are introduced.

axioms (1)

standard math The Pfaffian is defined for the skew-symmetric matrix produced by the stated construction from any rectangular X.
The identity assumes the construction yields a matrix to which the Pfaffian applies in the usual way.

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discussion (0)

Reference graph

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