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arxiv: 2302.02471 · v4 · submitted 2023-02-05 · 🧮 math.CA · math-ph· math.MP

Determinantally equivalent nonzero functions

Harry Sapranidis Mantelos This is my paper

Pith reviewed 2026-05-24 09:52 UTC · model grok-4.3

classification 🧮 math.CA math-phmath.MP
keywords determinantal equivalenceprincipal minorskernel transformationsnon-symmetric kernelsgraph cyclescombinatorial proof
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The pith

The conjecture classifying determinantally equivalent functions holds without symmetry under simple additional assumptions.

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

The paper studies pairs of functions K and Q such that every finite submatrix has the same determinant. It first exhibits a counterexample showing that the two conjectured transformations fail to cover all cases once symmetry is dropped. It then proves the same two transformations still give the complete classification when the functions obey minor extra conditions that block the counterexamples. The argument models the determinant condition as a weighted graph and derives the required relations from the algebraic identities on 3-cycles and 4-cycles, supplying an elementary combinatorial proof that also resolves the finite-matrix case.

Core claim

If two nonzero functions K and Q agree on all principal minors and satisfy the additional minor assumptions, then one is obtained from the other either by transposition or by pointwise multiplication with g(x) g(y)^{-1} for some nowhere-zero function g.

What carries the argument

Graph-theoretic encoding of the functions as edge-weighted graphs on vertex set Lambda, where agreement of determinants translates into identities on 3-cycles and 4-cycles that force the allowed transformations.

If this is right

  • The classification of equivalent kernels now applies directly to non-symmetric determinantal point processes that meet the extra conditions.
  • The finite-matrix version of the problem receives a purely combinatorial proof using only cycle identities rather than linear-algebraic machinery.
  • The same cycle-based technique yields an explicit description of all transformations preserving the determinant sequence for any finite collection of points.

Where Pith is reading between the lines

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

  • The counterexamples likely arise only when Lambda is infinite and the functions take values that cannot be realized by finite or symmetric data.
  • Analogous cycle arguments may classify transformations preserving other matrix invariants such as permanents on the same graph model.
  • The minor assumptions may be verifiable by checking finitely many small subsets, turning the global classification into a local condition.

Load-bearing premise

The functions satisfy additional minor assumptions that rule out the graph-theoretic counterexamples constructed in the paper.

What would settle it

An explicit pair of nonzero functions K and Q on some set Lambda that agree on every principal minor, obey the minor assumptions, yet are not related by transposition or by scaling with any nowhere-zero g.

Figures

Figures reproduced from arXiv: 2302.02471 by Harry Sapranidis Mantelos.

Figure 1
Figure 1. Figure 1: In green is the 2-cycle (p1, p3, p1) and in red is the 2-cycle (p0, p2, p0). In the below figure we provide a pictorial proof of the lemma: p0 p3 p1 p2 p0 p3 p1 p2 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: On the LHS graph, in red is the cycle s (1), and in blue is the cycle s (2). On the RHS graph, in red is the cycle r (1), and in blue is the cycle r (2) . Lemma 5.3 Let p := (pi) 3 i=0 be a 3-cycle in M, where |M| = 4, and let the two-variable function h : M2 → F be non-zero-valued except possibly on the set {(x, x) : x ∈ M}. Let p4 ∈ M be the unique p4 ∈ { / p0, p1, p2}. We can then write h[p] as a produc… view at source ↗
Figure 3
Figure 3. Figure 3: In red is the 2-cycle (p0, p4, p0), in green is the 2-cycle (p2, p4, p2), and in blue is the 2-cycle (p1, p4, p1). In the below figure we provide a pictorial proof of the lemma. p2 p1 p0 p4 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: In red is the 3-cycle s (1), in green is the 3-cycle s (2), and in blue is the 3-cycle s (3) . 6 Proof of the Main Result As is usually the case in mathematics, in every proof of a theorem there is one key trick which said proof is revolved around. In our case, it’s the following surprisingly elementary lemma. Lemma 6.1 Let a, b, a′ , b′ ∈ F be constants satisfying a + b = a ′ + b ′ (18) and ab = a ′ b ′ .… view at source ↗
read the original abstract

We study the problem raised in [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples. The problem is plainly stated as follows: Let $\Lambda$ be a set and $\mathbb{F}$ a field, and suppose that $K,Q:\Lambda^2\to\mathbb{F}$ are two functions such that for any $n\in\mathbb{N}$ and $x_1,x_2,\ldots,x_n\in\Lambda$, the determinants of matrices $(K(x_i,x_j))_{1\leq i,j\leq n}$ and $(Q(x_i,x_j))_{1\leq i,j\leq n}$ agree. What are all the possible transformations that transform $Q$ into $K$? In [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] the following two were conjectured: $(Tf)(x,y)=f(y,x)$; and $(Tf)(x,y)=g(x)g(y)^{-1}f(x,y)$ for some nowhere-zero function $g$. In the same paper, this conjectured classification is verified in the case of symmetric functions $K$ and $Q$. By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed. By taking $\Lambda$ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23--64]. Hence, our paper presents a simple non-linear-algebraic proof that uses only some elementary combinatorics and three simple algebraic identities involving $3$-cycles and $4$-cycles.

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

0 major / 1 minor

Summary. The paper constructs a counterexample to the conjecture (from Stevens 2021) that the only transformations mapping a function Q to a determinantally equivalent function K are transposition and diagonal scaling by a nowhere-zero g, when K and Q are not required to be symmetric. It then proves that the classification holds under additional minor assumptions that rule out the counterexamples, by extending graph-theoretic techniques, using elementary 3-cycle and 4-cycle identities, and reducing the finite case to a combinatorial argument that avoids the full machinery of Loewy (1986).

Significance. If the result holds, the work delineates the precise boundary between the symmetric case (already settled) and the general case, showing that the conjecture survives under natural extra conditions while failing without them. The elementary combinatorial proof is a clear strength, as is the explicit counterexample construction; both make the classification more accessible for applications in determinantal point processes.

minor comments (1)
  1. The abstract states that the additional assumptions 'preclude such counterexamples' but does not name them; a one-sentence characterization in the abstract would improve readability without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation to accept the manuscript. The report accurately captures the contribution of the counterexample and the elementary combinatorial proof under the stated assumptions.

Circularity Check

0 steps flagged

No circularity: derivation extends external graph techniques and uses standard cycle identities plus reduction to independent 1986 matrix result

full rationale

The paper constructs an explicit counterexample to the unrestricted conjecture from the 2021 Stevens paper (different author) and then classifies the transformations under added assumptions using elementary 3-cycle/4-cycle identities and graph techniques extended from that external source. The finite-matrix reduction is invoked only as supporting context and points to the independent Loewy 1986 result on principal minors; no parameter is fitted to data, no prediction is defined by construction from its own inputs, and no load-bearing step reduces to a self-citation or ansatz smuggled from the authors' prior work. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard mathematical axioms for fields and determinants together with the graph representation of kernels introduced in the cited work; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Determinant is well-defined and multilinear over any field F
    Invoked when equating det(K(x_i,x_j)) and det(Q(x_i,x_j)) for all finite subsets.
  • domain assumption Graph-theoretic encoding of kernel entries extends from the symmetric case
    Used to extend the techniques of the cited Stevens paper to the non-symmetric setting.

pith-pipeline@v0.9.0 · 5915 in / 1363 out tokens · 30338 ms · 2026-05-24T09:52:00.497540+00:00 · methodology

discussion (0)

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Reference graph

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