An algebraic proof of Colombo's difference-power determinant conjecture

AN ALGEBRAIC PROOF OF COLOMBO’S DIFFERENCE-POWER DETERMINANT CONJECTURE KUN LI* , LI TIE, PENG WANG, AND ZIHAN LIU Abstract. Let n ≥ 2 be even, let λ = (λ1 , . . . , λn ) ∈ Rn have pairwise distinct coordinates, and define the difference-power matrix

n

Ad (λ) := (λr − λs )d r,s=1 ,

arXiv:2608.28274v1 [cs.LG] 28 Aug 2026



d ∈ N.

In 1928, Colombo proved that det An−1 (λ) ̸= 0—and hence det An−1 (λ) > 0—and that rank Ad (λ) = d + 1 for 0 ≤ d < n − 1. He conjectured that det Ad (λ) ̸= 0

for every d ≥ n − 1.

For even d, the conjectured nonsingularity follows from previously published results on distancepower matrices. The remaining open cases were therefore the supercritical odd exponents d ≥ n+1. We prove nonsingularity for all these odd exponents, thereby completing Colombo’s conjecture. Consequently, rank Ad (λ) = min{n, d + 1} (d ∈ N). Our proof converts a hypothetical kernel vector into a real binary form having more projective real linear factors, counted with multiplicity, than its real Waring length permits.

Contents 1. Origin, historical progress, and main results 1.1. Origin and formulation of Colombo’s conjecture 1.2. Existing even-exponent methods and the present result 1.3. Outline of the proof 2. Binary forms, apolarity, and real Waring rank 2.1. The apolar pairing and evaluation 2.2. Vandermonde independence of pure powers 2.3. Real factors and real Waring rank 3. Odd exponents: the apolar–Waring contradiction 4. Completion and further consequences Appendix A. Correspondence with the Lean formalization Declaration on the use of generative AI and AI-assisted technologies Acknowledgements References

2 2 2 3 3 4 5 5 5 6 7 7 7 7

Date: 26 August 2026. 2020 Mathematics Subject Classification. Primary 15A15; Secondary 15A03, 15A18, 15B57, 13A50. Key words and phrases. pairwise-difference matrix, apolarity, binary form, real Waring rank. * Corresponding author. E-mail addresses: galoispure@163.com (K. LI), ttieli@icloud.com (L. TIE), wptest@qq.com (P. WANG), chelsealzh@163.com (Z. LIU).

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KUN LI* , LI TIE, PENG WANG, AND ZIHAN LIU

1. Origin, historical progress, and main results 1.1. Origin and formulation of Colombo’s conjecture. The conjecture considered here originates in Colombo’s 1928 study of a generalized Goursat problem [1, pp. 34–36]. Maclaurin expansion leads, at each degree d ∈ N, to the coefficient system n X

asd (λr − λs )d = brd ,

1 ≤ r ≤ n.

(1.1)

s=1

Here λs is the parameter of the s-th characteristic line, asd is an unknown Maclaurin coefficient, and brd is the corresponding prescribed coefficient. Abstracting the coefficient matrix, let n ≥ 2, let d ∈ N, and let λ = (λ1 , . . . , λn ) ∈ Rn have pairwise distinct coordinates. Define n

Ad (λ) = (λr − λs )d r,s=1 . 

(1.2)

For d = 0 we follow Colombo’s convention 00 = 1, so that A0 is the all-ones matrix. For fixed d, Ad (λ) is precisely the coefficient matrix of (1.1). Colombo [1, pp. 35–36, properties I–VIII] listed the following properties, paraphrased in modern notation. I. The determinant is a symmetric homogeneous polynomial of total degree nd, of degree at most 2d in each node. d II. The coefficient matrix is symmetric for even d and skew-symmetric for odd d: AT d = (−1) Ad . III. For odd d, odd order forces the determinant to vanish, whereas for even order the determinant is the square of a polynomial. IV. Within either parity class of exponents, unless the determinant vanishes identically throughout that class, it vanishes for only finitely many exponents. Q V. The square of the Vandermonde product r<s (λs − λr )2 divides the determinant. VI. At the threshold d = n − 1, the determinant is always nonzero. VII. For d < n − 1, the determinant vanishes identically and its matrix rank is exactly d + 1. VIII. The final item concerns an asymptotic limit in the exponent; it is not used in the finite-d proof. Colombo [1, p. 36] further reported verification for n = 4 and n = 6, while his methods had not settled all even orders. His remaining conjecture is that, for every even n, pairwise distinct real nodes, and integer d ≥ n − 1, one has det Ad (λ) ̸= 0. 1.2. Existing even-exponent methods and the present result. For positive even d, (λr − λs )d = |λr − λs |d , so Ad (λ) is a Hadamard power of a one-dimensional Euclidean distance matrix. Dyn–Goodman– Micchelli [2] analyzed such positive powers, and Auer [3] classified invertible Hadamard powers of distance matrices. Their results give nonsingularity and the relevant inertia in the even branch. Bhatia–Jain [4, Theorem 1(iv)] and Kapil–Mandeep–Singh [5] later gave an equivalent viewpoint through Kwong-matrix inertia and congruence with distance powers. These methods do not address the skew-symmetric higher odd-exponent branch. A recent unreviewed entry in the ICM Conjectures catalogue records that the citable literature located there left precisely the cases with even n ≥ 8 and odd d ≥ n + 1 unresolved [6]. This identifies the remaining branch addressed in the present paper. For the higher odd exponents, a hypothetical kernel vector becomes a real binary form. The matrix equations force prescribed real linear factors, whereas the Sylvester–Reznick bound

COLOMBO’S DIFFERENCE-POWER DETERMINANT

3

limits their number by the length of the same power-sum representation. The two counts are incompatible. Combining this argument with Colombo’s threshold result and the published even-exponent classification covers the full conjectural range. Theorem 1.1 (Colombo’s determinant conjecture). Let n ≥ 2 be even, let λ = (λ1 , . . . , λn ) ∈ Rn have pairwise distinct coordinates, and let d ≥ n − 1 be an integer. Then det Ad (λ) ̸= 0. Corollary 1.2 (Complete rank formula). Under the hypotheses on n and λ above, for every integer d ≥ 0, rank Ad (λ) = min{n, d + 1}. 1.3. Outline of the proof. Colombo’s property VI supplies the threshold case d = n − 1, and his property VII supplies the rank formula below the threshold [1, pp. 35–36]. Section 2 collects the classical binary-form tools used in the remaining odd branch. In Section 3, singularity gives P a nonzero form, where X and Y are commuting indeterminates, F = j cj (X + λj Y )d with at least n + 1 real projective linear factors, while the Sylvester–Reznick inequality gives at most n; this contradiction proves every supercritical odd exponent. Section 4 invokes Dyn–Goodman– Micchelli and Auer for the even exponents, completes Theorem 1.1 and Corollary 1.2, and records the determinant sign as a further result. Appendix A records the correspondence between the odd-exponent derivation and its Lean formalization. 2. Binary forms, apolarity, and real Waring rank Matrices are denoted by bold uppercase letters, vectors by bold lowercase or Greek letters, and scalar coordinates by ordinary italic letters. As above, X and Y are commuting indeterminates. Let ( ) Pd = R[X, Y ]d =

d X

fa X d−a Y a : fa ∈ R .

(2.1)

a=0

The usual monomial basis of Pd is X d , X d−1 Y, . . . , Y d . For the apolar and Vandermonde calculations, we instead use the binomially normalized monomial basis ! ! ! d d d d d−1 X , X Y, . . . , Y d. (2.2) 0 1 d Thus every F ∈ Pd has unique normalized coordinates F (X, Y ) =

d X a=0

!

d fa X d−a Y a , a

and (f0 , . . . , fd ) is its coordinate vector in (2.2). For example, X 2 − Y 2 ∈ P2 , whereas X 2 + Y is not homogeneous and belongs to no Pd . A nonzero real linear form is L = αX + βY with (α, β) ̸= (0, 0). Two such forms determine Q m the same real projective factor if they differ by a nonzero scalar. If F = j Lj j G, where the Lj are pairwise nonproportional and G has no real linear factor, define τ (F ) =

X j

mj .

(2.3)

KUN LI* , LI TIE, PENG WANG, AND ZIHAN LIU

4

Thus τ (F ) counts real projective linear factors with multiplicity. The real Waring length of nonzero F ∈ Pd is  

r X



j=1

LR (F ) = min r : F =

 

cj (αj X + βj Y )d , cj ∈ R \ {0}, (αj , βj ) ̸= (0, 0) .

(2.4)



These conventions agree with Reznick [7, §1] and Tokcan [8, Introduction]. Example 2.1 (Factor count and Waring length). For F = X 2 − Y 2 = (X − Y )(X + Y ), one has τ (F ) = 2 and LR (F ) = 2: the displayed difference of squares is a length-two representation, while F is not the square of one real linear form. The two quantities need not agree. For G = X 2 + Y 2 , τ (G) = 0 because G has no real projective zero, whereas LR (G) = 2 because G = X 2 + Y 2 and G is not a real square. If p ∈ R[x] has degree at most d, its degree-d homogenization is p[d] (X, Y ) = Y d p(X/Y ) ∈ Pd ,

(2.5)

expanded so that no division by Y remains. We write τ (p) = τ (p[d] ) when d is fixed by context. If deg p < d, the homogenization has d − deg p factors Y , which represent the projective root at infinity. For example, if p(x) = x2 − 1 and d = 3, then p[3] (X, Y ) = X 2 Y − Y 3 = Y (X − Y )(X + Y ),

τ (p) = 3.

Here the factor Y is precisely the projective root at infinity. The same example also has real Waring length three, since 1 1 4 p[3] (X, Y ) = (X + Y )3 − (X − Y )3 − Y 3 . 6 6 3


[3] The displayed representation gives LR p ≤ 3, while Theorem 2.4 gives 3 = τ p[3] ≤ LR p[3] .
Hence LR p[3] = 3. 2.1. The apolar pairing and evaluation. For P =

d X a=0

!

d pa X d−a Y a , a

Q=

d X a=0

!

d qa X d−a Y a , a

define the normalized top-transvectant (apolar) pairing by ⟨P, Q⟩d =

d X

!

a

(−1)

a=0

d pa qd−a . a

(2.6)

This is the top transvectant in normalized coordinates; see Olver [9, Chapter 5] and Ehrenborg– Rota [10]. Dolgachev–Kanev [11, Lemma 1.6] give the general pure-power evaluation property. Swapping the arguments in (2.6) shows that the pairing is symmetric for even d and alternating for odd d. For t ∈ R, define Lt (X, Y ) := X + tY ∈ P1 ,

Ldt (X, Y ) := (X + tY )d ∈ Pd .

In the normalized basis (2.2), the coordinate vector of Ldt is (1, t, . . . , td ). Lemma 2.2 (Pure powers and evaluation). For all u, v, t ∈ R and all F ∈ Pd , ⟨Ldu , Ldv ⟩d = (v − u)d ,

(2.7)

⟨Ldt , F ⟩d = F (−t, 1).

(2.8)

COLOMBO’S DIFFERENCE-POWER DETERMINANT

5

Proof. The normalized coefficient vector of Ldu is (1, u, . . . , ud ). Hence the binomial theorem gives ⟨Ldu , Ldv ⟩d =

d X

!

(−1)

a

a=0

If F =

d a d−a u v = (v − u)d . a

d
d−b Y b , then b b fb X

P

⟨Ldt , F ⟩d =

d X

d X d a d t fd−a = fb (−t)d−b = F (−t, 1). a b b=0

!

(−1)a

a=0

!

□ 2.2. Vandermonde independence of pure powers. Lemma 2.3 (Classical Vandermonde independence). If m ≤ d + 1 and t1 , . . . , tm are pairwise distinct real numbers, then Ldt1 , . . . , Ldtm are linearly independent in Pd . Proof. In the normalized monomial basis, the first m coefficient coordinates form the Vandermonde matrix. Its classical determinant formula [12] is Y

det taj 0≤a≤m−1 =  

1≤j≤m

(tj − ti ) ̸= 0,

1≤i<j≤m

□

where the final inequality follows from pairwise distinctness.

2.3. Real factors and real Waring rank. The external root-count result used below is the following homogeneous form of Sylvester’s rule of signs. Theorem 2.4 (Sylvester–Reznick). Let F ∈ R[X, Y ]d be nonzero and not a d-th power of a real linear form. Then τ (F ) ≤ LR (F ). Source and exact hypothesis. Reznick [7, Theorem 3.2] proves the sharper inequality τ (F ) ≤ σ ≤ r for an honest real representation by r ≥ 2 distinct projective directions, where σ is a cyclic sign-variation count. Applying it to a minimal real Waring representation gives the stated form. Tokcan [8, Introduction, pp. 1–2] records the same root-counting consequence. □ Remark 2.5 (Multiplicity matters). The quantity τ (F ) counts factors with multiplicity. Thus an additional factor increases τ (F ) even if it coincides with one of the already prescribed factors. For example, (X − Y )2 (X + Y ) has only two distinct real projective directions but has τ = 3, because X − Y occurs twice. 3. Odd exponents: the apolar–Waring contradiction Throughout this section, n is even, d > n − 1 is odd, and the nodes are pairwise distinct. Proposition 3.1. The matrix Ad (λ) is nonsingular. Proof. Suppose to the contrary that Ad (λ) is singular, and choose a nonzero kernel vector c = (c1 , . . . , cn )T . Define F =

n X

cj Ldλj ∈ Pd .

(3.1)

j=1

Since n ≤ d + 1, Lemma 2.3 gives F ̸= 0. For every r, Lemma 2.2 and the kernel equation give F (−λr , 1) = ⟨Ldλr , F ⟩d =

n X j=1

cj (λj − λr )d = − Ad c r = 0.

KUN LI* , LI TIE, PENG WANG, AND ZIHAN LIU

6

Consequently, X + λr Y | F (1 ≤ r ≤ n). Indeed, after the linear change U = X + λr Y , V = Y , the value at U = 0 is V d F (−λr , 1) = 0, so every term contains U . The factors are pairwise nonproportional. Hence Q(X, Y ) :=

n Y

(X + λr Y )

divides

F (X, Y ).

(3.2)

r=1

There is therefore a homogeneous H ∈ R[X, Y ]d−n such that F = QH. Since d > n − 1 and d, n have opposite parity, d − n is a positive odd integer. Every nonzero real binary form of odd degree has a real projective zero: on the unit circle, H(−v) = −H(v), so the intermediate value theorem along a semicircle gives a zero. Thus H has a real linear factor. For the concrete example H = X 3 + Y 3 , the projective zero [1 : −1] corresponds to the factor X + Y . The factor Q supplies n pairwise distinct real linear factors, and H supplies one more occurrence, even if it repeats one of them. Remark 2.5 gives τ (F ) ≥ n + 1.

(3.3)

Because Q contains at least two nonproportional factors, F is not a pure d-th power. After zero coefficients are omitted from (3.1), the same representation gives LR (F ) ≤ | supp(c)| ≤ n, where supp(c) = {j : cj ̸= 0}. Theorem 2.4 now yields n + 1 ≤ τ (F ) ≤ LR (F ) ≤ n, □

a contradiction. 4. Completion and further consequences

Proof of Theorem 1.1. If d = n − 1, Colombo’s property VI gives nonsingularity [1, pp. 35–36]. If d > n − 1 is odd, apply Proposition 3.1. If d is even, then Ad = [|λr − λs |d ] is a one-dimensional distance-power matrix. The results of Dyn–Goodman–Micchelli [2] and Auer [3] show that it is nonsingular in this range. These cases exhaust all integers d ≥ n − 1. □ Proof of Corollary 1.2. For 0 ≤ d < n − 1, Colombo’s property VII gives rank Ad = d + 1 [1, pp. 35–36]. For d ≥ n − 1, Theorem 1.1 gives rank Ad = n. □ Proposition 4.1 (Sign of the determinant). Let n ≥ 2 be even, let λ ∈ Rn have pairwise distinct coordinates, and let d ≥ n − 1. Then det Ad (λ) > 0

(d odd),

sgn det Ad (λ) = (−1)n/2

(d even).

(4.1)

Proof. For a real skew-symmetric matrix M of even order, its Pfaffian Pf(M) satisfies the standard identity det M = Pf(M)2 ; see Ishikawa–Wakayama [13, §2]. If d is odd, Ad is skew-symmetric and nonsingular by Theorem 1.1, so its determinant is strictly positive. If d is even, Dyn–Goodman–Micchelli [2] and Auer [3] give the distance-power inertia and hence the second sign in (4.1). □

COLOMBO’S DIFFERENCE-POWER DETERMINANT

7

Appendix A. Correspondence with the Lean formalization Formal verification. The odd-exponent derivation in Section 3 has been formalized and kernelchecked in Lean 4.32.1 with mathlib v4.32.1. The corresponding formalization source, including code and detailed documentation, is hosted at https://github.com/ttieli/Colombo1928. The relevant source dependency is Basic −→ Vandermonde −→ Apolar −→ ProjectiveRoots −→ SylvesterReznick −→ OddBranch. Table 1: Correspondence between paper results and Lean proof terms. Paper result

Lean file

Lean theorem

Formal role

Threshold exponent

OddBranch

threshold_nonsingular

Odd root-count bound

OddBranch

projectiveRootCount_powe rSum_le_support

All odd exponents

OddBranch

odd_nonsingular

Independently verifies Colombo’s threshold result. Formalizes the Sylvester–Reznick support bound. Matches the argument of Section 3.

The formalization contains no project-specific axiom and no sorry or admit placeholder. Its axiom audit reports only propext, Classical.choice, and Quot.sound. Kernel acceptance establishes closure of the formal proof term; it does not replace external peer review or determine literature priority. Declaration on the use of generative AI and AI-assisted technologies During the research and preparation of this manuscript, the authors used generative-AI systems, including large language models, under human direction to assist with mathematical exploration, the development of alternative proof routes, Lean 4 proof engineering and debugging, literature search and organization, and bilingual drafting and LaTeX editing. These systems are not authors and bear no responsibility for the work. The authors determined the final mathematical statements and exposition, checked the cited sources and the computational and formal outputs, and take full responsibility for the accuracy, originality, and integrity of the manuscript. Acknowledgements The authors thank the authors of the cited works and the Lean/mathlib community for the mathematical and formal infrastructure used here. References [1] Bonaparte Colombo. Problemi per le equazioni alle derivate parziali di tipo iperbolico. In Atti del Congresso Internazionale dei Matematici, Bologna, 3–10 settembre 1928, volume 3, pages 31–36, 1928. [2] Nira Dyn, Timothy Goodman, and Charles A. Micchelli. Positive powers of certain conditionally negative definite matrices. Indagationes Mathematicae (Proceedings), 89(2):163–178, 1986.

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KUN LI* , LI TIE, PENG WANG, AND ZIHAN LIU

[3] John W. Auer. On the invertibility of Hadamard powers of distance matrices. Linear and Multilinear Algebra, 45(2–3):235–245, 1998. [4] Rajendra Bhatia and Tanvi Jain. Inertia of Kwong matrices. Journal of Spectral Theory, 14(1):245–254, 2024. [5] Yogesh Kapil, Mandeep, and Mandeep Singh. On a question of Bhatia and Jain III. Results in Mathematics, 79(2):Article 51, 2024. [6] ICM Conjectures. Colombo’s difference-power determinant conjecture (entry 1928-alg-168). Online conjecture catalogue, 2026. Unreviewed catalogue record; dataset updated 18 August 2026; accessed 20 August 2026. [7] Bruce Reznick. On the length of binary forms. In Krishnaswami Alladi, Manjul Bhargava, David Savitt, and Pham Huu Tiep, editors, Quadratic and Higher Degree Forms, volume 31 of Developments in Mathematics, pages 207–232. Springer, New York, 2013. [8] Neriman Tokcan. On the Waring rank of binary forms. Linear Algebra and its Applications, 524:250–262, 2017. [9] Peter J. Olver. Classical Invariant Theory, volume 44 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999. [10] Richard Ehrenborg and Gian-Carlo Rota. Apolarity and canonical forms for homogeneous polynomials. European Journal of Combinatorics, 14(3):157–181, 1993. [11] Igor Dolgachev and Vassil Kanev. Polar covariants of plane cubics and quartics. Advances in Mathematics, 98(2):216–301, 1993. [12] Nathaniel Macon and Abraham Spitzbart. Inverses of Vandermonde matrices. The American Mathematical Monthly, 65(2):95–100, 1958. [13] Masao Ishikawa and Masato Wakayama. Minor summation formula of Pfaffians. Linear and Multilinear Algebra, 39(3):285–305, 1995.

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