Source-linked AI summary
An algebraic proof of Colombo's difference-power determinant conjecture
Kun Li, Li Tie, Peng Wang, Zihan Liu
TL;DR
The paper addresses Colombo’s conjecture that difference-power matrices are nonsingular for all d ≥ n − 1 under even n and distinct real nodes, especially the unresolved supercritical odd exponents. It converts a hypothetical kernel vector into a real binary form whose required number of projective real factors exceeds its Waring-length bound, and combines this proof with existing even-exponent results. The result is nonsingularity throughout the conjectured range and the complete rank formula for all d ≥ 0.
Problem
Colombo’s conjecture asks whether det A_d(λ) ≠ 0 for every integer d ≥ n − 1 with even n and pairwise distinct real nodes, with supercritical odd exponents remaining unresolved.
Method
For higher odd exponents, a hypothetical kernel vector is converted into a real binary form whose prescribed real linear factors outnumber the factors allowed by the Sylvester–Reznick Waring-length bound.
Results
The paper proves nonsingularity for all d ≥ n − 1, combining the odd-exponent proof with Colombo’s threshold case and published even-exponent results.
Takeaways & Limitations
Consequently, rank A_d(λ) = min{n, d + 1} for every integer d ≥ 0.
Takeaways & Limitations
The odd-exponent argument is developed under the assumptions that n is even, d > n − 1 is odd, and the nodes are pairwise distinct.
Abstract
from arXiv · showhide
Let $n\ge2$ be even, let $λ=(λ_1,\ldots,λ_n)\in\mathbb{R}^n$ have pairwise distinct coordinates, and define the difference-power matrix \[ A_d(λ) := \bigl[(λ_r-λ_s)^d\bigr]_{r,s=1}^n, \qquad d\in\mathbb{N}. \] In 1928, Colombo proved that $\det A_{n-1}(λ)\ne0$---and hence $\det A_{n-1}(λ)>0$---and that $\operatorname{rank} A_d(λ)=d+1$ for $0\le d<n-1$. He conjectured that \[ \det A_d(λ)\ne0 \qquad\text{for every } d\ge n-1. \] For even $d$, the conjectured nonsingularity follows from previously published results on distance-power matrices. The remaining open cases were therefore the supercritical odd exponents $d\ge n+1$. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, \[ \operatorname{rank} A_d(λ)=\min\{n,d+1\} \qquad(d\in\mathbb{N}). \] 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.
1. Origin, historical progress, and main results
The paper completes Colombo’s determinant conjecture by resolving the higher odd-exponent cases with a binary-form contradiction, while incorporating established results for even exponents. It consequently establishes the complete rank formula for all nonnegative exponents.
- Origin and historical progress: Colombo’s conjecture asks whether det A_d(λ) ≠ 0 for every integer d ≥ n − 1 when n is even and the nodes are pairwise distinct.The conjecture arose from Colombo’s 1928 study and remained unresolved in the higher odd branch.
- Existing even-exponent methods: For even d, published results on one-dimensional Euclidean distance-power matrices provide nonsingularity and relevant inertia.These methods do not address the skew-symmetric higher odd-exponent branch.
- Present proof strategy: A hypothetical kernel vector for a higher odd exponent yields a real binary form with at least n + 1 real projective linear factors.The argument uses the factorization F = QH and the existence of a real linear factor in the positive odd-degree form H.
- Present proof strategy: The Sylvester–Reznick bound permits at most n such factors from the corresponding real Waring representation, producing a contradiction.This proves nonsingularity for the remaining supercritical odd exponents.
- Main results: Theorem 1.1 establishes det A_d(λ) ≠ 0 for every integer d ≥ n − 1 under the stated hypotheses.The proof combines Colombo’s threshold case, the odd-exponent proposition, and published even-exponent results.
- Main results: The complete rank formula is rank A_d(λ) = min{n, d + 1} for every integer d ≥ 0.Below the threshold, Colombo’s property VII gives rank d + 1; at and above it, nonsingularity gives rank n.
2. Binary forms, apolarity, and real Waring rank
The section develops binary-form tools for counting real projective factors and measuring real Waring length, together with apolar evaluation and Vandermonde independence. These ingredients support the later contradiction argument.
- 2. Binary forms, apolarity, and real Waring rank: Binary forms are homogeneous degree-d polynomials in commuting indeterminates X and Y, represented in monomial or binomially normalized bases.The normalized basis is used for apolar and Vandermonde calculations.
- 2. Binary forms, apolarity, and real Waring rank: A real projective linear factor is identified up to nonzero scalar multiples, and τ(F) counts such factors with multiplicity.Repeated proportional factors contribute repeatedly to τ(F).
- 2. Binary forms, apolarity, and real Waring rank: The real Waring length LR(F) is the minimum number of real d-th powers of linear forms needed to represent a nonzero binary form.The examples show that factor count and Waring length need not coincide.
- 2. Binary forms, apolarity, and real Waring rank: Degree-d homogenization preserves a polynomial while adding Y factors for roots at infinity when its degree is below d.For p(x)=x^2−1 and d=3, p[3](X,Y)=Y(X−Y)(X+Y), so τ(p)=3.
- 2.1. The apolar pairing and evaluation.: The normalized top-transvectant, or apolar, pairing is symmetric for even d and alternating for odd d.Pure powers admit an evaluation property used in subsequent calculations.
- 2.2. Vandermonde independence of pure powers.: At most d+1 pure powers associated with distinct real parameters are linearly independent in P_d by the classical Vandermonde determinant.The first m normalized coefficient coordinates form a Vandermonde matrix whose determinant is nonzero under pairwise distinctness.
- 2.3. Real factors and real Waring rank.: Sylvester–Reznick bounds the number of real projective factors of a nonzero form that is not a real d-th power by its real Waring length.The sharper formulation inserts a cyclic sign-variation count between factor count and representation length.
3. Odd exponents: the apolar–Waring contradiction
For even n and odd d>n−1, singularity would produce a nonzero binary form divisible by the prescribed n linear factors and one additional real factor. This exceeds the form’s Waring-length bound, proving nonsingularity.
- 3. Odd exponents: the apolar–Waring contradiction: For even n, pairwise distinct nodes, and odd d>n−1, Proposition 3.1 states that A_d(λ) is nonsingular.These are the standing hypotheses for the higher odd-exponent argument.
- 3. Odd exponents: the apolar–Waring contradiction: Assuming singularity, a nonzero kernel vector c defines a degree-d real binary form F through the corresponding power-sum representation.Because n≤d+1, Vandermonde independence ensures F is nonzero.
- 3. Odd exponents: the apolar–Waring contradiction: The kernel equations force every factor X+λ_rY to divide F, and pairwise distinct nodes make these factors pairwise nonproportional.The divisibility follows by evaluating F after the linear change U=X+λ_rY, V=Y.
- 3. Odd exponents: the apolar–Waring contradiction: Writing F=QH leaves H with positive odd degree, so H has a real projective zero and contributes one additional real linear-factor occurrence.The existence of the zero follows from antipodal sign reversal on the unit circle and the intermediate value theorem.
- 3. Odd exponents: the apolar–Waring contradiction: The factor count reaches at least n+1, while the kernel representation gives LR(F)≤|supp(c)|≤n, contradicting the Sylvester–Reznick bound.The additional factor counts even when it repeats one of the prescribed factors; Q also prevents F from being a pure d-th power.
4. Completion and further consequences
The proof completes Colombo’s conjecture by combining the odd and even exponent cases, then derives the full rank formula and determinant-sign information.
- Completion of the conjecture: All integers d ≥ n−1 are covered: d=n−1 follows from Colombo, odd d>n−1 from Proposition 3.1, and even d from distance-power matrix results.These cases establish nonsingularity throughout the conjectured range.
- Rank consequence: For 0 ≤ d < n−1, Colombo’s property VII gives rank A_d=d+1, while Theorem 1.1 gives rank A_d=n for d≥n−1.Together, the two ranges yield rank A_d=min{n,d+1}.
- Determinant sign: For even n and d≥n−1, Proposition 4.1 states det A_d(λ)>0 for odd d and sgn det A_d(λ)=(−1)^(n/2) for even d.The determinant sign is an additional consequence beyond nonsingularity.
- Determinant sign: For odd d, skew-symmetry and nonsingularity imply positive determinant via det M=Pf(M)^2; even-d signs come from distance-power inertia.The Pfaffian identity applies to real skew-symmetric matrices of even order.
Appendix A. Correspondence with the Lean formalization
The appendix documents a Lean 4.32.1 and mathlib v4.32.1 formalization of the odd-exponent derivation, including its source, proof-term correspondence, and axiom audit.
- Formal verification: The odd-exponent derivation was formalized and kernel-checked in Lean 4.32.1 with mathlib v4.32.1.The corresponding source code and documentation are hosted in the Colombo1928 repository.
- Proof-term correspondence: Table 1 records the correspondence between the paper’s results and Lean proof terms.The table provides the appendix’s organizing link between mathematical statements and formal artifacts.
- Formal verification: The formalization uses no project-specific axiom and no sorry or admit placeholder; its axiom audit lists only propext, Classical.choice, and Quot.sound.Kernel acceptance verifies closure of the formal proof term but does not replace peer review or establish literature priority.
Declaration on the use of generative AI and AI-assisted technologies
The authors report using generative-AI systems under human direction for mathematical exploration, proof engineering, literature work, and drafting, while retaining responsibility for the final work.
- Use of AI-assisted technologies: Generative-AI systems assisted with mathematical exploration, alternative proof routes, Lean proof engineering, literature organization, and bilingual drafting and LaTeX editing.The systems are described as tools rather than authors.
- Author responsibility: The authors state that they determined the final mathematical statements and exposition, checked cited sources and outputs, and take full responsibility for the work.