cs.LGAug 28, 2026

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

Authors: Kun Li, Li Tie, Peng Wang, Zihan Liu

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Abstract

Let n≥2n\ge2 be even, let λ=(λ1,…,λn)∈Rnλ=(λ_1,\ldots,λ_n)\in\mathbb{R}^n have pairwise distinct coordinates, and define the difference-power matrix

Ad(λ):=[(λr−λs)d]r,s=1n,d∈N.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⁡An−1(λ)≠0\det A_{n-1}(λ)\ne0---and hence det⁡An−1(λ)>0\det A_{n-1}(λ)>0---and that rank⁡Ad(λ)=d+1\operatorname{rank} A_d(λ)=d+1 for 0≤d<n−10\le d<n-1. He conjectured that

det⁡Ad(λ)≠0for every d≥n−1.\det A_d(λ)\ne0 \qquad\text{for every } d\ge n-1.

For even dd, the conjectured nonsingularity follows from previously published results on distance-power matrices. The remaining open cases were therefore the supercritical odd exponents d≥n+1d\ge 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).\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.

Figures & tables

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Paper resultLean fileLean theoremFormal role
Threshold exponentOddBranchthreshold_nonsingularIndependently verifies Colombo’s threshold result.
Odd root-count boundOddBranchprojectiveRootCount_powerSum_le_supportFormalizes the Sylvester–Reznick support bound.
All odd exponentsOddBranchodd_nonsingularMatches the argument of Section 3 .
Appendix
Table 1. Correspondence between paper results and Lean proof terms.

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