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PreprintCC BY 4.0arXiv

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

  • Kun Li
  • Li Tie
  • Peng Wang
  • Zihan Liu
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Abstract

Let n≥2 be even, let λ=(λ1,…,λn)∈ℝn have pairwise distinct coordinates, and define the difference-power matrix Ad(λ) := [(λr−λs)d]r,s=1n, 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 distance-power 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.

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Cite

BibTeXRIS
BibTeX
@misc{li2026algebraic,
  title = {{An algebraic proof of Colombo's difference-power determinant conjecture}},
  author = {Kun Li and Li Tie and Peng Wang and Zihan Liu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28274},
  eprint = {2608.28274},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28274},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An algebraic proof of Colombo's difference-power determinant conjecture
AU  - Kun Li
AU  - Li Tie
AU  - Peng Wang
AU  - Zihan Liu
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28274
UR  - https://arxiv.org/abs/2608.28274
AB  - Let n≥2 be even, let λ=(λ_1,…,λ_n)∈ℝ^n have pairwise distinct coordinates, and define the difference-power matrix A_d(λ) := [(λ_r−λ_s)^d]_r,s=1^n, d∈ℕ. In 1928, Colombo proved that det A_n−1(λ)≠0---and hence det A_n−1(λ)>0---and that rank A_d(λ)=d+1 for 0≤ d<n−1. He conjectured that det A_d(λ)≠0 for every d≥ 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≥ n+1. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, rank A_d(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.
ER  -