Nonsingularity in the odd-exponent Colombo matrix conjecture
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For even N=2m≥2, distinct real nodes x₁,…,x₂ₘ and odd D≥N−1, must the matrix [(xⱼ−xᵢ)^D] be nonsingular? Colombo’s boundary exponent and the even-exponent case leave this odd-exponent range.
Why it matters and what counts as progress
Why it matters: The odd-exponent range completes a determinant question whose boundary and even-exponent cases were already known.
What counts as progress: Prove nonsingularity throughout the odd range or a sharper Pfaffian sign formula after ordering the nodes.
How the problem developed
Giuseppe Colombo In his 1928 ICM communication, recorded in the 1930 proceedings, established the rank obstruction and nonsingular endpoint D = N−1, described parity cases, and conjectured nonsingularity for even N and D ≥ N−1.
Nira Dyn, Timothy Goodman and Charles A. Micchelli Earlier distance-matrix results settled the even-exponent branch, leaving the supercritical odd exponents as the remaining cases.
Qianli Ma; WuJie AI and other agent systems, DeepSeek, Qwen, Kimi, GPT and other LLMs substantially contributed to the proof strategy and initial draft Proved the remaining odd-exponent cases for even N and supplied a Lean 4 formalization of the new result.
- 28 Aug 2026 · Independent proof [3]
Kun Li, Li Tie, Peng Wang and Zihan Liu Published an independent algebraic proof of nonsingularity for the remaining odd exponents.
AI contribution
AI-assisted proof settles the odd-exponent Colombo case
For even N=2m≥2, pairwise distinct real numbers x₁,…,x₂ₘ and odd D=2r+1≥N−1, Ma proves that the matrix [(xⱼ−xᵢ)^D] is nonsingular. After sorting the nodes, the proof gives the stronger sign (−1)^(m choose 2) Pf[((xⱼ−xᵢ)^D)]>0.
Sources
- Colombo, Problemi per le equazioni alle derivate parziali di tipo iperbolico
- Ma, Colombo’s determinant problem (arXiv version)
- Li, Tie, Wang and Liu, An algebraic proof of Colombo’s difference-power determinant conjecture
- Dyn, Goodman and Micchelli, Positive powers of certain conditionally negative definite matrices
- Ma, Colombo’s determinant problem (Zenodo record)
- Colombo determinant review
- Ma’s odd-exponent proof
- Pinned Lean theorem and check
- Matching Palomar verification record
- Independent odd-exponent proof
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