AI 中文总结
该研究完成了Colombo关于差幂行列式非奇异性的猜想，证明了超临界奇指数d≥n+1时差幂矩阵的非奇异性，得出其秩的统一公式。
AI 中文摘要
设n≥2为偶数，λ=(λ₁,…,λₙ)∈ℝⁿ的坐标两两不同，定义差幂矩阵A_d(λ):=[(λᵣ−λₛ)ᵈ]ᵣ,ₛ=1ⁿ，其中d∈ℕ。1928年，Colombo证明det A_{n−1}(λ)≠0（故det A_{n−1}(λ)>0），且对0≤d<n−1有rank A_d(λ)=d+1，他猜想对所有d≥n−1都有det A_d(λ)≠0。对于偶数d，该猜想的非奇异性可由已发表的关于距离幂矩阵的结果推出，剩余未解决的是超临界奇指数d≥n+1的情况。我们证明了所有这些奇指数下的非奇异性，从而完成了Colombo猜想，进而得到rank A_d(λ)=min{n,d+1}（d∈ℕ）。我们的证明将假设的核向量转化为一个实二元型，其具有的射影实线性因子（计重数）多于其实Waring长度允许的数量。
英文摘要
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.
