证明了差幂矩阵在奇数阶时的非奇异性质，完成百年猜想。
An algebraic proof of Colombo's difference-power determinant conjecture
- 通过代数方法将零空间向量转化为超多重线性因子的实二元型
- 证实所有奇数阶差幂矩阵行列式非零，涵盖此前未解情形
- 适用于代数几何与矩阵理论研究者，解决经典数学猜想
设 $n\ge2$ 为偶数，$\lambda=(\lambda_1,\ldots,\lambda_n)\in\mathbb{R}^n$ 具有互异坐标，定义差幂矩阵 $ A_d(\lambda) := \bigl[(\lambda_r-\lambda_s)^d\bigr]_{r,s=1}^n $，其中 $ d\in\mathbb{N} $。1928年，Colombo证明了 $ \det A_{n-1}(\lambda)\ne0 $（进而 $ \det A_{n-1}(\lambda)>0 $），且当 $ 0\le d<n-1 $ 时 $ \operatorname{rank} A_d(\lambda)=d+1 $。他提出猜想：对任意 $ d\ge n-1 $，有 $ \det A_d(\lambda)\ne0 $。对于偶数 $ d $，该结论由已有距离幂矩阵结果可得。未解情形为超临界奇数指数 $ d\ge n+1 $。本文证明了这些奇数指数下行列式仍非零，从而完全证实该猜想。由此推出：$ \operatorname{rank} A_d(\lambda)=\min\{n,d+1\} $（对所有 $ d\in\mathbb{N} $）。证明的核心是将假设的零空间向量转化为具有超过其实Waring长度允许的投影实线性因子数量的实二元形式。
原文摘要 · Abstract (English)
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.
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