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

arXiv · AI, language, vision and robotics · article · Aug 28, 2026 · UTC

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 supercriti

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