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

Abstract

Let $n\ge 2$ be even, let $\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{R}}^n$ 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 $\det A_{n-1}(\lambda )\ne 0$---and hence $\det A_{n-1}(\lambda )>0$---and that $\mathrm{rank}{A}_d(\lambda )=d+1$ 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.