Year: 2026
Type: preprint
Abstract: Let $n\ge2$ be even, let $\lambda=(\lambda_1,\ldots,\lambda_n)\in\mathbb{R}^n$ have pairwise distinct coordinates, and define the difference‑power matrix $$ A_d(\lambda) := \bigl[(\lambda_r-\lambda_s)^d\bigr]_{r,s=1}^n, \qquad d\in\mathbb N. $$ In 1928, Colombo proved that $\det A_{n-1}(\lambda)\ne0$—and hence $\det A_{n-1}(\lambda)>0$—and that $\operatorname{rank} A_d(\lambda)=d+1$ for $0\le d<n-1$. He conjectured that $$ \det A_d(\lambda)\ne0 \qquad\text{for every } d\ge n-1. $$ 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, $$ \operatorname{rank} A_d(\lambda)=\min\{n,d+1\} \qquad(d\in\mathbb N). $$ 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.
Source: arXiv (Cornell University)
Authors: Li, Kun, Tie, Li, Wang, Peng, Liu, Zihan
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204443264