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Colombo's Determinant Problem
Abstract
We completely solve Colombo's 1928 determinant problem. For distinct real x1,…,xN, N≥ 2, and an integer D≥ 1, we prove that det[(xj−xi)D]≠ 0 if and only if D≥ N−1 and either N is even or D is even. The even-exponent case follows from Dyn--Goodman--Micchelli (1986); the remaining odd case is proved by a strict Pfaffian sign theorem. The new odd-exponent theorem and its complete proof chain have also been formalized in Lean 4.
Versions
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BibTeX
@misc{ma2026colombos,
title = {{Colombo's Determinant Problem}},
author = {Qianli Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00101},
eprint = {2609.00101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00101},
}
RIS
TY - UNPB
TI - Colombo's Determinant Problem
AU - Qianli Ma
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00101
UR - https://arxiv.org/abs/2609.00101
AB - We completely solve Colombo's 1928 determinant problem. For distinct real x_1,…,x_N, N≥ 2, and an integer D≥ 1, we prove that det[(x_j−x_i)^D]≠ 0 if and only if D≥ N−1 and either N is even or D is even. The even-exponent case follows from Dyn--Goodman--Micchelli (1986); the remaining odd case is proved by a strict Pfaffian sign theorem. The new odd-exponent theorem and its complete proof chain have also been formalized in Lean 4.
ER -