Colombo's Determinant Problem

Abstract

We completely solve Colombo's 1928 determinant problem. For distinct real $x_1,\ldots,x_N$, $N\geq 2$, and an integer $D\geq 1$, we prove that $\det[(x_j-x_i)^D]\neq 0$ if and only if $D\geq N-1$ 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.