Colombo's Difference-Power Determinant Conjecture Fully Proven

In 1928, mathematician Colombo made significant contributions to the understanding of difference-power matrices, proving that for an even $n \ge 2$ and distinct coordinates $\lambda$, the determinant of $A_{n-1}(\lambda)$ is non-zero and positive, and that the rank of $A_d(\lambda)$ is $d+1$ for $0 \le d < n-1$. He then conjectured that the determinant of $A_d(\lambda)$ would be non-zero for all $d \ge n-1$. For even values of $d$, the nonsingularity aspect of this conjecture had already been established through prior work on distance-power matrices. This left the cases of supercritical odd exponents, specifically $d \ge n+1$, as the remaining open problem. The current research presents an algebraic proof that successfully demonstrates the nonsingularity for all these remaining odd exponents, thereby fully completing Colombo's conjecture. A key outcome of this proof is the definitive establishment that the rank of $A_d(\lambda)$ is $\min\{n, d+1\}$ for all natural numbers $d$. The proof method involves converting a hypothetical kernel vector into a real binary form with more projective real linear factors than its real Waring length allows.