Colombo’s difference-power determinant conjecture

For pairwise distinct real numbers $x_1,\ldots,x_N$ and an integer $D\ge1$, Ma proves the complete classification $$\det\bigl[(x_j-x_i)^D\bigr]_{i,j=1}^{N}\ne0 \quad\Longleftrightarrow\quad D\ge N-1\ \text{and}\ \bigl(N\text{ is even or }D\text{ is even}\bigr).$$ This fully resolves Colombo’s original conjecture for even $N$. The new part is the even-size, odd-exponent branch, strengthened to the strict Pfaffian sign theorem $$(-1)^{\binom m2}\operatorname{Pf} \bigl[(x_j-x_i)^{2r+1}\bigr]_{i,j=1}^{2m}>0 \qquad(r\ge m-1).$$ The even-exponent branch follows from the classical work of Dyn–Goodman–Micchelli. A concurrent independent proof of the odd branch by Kun Li, Li Tie, Peng Wang and Zihan Liu is linked below.

For pairwise distinct real numbers $x_1,\ldots,x_N$ and an integer $D\ge1$, Ma proves the complete classification $$\det\bigl[(x_j-x_i)^D\bigr]_{i,j=1}^{N}\ne0 \quad\Longleftrightarrow\quad D\ge N-1\ \text{and}\ \bigl(N\text{ is even or }D\text{ is even}\bigr).$$ This fully resolves Colombo’s original conjecture for even $N$. The new part is the even-size, odd-exponent branch, strengthened to the strict Pfaffian sign theorem $$(-1)^{\binom m2}\operatorname{Pf} \bigl[(x_j-x_i)^{2r+1}\bigr]_{i,j=1}^{2m}>0 \qquad(r\ge m-1).$$ The even-exponent branch follows from the classical work of Dyn–Goodman–Micchelli. A concurrent independent proof of the odd branch by Kun Li, Li Tie, Peng Wang and Zihan Liu is linked below.