Vandermonde polynomial

In algebra, the Vandermonde polynomial of an ordered set of n variables ${\displaystyle X_{1},\dots ,X_{n}}$, named after Alexandre-Théophile Vandermonde, is the polynomial:

${\displaystyle V_{n}=\prod _{1\leq i<j\leq n}(X_{j}-X_{i}).}$

(Some sources use the opposite order ${\displaystyle (X_{i}-X_{j})}$, which changes the sign ${\displaystyle {\binom {n}{2}}}$ times: thus in some dimensions the two formulas agree in sign, while in others they have opposite signs.)

It is also called the Vandermonde determinant, as it is the determinant of the Vandermonde matrix.

The value depends on the order of the terms: it is an alternating polynomial, not a symmetric polynomial.