Vandermonde polynomial

In algebra, the Vandermonde polynomial of an ordered set of n variables X 1 , … , X n {displaystyle X_{1},dots ,X_{n}} , named after Alexandre-Théophile Vandermonde, is the polynomial: In algebra, the Vandermonde polynomial of an ordered set of n variables X 1 , … , X n {displaystyle X_{1},dots ,X_{n}} , named after Alexandre-Théophile Vandermonde, is the polynomial: (Some sources use the opposite order ( X i − X j ) {displaystyle (X_{i}-X_{j})} , which changes the sign ( n 2 ) {displaystyle {inom {n}{2}}} times: thus in some dimensions the two formulae 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. The defining property of the Vandermonde polynomial is that it is alternating in the entries, meaning that permuting the X i {displaystyle X_{i}} by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the polynomial – in fact, it is the basic alternating polynomial, as will be made precise below. It thus depends on the order, and is zero if two entries are equal – this also follows from the formula, but is also consequence of being alternating: if two variables are equal, then switching them both does not change the value and inverts the value, yielding V n = − V n , {displaystyle V_{n}=-V_{n},} and thus V n = 0 {displaystyle V_{n}=0} (assuming the characteristic is not 2, otherwise being alternating is equivalent to being symmetric). Conversely, the Vandermonde polynomial is a factor of every alternating polynomial: as shown above, an alternating polynomial vanishes if any two variables are equal, and thus must have ( X i − X j ) {displaystyle (X_{i}-X_{j})} as a factor for all i ≠ j {displaystyle i eq j} . Thus, the Vandermonde polynomial (together with the symmetric polynomials) generates the alternating polynomials. Its square is widely called the discriminant, though some sources call the Vandermonde polynomial itself the discriminant.