Vandermonde polynomial

The defining property of the Vandermonde polynomial is that it is alternating in the entries, meaning that permuting the ${\displaystyle X_{i}}$ 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 ${\displaystyle V_{n}=-V_{n},}$ and thus ${\displaystyle V_{n}=0}$ (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 ${\displaystyle (X_{i}-X_{j})}$ as a factor for all ${\displaystyle i\neq j}$.

Its square is widely called the discriminant, though some sources call the Vandermonde polynomial itself the discriminant.

The discriminant (the square of the Vandermonde polynomial: ${\displaystyle \Delta =V_{n}^{2}}$) does not depend on the order of terms, as ${\displaystyle (-1)^{2}=1}$, and is thus an invariant of the unordered set of points.

If one adjoins the Vandermonde polynomial to the ring of symmetric polynomials in n variables ${\displaystyle \Lambda _{n}}$, one obtains the quadratic extension ${\displaystyle \Lambda _{n}[V_{n}]/\langle V_{n}^{2}-\Delta \rangle }$, which is the ring of alternating polynomials.

Given a polynomial, the Vandermonde polynomial of its roots is defined over the splitting field; for a non-monic polynomial, with leading coefficient a, one may define the Vandermonde polynomial as

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

(multiplying with a leading term) to accord with the discriminant.

Over arbitrary rings, one instead uses a different polynomial to generate the alternating polynomials – see (Romagny, 2005).

The Vandermonde determinant is a very special case of the Weyl denominator formula applied to the trivial representation of the special unitary group ${\displaystyle \mathrm {SU} (n)}$.

• Capelli polynomial (ref)

• The fundamental theorem of alternating functions, by Matthieu Romagny, September 15, 2005