In algebra, the Vandermonde polynomial of an ordered set of n variables
Contents
- Alternating
- Alternating polynomials
- Discriminant
- Vandermonde polynomial of a polynomial
- Generalizations
- Weyl character formula
- References
(Some sources use the opposite order
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.
Alternating
The defining property of the Vandermonde polynomial is that it is alternating in the entries, meaning that permuting the
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
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
Alternating polynomials
Thus, the Vandermonde polynomial (together with the symmetric polynomials) generates the alternating polynomials.
Discriminant
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:
If one adjoins the Vandermonde polynomial to the ring of symmetric polynomials in n variables
Vandermonde polynomial of a polynomial
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
(multiplying with a leading term) to accord with the discriminant.
Generalizations
Over arbitrary rings, one instead uses a different polynomial to generate the alternating polynomials – see (Romagny, 2005).
Weyl character formula
(a vast generalization)
The Vandermonde polynomial can be considered a special case of the Weyl character formula, specifically the Weyl denominator formula (the case of the trivial representation) of the special unitary group