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Vandermonde polynomial

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Vandermonde polynomial

In algebra, the Vandermonde polynomial of an ordered set of n variables X_1,\dots, X_n, named after Alexandre-Théophile Vandermonde, is the polynomial:

V_n = \prod_{1\le i

(Some sources use the opposite order (X_i-X_j), which changes the sign \binom{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.

Alternating

The defining property of the Vandermonde polynomial is that it is alternating in the entries, meaning that permuting the 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, and thus 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) as a factor for all i \neq j.

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: \Delta=V_n^2) does not depend on the order of terms, as (-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 \Lambda_n, one obtains the quadratic extension \Lambda_n[V_n]/\langle V_n^2-\Delta\rangle, which is the ring of alternating polynomials.

Characteristic classes

In characteristic classes, the Vandermonde polynomial corresponds to the Euler class, and its square (the discriminant) corresponds to the top Pontryagin class. This is formalized in the splitting principle, which connects characteristic classes to polynomials.

In the language of stable homotopy theory, the Vandermonde polynomial (and alternating polynomials generally) is an unstable phenomenon, which corresponds to the fact that the Euler class is an unstable characteristic class. That is, the ring of symmetric polynomials in n variables can be obtained from the ring of symmetric polynomials in arbitrarily many variables by evaluating all variables above X_n to zero: symmetric polynomials are thus stable or compatibly defined. However, this is not the case for the Vandermonde polynomial or alternating polynomials: the Vandermonde polynomial in n variables is not obtained from the Vandermonde polynomial in n+1 variables by setting X_{n+1}=0.

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 Vandemonde polynomial as

V_n = a^{n-1}\prod_{1\le i

(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 SU(n).

See also

  • Capelli polynomial (ref)

References

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