Vandermonde polynomial
Encyclopedia
In algebra, the Vandermonde polynomial of an ordered set of n variables 
, named after Alexandre-Théophile Vandermonde
, is the polynomial:
(Some sources use the opposite order
, which changes the sign 
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
.
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
and thus 
(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
as a factor for all 
.
s) generates the alternating polynomial
s.
, though some sources call the Vandermonde polynomial itself the discriminant.
The discriminant (the square of the Vandermonde polynomial:
) does not depend on the order of terms, as 
, 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
, one obtains the quadratic extension 
, which is the ring of alternating polynomials.
, 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
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 
variables by setting 
.
; for a non-monic polynomial, with leading coefficient a, one may define the Vandemonde polynomial as
(multiplying with a leading term) to accord with the discriminant.
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
.
, named after Alexandre-Théophile VandermondeAlexandre-Théophile Vandermonde
Alexandre-Théophile Vandermonde was a French musician, mathematician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.Vandermonde was a violinist, and became engaged with...
, is the polynomial:
(Some sources use the opposite order
, which changes the sign 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
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the Vandermonde matrix.
The value depends on the order of the terms: it is an alternating polynomial
Alternating polynomial
In algebra, an alternating polynomial is a polynomial f such that if one switches any two of the variables, the polynomial changes sign:f = -f....
, not a symmetric polynomial
Symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...
.
Alternating
The defining property of the Vandermonde polynomial is that it is alternating in the entries, meaning that permuting the 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
and thus (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
as a factor for all .Alternating polynomials
Thus, the Vandermonde polynomial (together with the symmetric polynomialSymmetric polynomial
In mathematics, a symmetric polynomial is a polynomial P in n variables, such that if any of the variables are interchanged, one obtains the same polynomial...
s) generates the alternating polynomial
Alternating polynomial
In algebra, an alternating polynomial is a polynomial f such that if one switches any two of the variables, the polynomial changes sign:f = -f....
s.
Discriminant
Its square is widely called the discriminantDiscriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
, though some sources call the Vandermonde polynomial itself the discriminant.
The discriminant (the square of the Vandermonde polynomial:
) does not depend on the order of terms, as , 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
, one obtains the quadratic extension , which is the ring of alternating polynomials.Characteristic classes
In characteristic classes, the Vandermonde polynomial corresponds to the Euler classEuler class
In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
, and its square (the discriminant) corresponds to the top Pontryagin class
Pontryagin class
In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four...
. This is formalized in the splitting principle
Splitting principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct...
, which connects characteristic classes to polynomials.
In the language of stable homotopy theory
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
, 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
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 variables by setting .Vandermonde polynomial of a polynomial
Given a polynomial, the Vandermonde polynomial of its roots is defined over the splitting fieldSplitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
; for a non-monic polynomial, with leading coefficient a, one may define the Vandemonde 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
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
, specifically the Weyl denominator formula (the case of the trivial representation
Trivial representation
In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...
) of the special unitary group
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
.See also
- Capelli polynomial (ref)