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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, the Vandermonde polynomial of an ordered set of ''n'' variables X_1,\dots, X_n, named after
Alexandre-Théophile Vandermonde Alexandre-Théophile Vandermonde (28 February 1735 – 1 January 1796) was a French mathematician, musician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was b ...
, is the
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
: :V_n = \prod_ (X_j-X_i). (Some sources use the opposite order (X_i-X_j), which changes the sign \binom times: thus in some dimensions the two formulas agree in sign, while in others they have opposite signs.) It is also called the Vandermonde determinant, as it is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the
Vandermonde matrix In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \dots & x_2^\\ 1 & x_3 ...
. The value depends on the order of the terms: it is an alternating polynomial, not a
symmetric polynomial In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
.


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 In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total or ...
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 polynomial In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
s) generates the alternating polynomials.


Discriminant

Its square is widely called the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
, 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 In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
\Lambda_n _n\langle V_n^2-\Delta\rangle, which is the ring of
alternating polynomials In algebra, an alternating polynomial is a polynomial f(x_1,\dots,x_n) such that if one switches any two of the variables, the polynomial changes sign: :f(x_1,\dots,x_j,\dots,x_i,\dots,x_n) = -f(x_1,\dots,x_i,\dots,x_j,\dots,x_n). Equivalently, if o ...
.


Vandermonde polynomial of a polynomial

Given a polynomial, the Vandermonde polynomial of its roots is defined over the
splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
; for a non- monic polynomial, with leading
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
''a'', one may define the Vandermonde polynomial as :V_n = a^\prod_ (X_j-X_i), (multiplying with a leading term) to accord with the discriminant.


Generalizations

Over arbitrary
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, 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 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 . There is a closely related formula for the ch ...
, specifically the
Weyl denominator 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 . There is a closely related formula for the ch ...
(the case of the
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''. A trivial representation of an associative or Lie algebra is a ...
) of the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
\mathrm{SU}(n).


See also

*
Capelli polynomial Capelli is an Italian surname meaning hair (plural). Notable people with the surname include: *Adler Capelli (born 1973), Italian former track cyclist *Alfredo Capelli (1855–1910), Italian mathematician *Andy Capelli, fictional character on ''Gen ...

ref


References


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