138
If the j in Eq. C. 1 are consecutive integer powers of their arguments, indexed from
0 through n 1, i.e.
1 x, x x71
2 n-I
1 X2 X2 X2
An(x)= I V(x) = ", Eq. C.2
1 xn x x.
then the resulting special case alternant is known as a Vandermonde determinant. The
values of a Vandermonde determinant and its minors are closely related to a class of
polynomials in n variables referred to as the symmetric polynomials (and to a companion
class of polynomials referred to as alternating polynomials). The distinguishing feature of
the symmetric polynomials is invariance with respect to permutations of the argument list
(e.g. N(x, 2, 3) = x1 + x2 + x3 = W(x2, x1, X3)). Alternating polynomials reverse sign with
each transposition of variables.
Section C.2 below develops an expression for the value of the order n Vander-
monde determinant in Eq. C.2. The evaluation method employs the determinant form and
a polynomial remainder theorem due to Bezout. Section C.3 introduces formal definitions
of symmetric and alternating polynomials and a fundamental theorem which associates
them with Vandermonde determinants. Section C.4 generalizes the symmetric and alter-
nating polynomial notions to the form required by the discussion in Section 9.5.
C.2 Evaluation of Vandermonde Determinants
The value of the order n Vandermonde determinant can be deduced from its form
(Eq. C.2) and a polynomial remainder theorem due to Bezout. Let y(x) be an arbitrary
polynomial function in n variables (i.e. x = (x,, x2, x* n)) and let xa) be generated from x
by replacing x, with the value a. Then, the theorem states that if y(x) is divided by the
binomial (x a) the remainder is (x(a)) [MoSt64]. That is
(x)) = (xi a)(x)+ W(xi)