Page 289 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Alternatively,  (x ,..., x ) can be defined as the coefficient of x in the generating function
n-j
j 1 n
 (x x ). ...(13)

i
1 i n


For example, on four variables x , ..., x , the elementary symmetric polynomials are
1
4
 (x , x , x , x ) = x + x + x + x 4 ...(14)
2
1
3
1
4
1
2
3
 (x , x , x , x ) = x x + x x + x x + x x + x x + x x 4 ...(15)
2
1
2
4
1
2
1
3
4
2
3
1
3
4
2
3
 (x , x , x , x ) = x x x + x x x + x x x + x x x 4 ...(16)
3
1
2
4
4
3
2
1
4
1
2
3
2
1
3
3
 (x , x , x , x ) = x x x x ...(17)
4 1 2 3 4 1 2 3 4
The power sum S (x ,..., x ) is defined by
p 1 n
n
p
S (x , ..., x ) =  x . ...(18)
n
k
p
1
k 1

The relationship between * and  ,..., is given by the so-called Newton-Girard formulas. The
p
1
related function s ( , ...,  ) with arguments given by the elementary symmetric polynomials
1
p
n
(not x ) is defined by
n
s ( ,..., ) = (–1) S (x ,...,x ) ...(19)
p–1
p
p
n
1
1
n
n
p

= ( 1) p 1  x . ...(20)
k
k 1

It turns out that s ( , ..., ) is given by the coefficients of the generating function
n
1
p
 s
ln (1 +  t +  t +  t + ...) =  k t k ...(21)
2
3
1
2
k 1 k
3

1 1
2
3
3
2
t
=   2 (  2 2 )t  3 (  3   3 3 )t  ...
1
1
2
1
1
so the first few values are
s =  1 ...(22)
1
2
s =   2 2 ...(23)
1
2
s =  3 1 – 3   3 3 ...(24)
1
2
3
4
2
2
s =   4   2  4   4 4 . ...(25)
1
3
1
2
1
2
4
In general, s can be computed from the determinant
p
 1 1 0 0  0
2 2  1 1 0  0

s = ( 1) p 1 3 3 4  2 3  1 2  1 1  0 0 ...(26)


p

4

     1
p p  p 1  p 2  p 3   1



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