MATH 8300:19 NOV 09

PAUL HEWITT

Asymmetric function of n variables is one which satises the functional equations

(1) f(r

1

;:::;r

n

) = f(r

(1)

;:::;r

(n)

);

for every permutation of the indices f1;:::;ng.The elementary symmetric func-

tion of degree d in n variables is dened by the rule

(2) s

d

(r

1

;:::;r

n

) =

X

i

1

<<i

d

r

i

1

r

i

d

:

By convention,s

0

= 1.You are probably familiar with the following result.

Theorem 1.If

X

n

+a

n1

X

n1

+ +a

0

=

n

Y

j=1

(X r

j

)

then

a

k

= (1)

nk

s

nk

(r

1

;:::;r

n

):

Sir Isaac Newton proved the following.

Theorem2 (Newton).If f is a polynomial symmetric function of n variables then

there is a polynomial g such that

f(r

1

;:::;r

n

) = g(s

1

(r

1

;:::;r

n

);:::;s

n

(r

1

;:::;r

n

)):

Moreover,the coecients of g lie in the subring generated by the coecients of f.

Proof.(Sketch.) The proof is algorithmic:you can easily turn the proof into a

computer program.The algorithmwe describe can be applied to each homogeneous

component of f separately,owing to the fact that the s

i

are homogeneous.So we

assume that f is homogeneous of degree n.

We use a rather sophisticated induction to keep track of the progress of the

algorithm:we order the monomials according to the lexicographic ordering of the

tuples of their exponents.For example,r

1

r

3

2

r

4

> r

1

r

2

r

2

3

r

4

since the tuple (1;3;0;1)

appears after the tuple (1;1;2;1) in a standard dictionary.

Suppose f contains a monomial with exponents (e

1

;e

2

;:::).Since f is symmet-

ric it must also contains a monomial having the same coecient and exponents

(e

(1)

;e

(2)

;:::),for every permutation .Hence the maximal monomial of f (in

our lexicographic order) has the form cr

e

1

1

r

e

2

2

,where e

1

e

2

.

Now the maximal monomial of s

i

its exponents (1;:::;1;0;:::;0).Using a sort

of\gaussian elimination"on the tuples of exponents we nd that

f cs

e

n

n

s

e

n

e

n1

n1

s

e

n1

e

n2

n2

has a smaller maximal monomial than that of f.

1

2 PAUL HEWITT

Exercises.

(1) Prove Theorem 1 by induction on n.

(2) Fill in the details in the proof of Newton's Theorem.

(3) Write a computer program which implements the algorithm in the proof of

Newton's Theorem.

(4) Let

p

k

(r

1

;:::;r

n

) =

n

X

j=1

r

k

j

:

Prove Newton's identities:

k

X

j=0

(1)

j

p

j

s

kj

= 0:

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