Newton's theorem on symmetric polynomials - The Live Toad

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Oct 13, 2013 (3 years and 8 months ago)

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MATH 8300:19 NOV 09
PAUL HEWITT
Asymmetric function of n variables is one which satises 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 dened 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
n1
X
n1
+   +a
0
=
n
Y
j=1
(X r
j
)
then
a
k
= (1)
nk
s
nk
(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 coecients of g lie in the subring generated by the coecients 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 coecient 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
n1
n1
s
e
n1
e
n2
n2
  
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
kj
= 0: