proof of Waring’s formula
The following is a proof of the Waring’s formula using formal power series. We will work with formal power series in indeterminate z with coefficients in the ring ℚ[x1,…,xn]. We also need the following equality
-log(1-z)=∞∑j=1zjj. |
Taking log on both sides of
1-σ1z+…+(-1)nσnzn=n∏m=1(1-xmz), |
we get
log(1-σ1z+…+(-1)nσnzn)=n∑m=1log(1-xmz), | (1) |
Waring’s formula will follow by comparing the coefficients on both sides.
The right hand side of the above equation equals
n∑m=1∞∑j=1xjmjzj |
or
∞∑j=1(n∑m=1xjm)zjj |
The coefficient of zk is equal to Sk/k.
On the other hand, the left hand side of (1) can be written as
∞∑j=11j(σ1z-σ2z2+…+(-1)n-1σnzn)j. |
For each j, the coefficient of zk in
(σ1z-σ2z2+…+(-1)n-1σnzn)j |
is
∑i1,…,in(-1)i2+i4+i6+…j!i1!⋯in!σi11⋯σinn, |
where the summation is extended over all n-tuple (i1,…,in) whose entries are non-negative integers, such that
i1+i2+…+in=j | ||
i1+2i2+…+nin=k. |
So the coefficient of zk in the left hand side of (1) is
∞∑j=1∑i1,…,in(-1)i2+i4+i6+…(j-1)!i1!⋯in!σi11⋯σinn, |
or
∑(-1)i2+i4+i6+…(i1+…+in-1)!i1!⋯in!σi11⋯σinn. |
The last summation is over all (i1,…,in)∈ℤn with non-negative entries such that i1+2i2+…+nin=k.
Title | proof of Waring’s formula |
---|---|
Canonical name | ProofOfWaringsFormula |
Date of creation | 2013-03-22 15:34:29 |
Last modified on | 2013-03-22 15:34:29 |
Owner | kshum (5987) |
Last modified by | kshum (5987) |
Numerical id | 7 |
Author | kshum (5987) |
Entry type | Proof |
Classification | msc 11C08 |