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 $\mathbb{Q}[x_{1},\ldots,x_{n}]$. We also need the following equality

 $-\log(1-z)=\sum_{j=1}^{\infty}\frac{z^{j}}{j}.$

Taking log on both sides of

 $1-\sigma_{1}z+\ldots+(-1)^{n}\sigma_{n}z^{n}=\prod_{m=1}^{n}(1-x_{m}z),$

we get

 $\log(1-\sigma_{1}z+\ldots+(-1)^{n}\sigma_{n}z^{n})=\sum_{m=1}^{n}\log(1-x_{m}z),$ (1)

Waring’s formula will follow by comparing the coefficients on both sides.

The right hand side of the above equation equals

 $\sum_{m=1}^{n}\sum_{j=1}^{\infty}\frac{x_{m}^{j}}{j}z^{j}$

or

 $\sum_{j=1}^{\infty}\left(\sum_{m=1}^{n}x_{m}^{j}\right)\frac{z^{j}}{j}$

The coefficient of $z^{k}$ is equal to $S_{k}/k$.

On the other hand, the left hand side of (1) can be written as

 $\sum_{j=1}^{\infty}\frac{1}{j}(\sigma_{1}z-\sigma_{2}z^{2}+\ldots+(-1)^{n-1}% \sigma_{n}z^{n})^{j}.$

For each $j$, the coefficient of $z^{k}$ in

 $(\sigma_{1}z-\sigma_{2}z^{2}+\ldots+(-1)^{n-1}\sigma_{n}z^{n})^{j}$

is

 $\sum_{i_{1},\ldots,i_{n}}(-1)^{i_{2}+i_{4}+i_{6}+\ldots}\frac{j!}{i_{1}!\cdots i% _{n}!}\sigma_{1}^{i_{1}}\cdots\sigma_{n}^{i_{n}},$

where the summation is extended over all $n$-tuple $(i_{1},\ldots,i_{n})$ whose entries are non-negative integers, such that

 $\displaystyle i_{1}+i_{2}+\ldots+i_{n}=j$ $\displaystyle i_{1}+2i_{2}+\ldots+ni_{n}=k.$

So the coefficient of $z^{k}$ in the left hand side of (1) is

 $\sum_{j=1}^{\infty}\sum_{i_{1},\ldots,i_{n}}(-1)^{i_{2}+i_{4}+i_{6}+\ldots}% \frac{(j-1)!}{i_{1}!\cdots i_{n}!}\sigma_{1}^{i_{1}}\cdots\sigma_{n}^{i_{n}},$

or

 $\sum(-1)^{i_{2}+i_{4}+i_{6}+\ldots}\frac{(i_{1}+\ldots+i_{n}-1)!}{i_{1}!\cdots i% _{n}!}\sigma_{1}^{i_{1}}\cdots\sigma_{n}^{i_{n}}.$

The last summation is over all $(i_{1},\ldots,i_{n})\in\mathbb{Z}^{n}$ with non-negative entries such that $i_{1}+2i_{2}+\ldots+ni_{n}=k$.

Title proof of Waring’s formula ProofOfWaringsFormula 2013-03-22 15:34:29 2013-03-22 15:34:29 kshum (5987) kshum (5987) 7 kshum (5987) Proof msc 11C08