I want to write $P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$ in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the fundamental theorem of symmetric polynomials using the Newton identities.
First I pick out the 'biggest' monomial according to the lexicographical ordering: $yz^{3}$. Now I want to rewrite this as a polynomial in the elementary symmetric polynomials. I don't quite understand how to do this.
On
Edit: As per Bill's comment I would like to clarify that this is not related to Gauss' proof.
$$P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$$ $$=x^3(y+z+x-x)+y^3(x+z+y-y)+z^3(x+y+z-z)$$ $$=x^3(x+y+z)+y^3(x+y+z)+z^3(x+y+z)-x^4-y^4-z^4$$ $$=(x+y+z)(x^3+y^3+z^3)-(x^4+y^4+z^4)$$
Now you can use identities for power sums.
On
You stated:
I want to write $P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$ in terms of elementary symmetric polynomials,
The polynomial $P$ is homogeneous of degree $4$ and highest degree $3$. Therefore, using partitions of $4$ it can be written as a linear combination of the elementary symmetric functions $$e_1:=x+y+z,\; e_2:=xy+xz+yz,\; e_3:=xyz$$ of $x,y,z$ as $$ P(x,y,z) = S := a\,e_3e_1 +b\,e_2^2 +c\,e_2e_1^2 +d\,e_1^4. $$ Now $d=0$ since the highest degree of $P$ is $3$. Now, the expansion of $S$ has highest degree term of $x$ as $c\,x^3(y+z).$ Comparing this to $P$ implies $c=1.$ But now, the expansion of $S$ contains $(b+2)x^2z^2$ which implies $b=-2.$ Now the expansion of $S$ contains $(a+1)x^2yz$ which implies $a=-1.$ The final result is the equation $$ P(x,y,z) = -e_3e_1 -2\,e_2^2 +e_2e_1^2. $$
By Gauss's algorithm below, if $\rm\ z^a\,y^b\, x^c\ $ is the highest w.r.t. lex order $\rm\ z > y > x\ $ then we subtract $\rm\ s_1^{a-b}\ s_2^{b-c}\ s_3^c\,$ to decrease its order. In our example $\rm\, z^3\, y\, $ is highest, therefore we subtract $\rm\ s_1^{3-1}\ s_2^{1-0}\ s_3^0 = (x+y+z)^2\ (xy+yz+zx)\,$ from $\rm\,P.\,$ The result is smaller in lex-order, hence iterating this reduction rewrites $\rm\:P\:$ as a polynomial in elementary symmetric polynomials $\rm\,s_i.\,$ Here the algorithm terminates in two more steps - see below. See here for another worked example.
As I mentioned in a prior post, Gauss's algorithm is the earliest known example of using lex-order reduction as in the Grobner basis algorithm. For a nice exposition see Chapter 7 of Cox, Little, O'Shea: Ideals, Varieties and Algorithms. They also give generalizations to the ring of invariants of a finite matrix group $\rm G \subset GL(n,k)$. Below is an excerpt which, coincidentally, presents this example. You might find it helpful to first read the example at the end before reading the proof.