Symmetric polynomials and the Newton identities

By Gauss's algorithm, if $\rm\ z^a\ y^b\ x^c\ $ is the highest w.r.t. lex order $\rm\ z > y > x\ $ then you subtract $\rm\ s_1^{a-b}\ s_2^{b-c}\ s_3^c\:.\:$ Thus since $\rm\ z^3\ y $ is highest you 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, so iterating this reduction yields a representation of $\rm\:P\:$ in terms of elementary symmetric polynomials $\rm\:s_i\:.\:$ Here the algorithm terminates in two more steps.

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 $\rm G \subset GL(n,k)$. Here's an excerpt which, coincidentally, presents this example. You might find it helpful to first read the example at the end before reading the proof.

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.