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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.

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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.

alt text alt text

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  • $\begingroup$ Bill, I have to ask, given that it's Gauss: which algorithm? $\endgroup$ – J. M. is a poor mathematician Dec 12 '10 at 16:49
  • $\begingroup$ @J.M. Recall my prior post on Gauss's algorithm, which has a reference. Alas, it appears nobody read it since it got no votes. $\endgroup$ – Bill Dubuque Dec 12 '10 at 17:04
  • $\begingroup$ Thanks for the refresh; I would think I ran out of upvotes that day I commented on it though... $\endgroup$ – J. M. is a poor mathematician Dec 12 '10 at 17:09
  • $\begingroup$ @J.M. Ah, I only just now noticed that you too commented on the prior post. What a coincidence. $\endgroup$ – Bill Dubuque Dec 12 '10 at 17:17
  • $\begingroup$ The first paragraph is interesting; motivating towards some work of Gauss still in modern algebra. $\endgroup$ – p Groups Jul 6 '16 at 4:40
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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.

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    $\begingroup$ @Timothy: do you mind breaking the formula into two displayed lines? It is running out of the displayport on my small netbook. $\endgroup$ – Willie Wong Dec 12 '10 at 14:15
  • $\begingroup$ @Willie: Yeah I just did that after I saw how horrible it looked :) $\endgroup$ – Timothy Wagner Dec 12 '10 at 14:15
  • $\begingroup$ @Timothy: The OP's remarks make it clear that he is attempting to understand the classic Gauss proof using lex-order. But the above approach is of no help in this regard. $\endgroup$ – Bill Dubuque Dec 12 '10 at 16:52
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    $\begingroup$ @Bill Dubuque: I am not aware of Gauss' proof about this. I interpreted OP's statements to mean that the said proof was a possible approach and not necessarily the approach the OP had to use. $\endgroup$ – Timothy Wagner Dec 12 '10 at 16:54
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    $\begingroup$ @Bill: I never claimed it does. The OP's first statement says that he wants to write the given polynomial in terms of elementary symmetric functions. That is the question I answered. He explained his approach. I never claimed mine was the same approach. The OP did not claim that he wants to solve this problem using Gauss' proof. The OP said he tried solving this problem using Gauss' proof. $\endgroup$ – Timothy Wagner Dec 12 '10 at 17:41

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