How Can f and g Be Expressed Using Elementary Symmetric Polynomials?

[math_grl]

Let $$f, g \in \mathbb{Z}[x, y, z]$$ be given as follows: $$f = x^8 + y^8 + z^6$$ and $$g = x^3 +y^3 + z^3$$. Express if possible $$f$$ and $$g$$ as a polynomial in elementary symmetric polynomials in $$x, y, z$$.

Professor claims there is an algorithm we were supposed to know for this question on the midterm. I missed it. Any ideas?

 

[Eynstone]

g can be expressed as required using Newton's formula ; f is not even symmetric.

 

[math_grl]

By Newton's formula,
$$g = (\sigma_1^2 - 2\sigma_2)\sigma_1 - \sigma_1 \sigma_2 + 3\sigma_3 = \sigma_1^2 - 3\sigma_1 \sigma_2 + 3\sigma_3$$

where the $$\sigma_i$$'s are the elementary symmetric polynomials?

just trying to verify that I did it right?

and f is not of the form $$\sum^n_{i=1} x_i^k$$ for some $$k \in \mathbb{N}$$ so we can't use Newton's formula...but I was wondering one would know precisely that it's not possible to express it in terms of the e.s.p.'s?

 

[Eynstone]

and f is not of the form $$\sum^n_{i=1} x_i^k$$ for some $$k \in \mathbb{N}$$ so we can't use Newton's formula...but I was wondering one would know precisely that it's not possible to express it in terms of the e.s.p.'s?

Suppose that f = P(s1,s2,...) is expressible in terms of the e.s.p.'s. P won't change on switching y & z ; f will. A contradiction.