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Symmetric polynomial algorithm?

  1. Mar 29, 2010 #1
    Let [tex]f, g \in \mathbb{Z}[x, y, z][/tex] be given as follows: [tex]f = x^8 + y^8 + z^6[/tex] and [tex]g = x^3 +y^3 + z^3[/tex]. Express if possible [tex]f[/tex] and [tex]g[/tex] as a polynomial in elementary symmetric polynomials in [tex]x, y, z[/tex].

    Professor claims there is an algorithm we were supposed to know for this question on the midterm. I missed it. Any ideas?
  2. jcsd
  3. Mar 30, 2010 #2
    g can be expressed as required using Newton's formula ; f is not even symmetric.
  4. Mar 31, 2010 #3
    By newton's formula,
    [tex]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[/tex]

    where the [tex]\sigma_i[/tex]'s are the elementary symmetric polynomials?

    just trying to verify that I did it right?

    and f is not of the form [tex]\sum^n_{i=1} x_i^k[/tex] for some [tex]k \in \mathbb{N}[/tex] 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?
  5. Apr 6, 2010 #4
    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.
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