Symmetric polynomial algorithm?

  • Thread starter math_grl
  • Start date
  • #1
49
0

Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
336
0
g can be expressed as required using Newton's formula ; f is not even symmetric.
 
  • #3
49
0
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?
 
  • #4
336
0
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?
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.
 

Related Threads for: Symmetric polynomial algorithm?

Replies
2
Views
2K
Replies
1
Views
1K
Replies
1
Views
2K
  • Last Post
Replies
14
Views
7K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
13
Views
4K
  • Last Post
Replies
1
Views
3K
Replies
2
Views
14K
Top