Results 1 to 3 of 3

Math Help - Elementary symmetric polynomials

  1. #1
    Newbie
    Joined
    Aug 2008
    Posts
    6

    Elementary symmetric polynomials

    Let elementary symmetric polynomials (Vieta's formula)
    \Pi_k(x_1,x_2,\ldots x_n) be integers for every k=\overline{1,n}. Is the polynomial
    \Pi_2(x_1x_2,x_1x_3\ldots ,x_ix_j,\ldots x_{n-1}x_n) also integer, where 1\leq i<j\leq n?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by rodeo View Post
    Let elementary symmetric polynomials (Vieta's formula)
    \Pi_k(x_1,x_2,\ldots x_n) be integers for every k=\overline{1,n}. Is the polynomial
    \Pi_2(x_1x_2,x_1x_3\ldots ,x_ix_j,\ldots x_{n-1}x_n) also integer, where 1\leq i<j\leq n?
    In terms of the elementary symmetric polynomials E_k = \Pi_k(x_1,x_2,\ldots x_n), it looks as though \Pi_2(x_1x_2,x_1x_3,\ldots ,x_ix_j,\ldots x_{n-1}x_n) = E_1E_3 - E_4. So it will be an integer.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2008
    Posts
    6
    I think that I prove this by an easy induction.
    What about \Pi_k(x_1x_2,x_1x_3,\ldots,x_ix_j,\ldots,x_{n-1}x_n<br />
) for any 2 \leq k \leq n(n-1)/2?

    Thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Elementary Functions and Polynomials
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 5th 2011, 02:59 AM
  2. Symmetric Polynomials
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 11th 2010, 05:22 AM
  3. Symmetric Polynomials
    Posted in the Calculus Forum
    Replies: 0
    Last Post: January 3rd 2009, 04:41 AM
  4. Help with two results on symmetric polynomials
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 25th 2008, 12:22 PM
  5. Fundamental Theorem of Symmetric Polynomials
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 2nd 2008, 01:23 PM

Search Tags


/mathhelpforum @mathhelpforum