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Thread: Elementary symmetric polynomials

  1. #1
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    Elementary symmetric polynomials

    Let elementary symmetric polynomials (Vieta's formula)
    $\displaystyle \Pi_k(x_1,x_2,\ldots x_n)$ be integers for every $\displaystyle k=\overline{1,n}$. Is the polynomial
    $\displaystyle \Pi_2(x_1x_2,x_1x_3\ldots ,x_ix_j,\ldots x_{n-1}x_n)$ also integer, where $\displaystyle 1\leq i<j\leq n$?
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  2. #2
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    Quote Originally Posted by rodeo View Post
    Let elementary symmetric polynomials (Vieta's formula)
    $\displaystyle \Pi_k(x_1,x_2,\ldots x_n)$ be integers for every $\displaystyle k=\overline{1,n}$. Is the polynomial
    $\displaystyle \Pi_2(x_1x_2,x_1x_3\ldots ,x_ix_j,\ldots x_{n-1}x_n)$ also integer, where $\displaystyle 1\leq i<j\leq n$?
    In terms of the elementary symmetric polynomials $\displaystyle E_k = \Pi_k(x_1,x_2,\ldots x_n)$, it looks as though $\displaystyle \Pi_2(x_1x_2,x_1x_3,\ldots ,x_ix_j,\ldots x_{n-1}x_n) = E_1E_3 - E_4$. So it will be an integer.
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  3. #3
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    I think that I prove this by an easy induction.
    What about $\displaystyle \Pi_k(x_1x_2,x_1x_3,\ldots,x_ix_j,\ldots,x_{n-1}x_n
    )$ for any $\displaystyle 2 \leq k \leq n(n-1)/2$?

    Thank you!
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