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Thread: Polynomial

  1. #1
    Senior Member Sampras's Avatar
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    Polynomial

    Suppose $\displaystyle p(x) = a_0 + a_{1}x + \cdots + a_{m}x^m $. Define $\displaystyle \Gamma(p(x)) = a_{0}^{2}+ a_{1}^{2} + \cdots + a_{m}^{2} $. Find a polynomial $\displaystyle g(x) $, such that $\displaystyle g(0) = 0 $ and $\displaystyle \Gamma(p(x)^{n}) = \Gamma(g(x)^{n}) $.

    This would be a guess and check process? $\displaystyle \Gamma(p(x)) $ gives the "distance-squared" from the m-tuple $\displaystyle (a_0, \dots, a_m) $ to $\displaystyle (0, \dots, 0) $.
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Suppose $\displaystyle p(x) = a_0 + a_{1}x + \cdots + a_{m}x^m $. Define $\displaystyle \Gamma(p(x)) = a_{0}^{2}+ a_{1}^{2} + \cdots + a_{m}^{2} $. Find a polynomial $\displaystyle g(x) $, such that $\displaystyle g(0) = 0 $ and $\displaystyle \Gamma(p(x)^{n}) = \Gamma(g(x)^{n}) $.

    This would be a guess and check process? $\displaystyle \Gamma(p(x)) $ gives the "distance-squared" from the m-tuple $\displaystyle (a_0, \dots, a_m) $ to $\displaystyle (0, \dots, 0) $.

    $\displaystyle g(x)=a_0x+a_1x^2+....+a_nx^{n+1}$

    Tonio
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  3. #3
    Senior Member Sampras's Avatar
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    Quote Originally Posted by tonio View Post
    $\displaystyle g(x)=a_0x+a_1x^2+....+a_nx^{n+1}$

    Tonio
    Should have been $\displaystyle g(0) = 1 $.
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