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) $.