1. ## Polynomial

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

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

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

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

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

Tonio

3. Originally Posted by tonio
$g(x)=a_0x+a_1x^2+....+a_nx^{n+1}$

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