I have a quadratic equation in the form:
$a_{0}^{2}+a_{1}^{2}t^{2}+a_{2}^{2}t^{4}+...+a_{n}^{2}t^{2n}+2(a_0a_1t+a_0a_2t^2+...+a_0a_nt^n+...+a_na_{n-1}t^{2n-1})=\phi(c)$.
All $a_i$ are algebraically independent. I need to solve this equation at 2 different points for $t= \alpha, \beta $.
Now can I write this equation in terms of new variable $c_i$
$ \text {where $c_i $is the coefficient of }t^i$ as $c_i:=\sum_{j=0}^{i}a_ja_{i-j}$ and solve?
Thank you!