Finding a Polynominal
Let \begin{equation} x_1 + x_2 + \dots+ x_n = A \end{equation}
The value of $x_1$ to $x_n$ is not given. Suppose we have $n$ variables $y_1, y_2,\dots, y_n$. Is there any way to find $x_1y_1 + x_2y_2 + x_3y_3 + \dots+ x_ny_n$, based on the value of $A$ and $y_1$ to $y_n$?
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$\begingroup$In most cases no, the answer is yes iff $y_i=y_j\forall i,j\in\{1,2,\cdots,n\}$
Consider $x_1 + x_2 + \dots+ x_n = A$ and that the condition I stated are false.
In this case I can rewrite the $y_1x_1 +y_n x_2 + \dots+ y_nx_n=B$ it as $w_1u_1+w_{2}u_{2}+\cdots+w_\ell u_\ell=B$ where $1<\ell\le n$, $w_k=y_h$ for some $h$ and $u_k$ is combination of some values of $x_s$.
This is equation is the most reduce form, i.e. has the least amount of variables we can get, so unless I have only one variable I can't solve it, but one of my conditions is that not all of $y_i$ are equal, so I have at least 3 variables thus I can't solve this.
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