What can be said about the solutions of, resp. solving, the following system polynomial equations, in which the $x_i$ and $y_j$ are the variables and $c_{i,j},\,d_i\in\mathbb{R}$ are constants:
$$\begin{matrix} c_{0,1}x_1+\,\cdots+\,c_{0,n}x_n&=&d_0\\ c_{1,1}x_1y_1+\,\cdots+\,c_{1,n}x_ny_n&=&d_1\\ c_{2,1}x_1y_1^2+\,\cdots+\,c_{2,n}x_ny_n^2&=&d_2\\ \vdots \\ c_{i,1}x_1y_1^i+\,\cdots+\,c_{i,n}x_ny_n^i&=&d_i\\ \vdots\\ c_{m,1}x_1y_1^{m}+\,\dots+\,c_{m,n}x_ny_n^{m}&=&d_{m} \end{matrix} $$ knowing the conditions and algorithms for efficient numeric or symbolic calculation of the solutions would be the precondtion for an algorithm for non-polynomial splines as requested in this MO question Non-polynomial splines, a non-linear problem