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](https://mathoverflow.net/questions/352683/non-polynomial-splines-a-non-linear-problem)



Solvability of a system of polynomial equations
For the special case where all $c\_{i,j}$'s are equal to 1 and $m=2n-1$, take a look at Ramanujam's paper: <http://ramanujan.sirinudi.org/Volumes/published/ram03.pdf>. Needless to say, it is an ingenious method. The steps are the following:


(1) The key idea is to recognize that the coefficients (w.r.t $\theta$, upto the $2n-1$ order) of $\sum\_{k=1}^n\frac{x\_k}{1-\theta y\_k}$ in the series expansion would be the LHS of the nonlinear equations. 


(2) Now, $\sum\_{k=1}^n\frac{x\_k}{1-\theta y\_k} = d\_1 + d\_2\theta + \cdots + d\_{2n}\theta^{2n-1} + \cdots = \frac{A\_1 + A\_2\theta + \cdots + A\_{n-1}\theta^{n-1}}{B\_1 + B\_2\theta + \cdots + B\_{n-1}\theta^{n-1}}$.


(3) Multiplying by $B\_1 + B\_2\theta + \cdots + B\_{n-1}\theta^{n-1}$ on either side and comparing coefficients, gives linear equations in $A\_i$'s and $B\_i$'s.


(4) If a solution exists, one can then determine partial fractions (in $\theta$), and the coefficients would be the answers. 


For your case, you might want to think on the same lines with Pade approximants.


All this is only a suggestion for the most general case. Hope it helps.


