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 nonpolynomial splines as requested in this MO question Nonpolynomial splines, a nonlinear problem
1 Answer
For the special case where all $c_{i,j}$'s are equal to 1 and $m=2n1$, 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 $2n1$ 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, $\sum_{k=1}^n\frac{x_k}{1\theta y_k} = d_1 + d_2\theta + \cdots + d_{2n}\theta^{2n1} + \cdots = \frac{A_1 + A_2\theta + \cdots + A_{n1}\theta^{n1}}{B_1 + B_2\theta + \cdots + B_{n1}\theta^{n1}}$.
(3) Multiplying by $B_1 + B_2\theta + \cdots + B_{n1}\theta^{n1}$ 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.

1$\begingroup$ the linked paper seems to be the dooropener and appetizer for tackling the more general problem; thank you very much for unearthing the reference! $\endgroup$ Apr 19, 2020 at 16:38

$\begingroup$ The link in this answer seems to be broken now... $\endgroup$ Apr 10, 2022 at 15:59

$\begingroup$ this coding.yonsei.ac.kr/ramanujan_paper.pdf is an alternative link to Ramanujan's paper $\endgroup$ Aug 17, 2022 at 9:28