Solvability of a system of polynomial equations

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

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.

• the linked paper seems to be the door-opener and appetizer for tackling the more general problem; thank you very much for unearthing the reference! Apr 19, 2020 at 16:38
• The link in this answer seems to be broken now... Apr 10, 2022 at 15:59
• this coding.yonsei.ac.kr/ramanujan_paper.pdf is an alternative link to Ramanujan's paper Aug 17, 2022 at 9:28