Consider the following equation $$ \frac{e^{\sum_i \alpha_i x_i}}{\sum_i x_i}=\sum_k\frac{e^{\sum_i \alpha_i y_{i, k}}}{\sum_i y_{i, k}} $$ Is it possible to write $x_i$ as a function of the terms $y_{i,k}$? If not, is there any way of simplifying this relation?
My approach: Rewriting, we have $$ e^{\sum_i \alpha_i x_i-\log\left( \sum_i x_{i}\right)}=\sum_ke^{\sum_i \alpha_i y_{i,k}-\log\left( \sum_i y_{i,k}\right)} $$ but it still seems quite untraceable to relate $x_i$ and $x_{i,k}$. If, in a simpler example, we had the relation $$ e^{\sum_i x_i}=\prod_k e^{\sum_i y_{i,k}} $$ this could be simplified to $$ e^{\sum_i x_i}=e^{\sum_i \sum_k y_{i,k}} $$ and thus the relation $x_i=\sum_k y_{i,k}$ is a potential relation (though not necessarily unique). Can we say something similar about the general case? Could Taylor help?
