A similar question is this, but in my case I want to solve for the values in the Vandermonde matrix. That is, solve for $x_1, ..., x_n \in \mathbb{R}$ in $$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ x_1 & x_2 & x_3 & ... & x_n \\ x_1^2 & x_2^2 & x_3^2 & ... & x_n^2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & ... & x_n^{n-1}\end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n\end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \\ \vdots \\ b_n \end{pmatrix} $$ where $a_1, ..., a_n \in \mathbb{R}$ and $b_1, ..., b_n \in \mathbb{R}$ are known. In terms of algebraic geometry, it seems that I have a fixed linear combination of points on the rational normal curve, and I want to solve for the points.
I want to understand when the matrix equation has a solution, and ideally how to find a solution. Really anything helps. The equation boils down to a system of polynomial equations, but I hope that the extra structure helps.