If I have a polynomial equation in one variable $w$:
$$\sum_{n=0}^N a_n(x,y) w^n = 0$$
where $a_n(x,y)$ are the coefficients parameterised by the real variables $x$ and $y$, I can obtain solutions $w(x,y)$ (which, if it simplifies things, for my purposes are real for all values of $x$ and $y$). How in general can I find the coordinates $(x,y)$ for which two (or more) of the solutions $w(x,y)$ are degenerate?
For example, in the picture below, one can see plots of three solutions to a polynomial equation with $w(x,y)$ on the vertical axis and $x$ and $y$ on the horizontal axes. There are points where two of the solutions are degenerate, and I would like to find exactly (not numerically) where these points are.
This is also a resource recommendation request if anybody has any suggestions where I can read more about this type of problem. Thanks very much for any help.