One can rehomogenize your problem to deal with the polynomial $f(x,y,z)=z^{m+n}F(x/z,y/z)$.
Then what you want is the discriminant of the degree $m+n$ homogeneous polynomial $f$.
This is the same as the resultant of $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$. There is a complicated determinant formula for that due to Sylvester and Morley, see this paper.
There are quite a few monomials missing in your $F$ so there might be better formulas based on $A$-discriminants and resultants. For that I would look up the book by Gelfand, Kapranov and Zelevinsky.
Update: I just looked up my copy of "Using Algebraic Geometry" by Cox Little and O'Shea.
Exercise 15 of Ch. 3, Sec. 4 explains Sylvester's construction in a more pedagogical way.
Ch. 7 should help with the degree determination, although the absence of $c$ in front of the leading monomial $X^m Y^n$ could complicate things.