The boundry of the Mandelbrot set as a nonstandard algebraic curve

Suppose we set F_0 = z, and F_n = F_(n-1)^2 + z. Now let fa be the
result of substituting x+iy for z in F_n, and fb x-iy. Let G_n = fa*fb,
expanded out. Then we may define the nth Mandelbrot
curve M_n as the real plane algebraic curve of degree 2^(n+1) given by
G_n = 1.

The Mandelbrot curve has two ordinary 2^n-fold singularities, giving it
a genus of (2^n-1)^2. These are at infinity, but we can move them; if
we subsitute y=iy into M_n, and convert to projective form by
subsituting x=x/z, y=y/z, and then setting y=1, we get a curve in x and
z,
L_n, which has two ordinary 2^n-fold singularities at [-1,0] and [1,0].

If now we consider the Mandelbrot curve in a hyperreal field, we can
have a curve M_N, for N an infinite hyperreal integer. If we take the
standard parts of the real points of M_N, we get exactly the Mandelbrot
set.

This nonstandard algebraic curve has two ordinary 2^N-fold-point
singularities. What I'm wondering is what does that mean in terms of
the actual Mandelbrot set?

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