Generic Polynomial Lemniscate
In general, a polynomial lemniscate will not touch at the origin, and will have only two ordinary n-fold singularities, and hence a genus of (n − 1)2. As a real curve, it can have a number of disconnected components. Hence, it will not look like a lemniscate, making the name something of a misnomer.
An interesting example of such polynomial lemniscates are the Mandelbrot curves. If we set p0 = z, and pn = pn−12 + z, then the corresponding polynomial lemniscates Mn defined by |pn(z)| = ER converge to the boundary of the Mandelbrot set. If ER < 2 they are inside, if ER ≥ 2 they are outside of Mandelbrot set. The Mandelbrot curves are of degree 2n+1, with two 2n-fold ordinary multiple points, and a genus of (2n − 1)2.
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