Polynomial lemniscate
From Wikipedia, the free encyclopedia
In mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients of degree n.
For any such polynomial p and positive real number c, we may define a set of complex numbers by | p(z) | = c. This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve f(x,y)=c2 of degree 2n, which results from expanding out 
in terms of z = x + iy.
[edit] Erdős lemniscate
A conjecture of Erdős which has attracted considerable interest concerns the maximum length of a polynomial lemniscate f(x,y)=1 of degree 2n, which Erdős conjectured was attained when p(z)=zn-1. In the case when n=2, the Erdős lemniscate is the Lemniscate of Bernoulli
- (x2 + y2)2 = 2(x2 − y2)
and it has been proven that this is indeed the maximal length in degree four. The Erdős lemniscate has three ordinary n-fold points, one of which is at the origin, and a genus of (n-1)(n-2)/2. By inverting the Erdős lemniscate in the unit circle, one obtains a nonsingular curve of degree n.
[edit] 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.[citation needed]
