
Polynomial lemniscate

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 formula_1 This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve "ƒ"("x", "y") = "c"2 of degree 2"n", which results from expanding out formula_2 in terms of "z" = "x" + "iy".
When "p" is a polynomial of degree 1 then the resulting curve is simply a circle whose center is the zero of "p". When "p" is a polynomial of degree 2 then the curve is a Cassini oval.
Erdős lemniscate.
A conjecture of Erdős which has attracted considerable interest concerns the maximum length of a polynomial lemniscate "ƒ"("x", "y") = 1 of degree 2"n" when "p" is monic, which Erdős conjectured was attained when "p"("z") = z"n" − 1.
This is still not proved but Fryntov and Nazarov proved that "p" gives a
local maximum. In the case when "n" = 2, the Erdős lemniscate is the Lemniscate of Bernoulli
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".
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 "p"0 = "z", and "p""n" = "p""n"−12 + "z", then the corresponding polynomial lemniscates Mn defined by |"p""n"("z")| = 2 converge to the boundary of the Mandelbrot set.
The Mandelbrot curves are of degree 2n+1.