Polynomial lemniscate

[math]\displaystyle{ |z^6+z^5+z^4+z^3+ }[/math] [math]\displaystyle{ z^2+z+1|=1 }[/math]

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 [math]\displaystyle{ |p(z)| = c. }[/math] This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ƒ(x, y) = c² of degree 2n, which results from expanding out [math]\displaystyle{ p(z) \bar p(\bar z) }[/math] 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

Erdős lemniscate of degree ten and genus six

A conjecture of Erdős which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ƒ(x, y) = 1 of degree 2n when p is monic, which Erdős conjectured was attained when p(z) = zⁿ − 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

[math]\displaystyle{ (x^2+y^2)^2=2(x^2-y^2)\, }[/math]

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 p₀ = z, and p_n = p_n−1² + z, then the corresponding polynomial lemniscates M_n defined by |p_n(z)| = 2 converge to the boundary of the Mandelbrot set. The Mandelbrot curves are of degree 2ⁿ⁺¹.

References

• Alexandre Eremenko and Walter Hayman, On the length of lemniscates, Michigan Math. J., (1999), 46, no. 2, 409–415 [1]
• O. S. Kusnetzova and V. G. Tkachev, Length functions of lemniscates, Manuscripta Math., (2003), 112, 519–538 [2]