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 $|p(z)|\; =\; c.$ 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.

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 f(x,y)=1 of degree 2n when p is monic, which Erdős conjectured was attained when p(z)=zⁿ-1. In the case when n=2, the Erdős lemniscate is the Lemniscate of Bernoulli

:$(x^2+y^2)^2=2(x^2-y^2)$

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 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.Fact|date=June 2007}

References

*Alexandre Eremenko and Walter Hayman, "On the length of lemniscates", Michigan Math. J., (1999), 46, no. 2, 409–415 [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1030132418]
*O. S. Kusnetzova and V. G. Tkachev, "Length functions of lemniscates", Manuscripta Math., (2003), 112, 519-538 [http://arxiv.org/abs/math.CV/0306327]
* [http://www.mathcurve.com/courbes2d/cassinienne/cassinienne.shtml "Cassinian curve" at Encyclopédie des Formes Mathématiques Remarquables]