Polynomial lemniscate

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 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. first1=A| last1=Fryntov| first2=F| last2=Nazarov| title=New estimates for the length of the Erdos-Herzog-Piranian lemniscate| year=2008| journal=Linear and Complex Analysis| volume=226| pages=49–60| arxiv=0808.0717| bibcode=2008arXiv0808.0717F}} 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 p0 = 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.

Notes

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 https://arxiv.org/abs/math.CV/0306327

References

1. [https://www.desmos.com/calculator/coamqcajzq Desmos.com - The Mandelbrot Curves]
2. (2007). "High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction". Springer.