Polynomial lemniscate

| z 6 + z 5 + z 4 + z 3 + {\displaystyle |z^{6}+z^{5}+z^{4}+z^{3}+} z 2 + z + 1 | = 1 {\displaystyle z^{2}+z+1|=1}

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 . {\displaystyle |p(z)|=c.} 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 p ( z ) p ¯ ( z ¯ ) {\displaystyle p(z){\bar {p}}({\bar {z}})} 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.[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 ) {\displaystyle (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 pn = pn−12 + z, then the corresponding polynomial lemniscates Mn defined by |pn(z)| = 2 converge to the boundary of the Mandelbrot set.[2] The Mandelbrot curves are of degree 2n+1.[3]

Notes

  1. ^ Fryntov, A; Nazarov, F (2008). "New estimates for the length of the Erdos-Herzog-Piranian lemniscate". Linear and Complex Analysis. 226: 49–60. arXiv:0808.0717. Bibcode:2008arXiv0808.0717F.
  2. ^ Desmos.com - The Mandelbrot Curves
  3. ^ Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2007), High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction, Springer, p. 492, ISBN 9781402054563.

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]