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Polynomial lemniscate

{{Short description|Plane algebraic curve}}

{{more footnotes|date=December 2010}}

[[Image:Cyc7.png|thumb| |z^6+z^5+z^4+z^3+

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. 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) \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

Image:Erdos5.png

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.

{{cite journal|

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 pn = pn−12 + z, then the corresponding polynomial lemniscates Mn defined by |pn(z)| = 2 converge to the boundary of the Mandelbrot set.[https://www.desmos.com/calculator/coamqcajzq Desmos.com - The Mandelbrot Curves]

The Mandelbrot curves are of degree 2n+1.{{citation|title=High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction|publisher=Springer|year=2007|isbn=9781402054563|page=492|url=https://books.google.com/books?id=mbtCAAAAQBAJ&pg=PA492|first1=Vladimir G.|last1=Ivancevic|first2=Tijana T.|last2=Ivancevic}}.

Notes

{{Reflist}}

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]

{{DEFAULTSORT:Polynomial Lemniscate}}

Category:Plane curves

Category:Algebraic curves