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 ''ƒ''(''x'', ''y'') = ''c''² of degree 2''n'', 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

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

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

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 ''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..

Notes

References

*Alexandre Eremenko and Walter Hayman, ''On the length of lemniscates'', Michigan Math. J., (1999), 46, no. 2, 409–41

*O. S. Kusnetzova and V. G. Tkachev, ''Length functions of lemniscates'', Manuscripta Math., (2003), 112, 519–53


{{DEFAULTSORT:Polynomial Lemniscate
Plane curves
Algebraic curves