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 $p(z)\; \backslash bar\; p(\backslash bar\; z)$ in terms of z = x + iy.

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, which Erdős conjectured was attained when p(z)=z^n-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)| = 1 converge to the boundry of the Mandelbrot set. The Mandelbrot curves are of degree 2ⁿ⁺¹, with two 2ⁿ-fold ordinary multiple points, and a genus of (2ⁿ-1)².}

References

• Alexandre Eremenko and Walter Hayman, On the length of lemniscates, Michigan Math. J., (1999), 46, no. 2, 409–415 ^*
• O. S. Kusnetzova and V. G. Tkachev, Length functions of lemniscates, Manuscripta Math., (2003), 112, 519-538 ^*