Polynomial lemniscate

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

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.

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.

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. 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.