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 ${\displaystyle |p(z)|=c.}$ 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 ${\displaystyle p(z){\bar {p}}({\bar {z}})}$ 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.

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. In the case when n = 2, the Erdős lemniscate is the Lemniscate of Bernoulli

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