Polynomial Lemniscate

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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 <math>|p(z)| = c.</math> This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve *ƒ*(*x*, *y*) = *c*<sup>2</sup> of degree 2*n*, which results from expanding out <math>p(z) bar p(bar z)</math> 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<sup>*n*</sup> − 1. In the case when *n* = 2, the Erdős lemniscate is the Lemniscate of Bernoulli

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...

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For any such polynomial

When

- <math>(x^2+y^2)^2=2(x^2-y^2),</math>

and it has been proven that this is indeed the maximal length in degree four. The Erdős lemniscate has three ordinary

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