page_title: Polynomial lemniscate
section_title: Generic polynomial lemniscate
hierarchical_section_title: Polynomial lemniscate / Generic polynomial lemniscate
caption_attribution_description: English: Second Mandelbrot curve, of degree eight and genus nine.
context_page_description: 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 This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ƒ = c² of degree 2n, which results from expanding out 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.
context_section_description: 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 p₀ = z, and pₙ = pₙ−1² + z, then the corresponding polynomial lemniscates Mₙ defined by |pₙ(z)| = 2 converge to the boundary of the Mandelbrot set.
  The Mandelbrot curves are of degree 2ⁿ⁺¹.
caption: Mandelbrot curve M2 of degree eight and genus nine
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