Abstract: TBA

Abstract: We explore a family of polynomials whose roots are related to the Mandelbrot set. The roots correspond to the $k$-periodic points of the iteration defining the Mandelbrot set. The Mandelbrot polynomials are defined by $p_0(\zeta)=0$ and $p_{j+1}(\zeta)=\zeta p^2_{j}(\zeta)+1$. These polynomials give rise to a novel family of recursively constructed zero-one matrices whose eigenvalues are the roots of $p_k(\zeta)$. The LU decomposition of the resolvent of these matrices is highly structured, and one linear solve can be done in $O(n)$ operations. Krylov based eigenvalue solvers can then be used to compute the eigenvalues of these matrices in an efficient manner.