Factoring Gleason polynomials modulo 2

Abstract

Among the connected components of the interior of the Mandelbrot set are those that are hyperbolic. These components consist of parameters c∈ℂ for which the critical point z 0 =0 of f c :z↦z 2 +c is attracted to an attracting periodic cycle. Every hyperbolic component contains a unique center; that is, a parameter c for which the critical point z 0 is periodic. For a given n≥1, the Gleason polynomial for period n is the monic polynomial G n ∈ℤ[c] whose roots are exactly the centers of the hyperbolic components of period n. It is unknown if G n factors over ℤ. In this article, we factor G n modulo 2. We prove the following remarkable fact: the number of irreducible factors of G n modulo 2 is equal to the number of real roots of G n .