1.iii. Maximally Inflected Curves

    A maximally inflected (plane) curve is a real map f : P¹ --> P² all of whose ramification occurs at real points in RP¹. A priori, there is no guarantee that if a map has its ramification at real points that map must be real. However, there is a conjecture of Boris Shapiro and Michael Shapiro that (when translated into this setting) declares that every map with only real ramification is real. (See [So00b] for an elaboration on thier conjecture.)

Conjecture. A rational plane curve f : P¹ --> P² whose only ramification occurs at real points must be real. Specifically, we have
    Given ramification indices A₁:=(0,b₁,c₁), A₂:=(0,b₂,c₂), ..., A_n:=(0,b_n,c_n) for degree d rational curves f : P¹ --> P² with |A₁|+|A₂|+...+|A_n| = 3d-6, and any real points s₁, s₂, ..., s_n, then every rational curve f : P¹ --> P² of degree d that has ramification A_i at the point s_i, for i=1,2,...,n, is in fact real.

    Among the evidence for this is a result of Eremenko and Gabrielov ([EG]) proving the full conjectre of Shapiro and Shapiro for rational maps f : P¹ --> P¹, which also imples the case of quartics curves in the plane. Another important piece of evidence is the following result

Theorem. ([So99])     Given ramification indices A₁:=(0,b₁,c₁), A₂:=(0,b₂,c₂), ..., A_n:=(0,b_n,c_n) for degree d rational curves f : P¹ --> P² with |A₁|+|A₂|+...+|A_n| = 3d-6, then there exist real points s₁, s₂, ..., s_n, then every rational curve f : P¹ --> P² of degree d that has ramification A_i at the point s_i, for i=1,2,...,n, is in fact real.

    The remainder of this web page will concern itself with pictures of maximally inflected curves and some constructions.