# What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius transformations in both domain and range. For degree 1 and 2, there is only one equivalence class. For degree 3, there is a well-understood one-complex-parameter family, so the real challenge is for higher degrees.

Given a set of points to be the critical values [in the range], along with a covering space of the complement homeomorphic to a punctured sphere, the uniformization theorem says this Riemann surface can be parametrized by $S^2$, thereby defining a rational function. Is there a reasonable way to compute such a rational map?

Given a set of $2d-2$ points on $CP^1$ to be critical points [in the domain], it has been known since Schubert that there are Catalan(d) rational functions with those critical points. Is there a conceptual way to describe and identify them?

In the case that all critical points are real, Eremenko and Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry. Annals of Mathematics, v.155, p.105-129, 2002 gave a good description. They are determined by $f^{-1}(R)$, which is $R$ together with mirror-image subdivisions of the upper and lower half-plane by arcs. These correspond to the various standard things that are enumerated by Catalan numbers. Is there a global conceptual classification of this sort? And, is there a way to find a rational map with given critical points along with some kind of additional combinatorial data?

Note that for the case of polynomials, this is very trivial: the critical points are zeros of its derivative, so there is only one polynomial, which you get by integrating its the derivative.

Is there a complete characterization of the Schwarzian derivative for a rational map, starting with the generic case of $2d-2$ distinct critical points?

Cf. the recent question by Paul Siegel. The Schwarzian $q$ for a generic rational map has a double pole at each critical point. As a quadratic differential, it defines a metric $|q|$ on the sphere - critical points which is isometric to an infinitely long cylinder of circumference $\sqrt 6 \pi$ near each. Negative real trajectories of the quadratic differential go from pole to pole, defining a planar graph.

What planar graphs occur for Schwarzian derivatives of rational functions? What convex (or other) inequalities do they satisfy?

The map from the configuration space of $(2d-2)$ points together with branching data to the configuration space of $2d-2$ points, defined by mapping (configuration of critical values plus branched cover data) to (configuration of critical points) is a holomorphic map, which implies it is a contraction of the Teichmuller metric.

Is this map a contraction for other readily described metrics?

• The first question for three critical values seems to be the problem of "dessin d'enfant". A reasonable amount of calculation (some by hand, some by computer) has been done but the number of explicit examples is not large. – Torsten Ekedahl Sep 10 '10 at 17:02
• The first question (which, as TE says, is the problem of computing equations for dessins d'enfant when there are three branch points) appears to be hard. One person who's done a lot of work on it is Jean-Marc Couveignes; see math.univ-toulouse.fr/~couveig/publi/volk.pdf for a representative piece of work. – JSE Sep 12 '10 at 2:19
• Thanks TE and JSE for the pointer. I'm not yet convinced that the computation should be hard in principle, even if nobody has yet implemented an efficient process. The map from {rational functions/up to precomposition with Moebius} to {critical values, branching data} is biholomorphic, so at worst the inverse function theorem should be efficient once implemented, although annoying to implement because of the complexity of tracking the branch data, the braid group action, degeneracy relationships and orbifold singularities well enough to get good local coordinates, especially in the range. – Bill Thurston Sep 12 '10 at 18:13
• That link in JSE's comment is now math.u-bordeaux1.fr/~jcouveig/publi/volk.pdf – John R Ramsden Aug 4 '12 at 8:12
• It seems still to be there, but to avoid any further possibility of link rot, @JSE's link, as updated by @JohnRRamsden, is to Couveignes - Tools for the computation of families of coverings (MSN). – LSpice Apr 6 at 14:29

1. There is a characterization of Schwarzian derivatives of rational maps: section 3 in the text: http://www.math.purdue.edu/~eremenko/dvi/schwarz3.pdf There is something similar also in arXiv:math/0512370, chapter 2. All these descriptions are various systems of algebraic equations. One of them, the "Bethe ansatz equations for the Gaudin model", proved to be very useful, see Mukhin, Tarasov and Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. Math. 170, 2 2009, 863-15.

2. There is some cell decomposition of the sphere which can be intrinsically related to a ratonal function. It is described in the paper Bonk, Eremenko, Schlicht regions of entire and meromorphic functions, J. d'Analyse, 77, 1999, 69-104, Sections 7.8. For a given cell decomposition, a rational function can be recovered using an algorithm similar to Thurston's circle packing algorithm. However, with this description, critical points or critical valued cannot be prescribed, and the cell decomposition does not determine the rational function completely.

EDIT on 12.18.2015. Here is a recent preprint related to this question and to my answer: http://arxiv.org/abs/1511.04246. And another preprint of the same author: http://arxiv.org/abs/1502.04760. Let me mention that the contents of the second preprint was well known to German mathematicians in 1930-40, and for an exposition of this material in English one can look to this book MR2435270 Chapter VII, and references there.

The way that I view the shape of rational functions (and especially polynomials) is in terms of the configuration of their critical level curves. Assume throughout that $r$ is a rational function with a simple pole at $\infty$ (which can always be achieved by pre and post composing with Moebius functions). First lets classify the individual level curves of $r$.

THEOREM 1 Let $\Lambda$ be a level curve of $r$ (ie. a component of the set $\{z:|r(z)|=\epsilon\}$ for some $0<\epsilon<\infty$). There are two possibilities.

• If $\Lambda$ contains no critical points of $r$, then $\Lambda$ is a smooth Jordan curve in $\mathbb{C}$.

• If $\Lambda$ contains critical points of $r$, then $\Lambda$ is a piecewise smooth connected finite graph having vertices at the critical points, which forms a "complicated figure eight". That is, it satisfies the following properties:

1. There are evenly many (and more than two) edges of $\Lambda$ incident to each vertex of $\Lambda$.

2. Each edge of $\Lambda$ is incident to a bounded face of $\Lambda$ (if $r$ is a polynomial, now allowed to have any number of poles at $\infty$, there is the additional constraint that each edge is also incident to the unbounded face of $\Lambda$).

Call level curves of the second kind (ie. one containing critial points) critical level curves. The two properties together imply that a critical level curve $\Lambda$ of $r$ is a sort of figure eight graph, which can be formed from a figure eight by iteratively joining circles to the graph at single points (with the circle in either a bounded face or an unbounded face).

Let us define a figure eight type graph to be one which has the properties described in Theorem 1, and a polynomial figure eight type graph to be one with the additional property described in Item 2 of Theorem 1.

We will now classify the way in which the critical level curves can lie amongst each other.

THEOREM 2 Let $\Lambda$ be a critical level curve (in fact any level curve) of $r$, and let $D$ denote one of its bounded faces. One of the following obtain.

1. $r$ has a single distinct zero or pole in $D$.

2. There is some critical level curve $\Lambda_D$ which is "maximal" in $D$. That is, maximal in the sense all zeros and poles of $r$ in $D$ are contained in the bounded faces of $\Lambda_D$, and every critical point of $r$ in $D$ is either in $\Lambda_D$, or contained in a bounded face of $\Lambda_D$.

The picture that arises from Theorem 2 is that, if $A$ is a set containing all the critical level curves of $r$, along with the zeros and poles of $r$, then every component of $A^c$ in $\mathbb{C}$ is a topological annulus.

More can be said in fact.

THEOREM 3 If $D$ is a component of $A^c$. Then the following holds.

1. $D$ is topologically an annulus.

Let $D_-$ denote the bounded face of $D^c$. Let $Z$ and $P$ denote the number of zeros and poles of $r$ in $D_-$ respectively.

1. $r^{\frac{1}{Z-P}}$ is a conformal map from $D$ to an annulus centered at the origin.

We could rephrase Item 2. as:

1. There is conformal map $\phi$ from $D$ to an annulus centered at the origin such that $r=\phi^{Z-P}$ on $D$.

We see now a clear idea of the shape of the function $r$ in terms of its critical level curves. The critical level curves form a sort of skeleton, either one zero or pole, or one maximal critical level curve in each bounded face of any given level curve. In between the critical level curves there is a smooth sheet of the function, conformally just a pure power.

Since the rational function is so simple between its critical level curves, it should come as no surprise that these critical level curves determine the rational function. That is, the geometric "skeleton" of the critical level curves is a strong conformal invariant of the rational function.

If some additional data is appended, (like the arguments of $r$ at the vertices, the net change in $\arg(r)$ along each edge of the graph, and the magnitude of $|r|$ on each graph), then the configuration of the critical level curves of a rational function is a strong conformal invariant.

THEOREM 4 If any two rational functions $r_1$ and $r_2$ have the same configuration of critical level curves (up to orientation preserving homeomorphism), then they are conformally equivalent. That is, there is a Moebius map $M$ such that $$r_2=r_1\circ M$$ on $\overline{\mathbb{C}}$.

Finally, a natural question is to ask: Which such pictures arise from rational functions? The answer is completely known for polynomials, and the answer is all of them.

THEOREM 5 Every configuration of finitely many polynomial figure eight type graphs arranged according to the conclusion of Theorem 2 is the critical level curve configuration for some polynomial.

The corresponding fact for rational functions (ie. all configurations correspond to some rational function, where the additional restriction from Theorem 1 Item 2 is dropped) seems to be almost certainly true, but at the moment defies proof.

Unfortunately I do not know of any good methods for computing the critical level curve configuration for a given rational function (other than approximating it with ContourPlot in Mathematica), or to find a rational function with a given critical level curve configuration.

All the above may be found in detail in papers 2 and 3 on the Arxiv here, or on my website here. Background on level curve of analytic functions may be found in paper 4 on the Arxiv page linked to above.