What are the shapes of rational functions? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:18:32Z http://mathoverflow.net/feeds/question/38274 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38274/what-are-the-shapes-of-rational-functions What are the shapes of rational functions? Bill Thurston 2010-09-10T06:18:20Z 2013-01-14T17:19:02Z <p>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.</p> <blockquote> <p>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?</p> </blockquote> <p>I'm interested in ideas of good and bad ways to go about this. Computer code would also be most welcome.</p> <blockquote> <p>Given a set of $2d-2$ points on $CP^1$ to be critical <em>points</em> [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?</p> </blockquote> <p>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. <em>Annals of Mathematics</em>, 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?</p> <p>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.</p> <blockquote> <p>Is there a complete characterization of the Schwarzian derivative for a rational map, starting with the generic case of $2d-2$ distinct critical points?</p> </blockquote> <p>Cf. the recent <a href="http://mathoverflow.net/questions/38105/is-there-an-underlying-explanation-for-the-magical-powers-of-the-schwarzian-deriv/38125#38125" rel="nofollow">question</a> 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. </p> <blockquote> <p>What planar graphs occur for Schwarzian derivatives of rational functions? What convex (or other) inequalities do they satisfy?</p> </blockquote> <p>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.</p> <blockquote> <p>Is this map a contraction for other readily described metrics?</p> </blockquote> http://mathoverflow.net/questions/38274/what-are-the-shapes-of-rational-functions/103894#103894 Answer by Alexandre Eremenko for What are the shapes of rational functions? Alexandre Eremenko 2012-08-03T19:46:43Z 2012-08-03T20:24:55Z <ol> <li><p>There is a characterization of Schwarzian derivatives of rational maps: section 3 in the text: <a href="http://www.math.purdue.edu/~eremenko/dvi/schwarz.pdf" rel="nofollow">http://www.math.purdue.edu/~eremenko/dvi/schwarz.pdf</a> 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.</p></li> <li><p>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.</p></li> </ol> <p>Alex Eremenko.</p> http://mathoverflow.net/questions/38274/what-are-the-shapes-of-rational-functions/118905#118905 Answer by Karl Luttinger for What are the shapes of rational functions? Karl Luttinger 2013-01-14T17:19:02Z 2013-01-14T17:19:02Z <p>The algorithmic unsolvability of the general polynomial equation of degree greater than 4 implies that this is not possible in the general case. </p>