Catalan objects associated to a univariate polynomial - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:47:22Z http://mathoverflow.net/feeds/question/87548 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87548/catalan-objects-associated-to-a-univariate-polynomial Catalan objects associated to a univariate polynomial Hugh Thomas 2012-02-04T21:19:31Z 2012-02-05T00:15:49Z <p>Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta &lt; \pi$, there are natural constructions which associate to this data:</p> <p>a noncrossing matching on $2n$ vertices, and</p> <p>a triangulation of an $n+2$-gon. </p> <p>These objects are both counted by Catalan numbers, leading me to feel that there should be some connection. It is <i>not</i> the case that the fibres of the two maps are the same, and I will say more below about how they are clearly different. My question is: what is the connection? </p> <p>Let me now explain the constructions. Consider the points $z$ in $\mathbb C$ such that arg($\pm f(z)$)$=\theta$. Generically, this consists of $n$ unbounded curves, which, for large $|z|$, are evenly-spaces spokes of a wheel (with $2n$ spokes), and therefore can be interpreted as giving rise to a noncrossing matching. See <a href="http://arxiv.org/abs/math/0511248" rel="nofollow">Martin, Savitt, Singer</a> and <a href="http://arxiv.org/abs/math/0606169" rel="nofollow">Savitt</a>. As you tune $\theta$, the matchings change in a nicely controlled way, as pairs of components meet and reconnect. Tuning from $\theta=0$ to $\theta=\pi$ results in $n-1$ of these reconnections, and the total effect is to rotate the diagram by one step. </p> <p>Now we consider how to produce the triangulation. I found out about this from a sequence of papers by Cecotti, Vafa, and a varying list of others, studying a connection between BPS states of 4d supersymmetric quantum field theories and cluster algebras. For the present topic see, in particular, Sections 4 and 5 of <a href="http://arxiv.org/abs/1112.3984" rel="nofollow">Cecotti, Córdova, Vafa</a>. I found background provided in Section 4 of <a href="http://arxiv.org/abs/math-ph/9811024" rel="nofollow">Mulase and Penkava</a> helpful. The idea is to define a foliation of $\mathbb C$, where a curve $p(t)$ lies on a leaf of the foliation if $(p'(t))^2f(p(t))e^{2\theta i}$ is always real and positive. From a zero of $f$, there will be three equally-spaced curves coming out, and generically they will connect to points at infinity. For $|z|$ very large, the trajectories approach spokes of a wheel, with $n+2$ spokes. By connecting into a triangle the endpoints (on the circle at infinity) of the three curves coming out from each zero of $f$, you get a triangulation of an $n+2$-gon. As you tune $\theta$, the triangulation changes by diagonal flips. Again, the total effect of tuning from 0 to $\pi$ is to rotate the diagram by one step (though note that the "step" is a different fraction of $2\pi$ from the step that appears in the noncrossing matching). </p> <p>(Edited to add: I should mention that, in the terminology of the people who study these flows, $f(z)dz$ is called a "quadratic differential", though these are often considered in more complicated situations, where there might be poles, for instance.)</p> <p>If you prefer to complicate matters rather than simplifying them, you could also consider the question of whether there are any other Catalan objects associated to a monic polynomial and a phase (with different fibres than the above maps).</p>