From: h720@cpca6.uea.ac.uk (T.Ward) Subject: Re: Random polynomial Date: 20 May 1999 10:07:23 +0100 Newsgroups: sci.math Keywords: real roots of a random polynomial In article <3742BCCE.3F3F@cclix1.polito.it> Enrico Talinucci writes: >a_0,..,a_n are independent random variables, uniformly distributed >between -1 and 1. What is the probability that all the roots of the >polynomial > > a0_ + a_1 X + ... + a_n X^n > >are real? They deal with a rather different question (coefficients are standard normals), but it might be worth looking at the survey paper "How many zeros of a random polynomial are real?" by Edelman and Kostlan, Bull. AMS 32 (1995), 1-37. If you do not have access to a library you can probably find the full text at the AMS web site http://www.ams.org/bull/ Tom Ward ============================================================================== From: greg@math.ucdavis.edu (Greg Kuperberg) Subject: This week in the xxx mathematics archive (24 May - 28 May) Date: 1 Jun 1999 22:30:46 -0700 Newsgroups: sci.math.research Here are this week's titles in the xxx mathematics archive, available at: http://front.math.ucdavis.edu/ Instructions for contributing articles are available at: http://front.math.ucdavis.edu/submissions.html A couple of weeks ago Pavel Bleher gave a talk at Davis presenting the result of his two recent archive articles, math-ph/9903012 and math-ph/9904020. These articles discuss a spectacular generalization of a classic question in probability theory: Where are the zeroes of a random polynomial in the large-degree limit? Of course this question begs a preliminary question: What's a random polynomial? One approach is to say that the coefficients are identical, independent random variables, perhaps with the flat distribution on [-1,1], or better yet Gaussian variables. A fancier approach is to demand symmetry. It is natural to interpret the zeroes as points on the projective line RP^1, and correspondingly to consider homogeneous polynomials in two variables instead of inhomogeneous polynomials in one variable. One can consider the "round" model of RP^1, i.e., endowing it with a Riemannian metric whose rotational isometries are fractional linear transformations. Then among Gaussian measures on the vector space of polynomials of degree n, some are also "round", i.e., invariant under these rotational isometries. I had considered a related problem which I will mention below, and I had considered independent coefficients to be too arbitrary or unnatural. "Round" measures sounded more natural, but they seemed more complicated and they are not unique. In his talk, Bleher mentioned a standard observation which pleasantly surprised me, namely that there is a unique measure on polynomials of degree n which is Gaussian and round and for which the coefficients of the polynomial are independent (necessarily Gaussian non-identical) random variables. It is called the O(2) measure and it is in some sense the most natural measure on single-variable polynomials. You can compute the distribution of a random root of an O(2) random polynomial from symmetry alone; it has to equal Haar measure in the round model of RP^1. According to Bleher, is also relatively easy to find the expected number of real roots, which as it happens is exactly sqrt(N) for an O(2) polynomial of degree N. And people have derived much more delicate information in the vein of mathematical physics, namely the correlation function for the position of clusters of r roots for any fixed r in the limit as N goes to infinity. The O(2) measure comes from an even more fundamental measure on complex one-variable polynomials called the SU(2) measure. In this case there is a unique SU(2)-round Gaussian measure, and it automatically has the additional property that the coefficients are independent Gaussian variables. To put these measures in a broader context, there is an O(n+1) Gaussian measure on real polynomials and an SU(n+1) measure on complex polynomials in n variables (equivalently, homogeneous polynomials in n+1 variables). In math-ph/9903012 and math-ph/9904020, Bleher, Shiffman, and Zelditch consider something vastly more general still, namely a geometrically natural Gaussian measure on global sections of a positive line bundle L of a compact Kahler manifold. They generalize the classic correlation function results to sections of L^N in the limit as N goes to infinity. In the second article they do the same for simultaneous zeroes of several sections, which generalizes the model of random systems of polynomial equations in k variables. The related problem that I considered was the taxonomy of real algebraic curves in RP^2. This was studied by Russian algebraic geometers over a long period. They found, for example, that a smooth sextic curve consists of at most 11 circles (always one more than the genus), and they classified the way that the circles can nest. If there are at least 4 circles, there can be no triple nesting and there cannot be two nested pairs. There can only be a mommy circle with some n baby circles inside and k baby circles outside. They found that if there are 11 circles, n and k can only take the values 1, 5, and 9. Oleg Viro, who is now best known for the Turaev-Viro invariant in quantum topology, contructed the case n=k=5. My idea was to see what happens when you guess curves at random, possibly experimentally with a computer. Thanks to Bleher, Shiffman, and Zelditch, I can estimate the expected length of such a curve with respect to the round metric on RP^2, if the curve itself is chosen with respect to the SO(3) measure. But what is the expected number of circles and how do they usually nest?