How many zeros of a random polynomial are real?

We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve $(1,t,\ldots,t^n)$ projected onto the surface of the unit sphere, divided by $\pi$. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac's assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.Comment: 37 page

The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument

We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix.Comment: 14 pages, 3 figure

Random Triangle Theory with Geometry and Applications

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to $(0,0)$ or reformulation as a 2x2 matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage other to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: Triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the "square root ellipticity statistic" of random matrix theory. Another proof connects the Hopf map to the SVD of 2 by 2 matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight to both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed and applications are proposed