Equidistribution of zeros of random holomorphic sections

We study asymptotic distribution of zeros of random holomorphic sections of high powers of positive line bundles defined over projective homogenous manifolds. We work with a wide class of distributions that includes real and complex Gaussians. As a special case, we obtain asymptotic zero distribution of multivariate complex polynomials given by linear combinations of orthogonal polynomials with i.i.d. random coefficients. Namely, we prove that normalized zero measures of m i.i.d random polynomials, orthonormalized on a regular compact set $K\subset \Bbb{C}^m,$ are almost surely asymptotic to the equilibrium measure of $K$.Comment: Final version incorporates referee comments. To appear in Indiana Univ. Math.

On sums of powers of zeros of polynomials

Due to Girard's (sometimes called Waring's) formula the sum of the $r-$th power of the zeros of every one variable polynomial of degree $N$, $P_{N}(x)$, can be given explicitly in terms of the coefficients of the monic ${\tilde P}_{N}(x)$ polynomial. This formula is closely related to a known \par \noindent $N-1$ variable generalization of Chebyshev's polynomials of the first kind, $T_{r}^{(N-1)}$. The generating function of these power sums (or moments) is known to involve the logarithmic derivative of the considered polynomial. This entails a simple formula for the Stieltjes transform of the distribution of zeros. Perron-Stieltjes inversion can be used to find this distribution, {\it e.g.} for $N\to \infty$.\par Classical orthogonal polynomials are taken as examples. The results for ordinary Chebyshev $T_{N}(x)$ and $U_{N}(x)$ polynomials are presented in detail. This will correct a statement about power sums of zeros of Chebyshev's $T-$polynomials found in the literature. For the various cases (Jacobi, Laguerre, Hermite) these moment generating functions provide solutions to certain Riccati equations

Optimal non-linear transformations for large scale structure statistics

Recently, several studies proposed non-linear transformations, such as a logarithmic or Gaussianization transformation, as efficient tools to recapture information about the (Gaussian) initial conditions. During non-linear evolution, part of the cosmologically relevant information leaks out from the second moment of the distribution. This information is accessible only through complex higher order moments or, in the worst case, becomes inaccessible to the hierarchy. The focus of this work is to investigate these transformations in the framework of Fisher information using cosmological perturbation theory of the matter field with Gaussian initial conditions. We show that at each order in perturbation theory, there is a polynomial of corresponding order exhausting the information on a given parameter. This polynomial can be interpreted as the Taylor expansion of the maximally efficient "sufficient" observable in the non-linear regime. We determine explicitly this maximally efficient observable for local transformations. Remarkably, this optimal transform is essentially the simple power transform with an exponent related to the slope of the power spectrum; when this is -1, it is indistinguishable from the logarithmic transform. This transform Gaussianizes the distribution, and recovers the linear density contrast. Thus a direct connection is revealed between undoing of the non-linear dynamics and the efficient capture of Fisher information. Our analytical results were compared with measurements from the Millennium Simulation density field. We found that our transforms remain very close to optimal even in the deeply non-linear regime with \sigma^2 \sim 10.Comment: 11 pages, matches version accepted for publication in MNRA

On polynomials connected to powers of Bessel functions

The series expansion of a power of the modified Bessel function of the first kind is studied. This expansion involves a family of polynomials introduced by C. Bender et al. New results on these polynomials established here include recurrences in terms of Bell polynomials evaluated at values of the Bessel zeta function. A probabilistic version of an identity of Euler yields additional recurrences. Connections to the umbral formalism on Bessel functions introduced by Cholewinski are established

Polynomial Triangles Revisited

A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to fill this gap. We describe some aspects of these arrays, which generalize similar properties of the binomial coefficients. Some combinatorial models enumerated by polynomial coefficients, including lattice paths model, spin chain model and scores in a drawing game, are introduced. Several known binomial identities are then extended. In addition, we calculate recursively generating functions of column sequences. Interesting corollaries follow from these recurrence relations such as new formulae for the Fibonacci numbers and Hermite polynomials in terms of trinomial coefficients. Finally, properties of the entropy density function that characterizes polynomial coefficients in the thermodynamical limit are studied in details.Comment: 24 pages with 1 figure eps include