Polynomial Density Research Articles

A note on the squarefree density of polynomials

AbstractThe conjectured squarefree density of an integral polynomial in variables is an Euler product which can be considered as a product of local densities. We show that a necessary and sufficient condition for to be 0 when is a polynomial in variables over the integers, is that either there is a prime such that the values of at all integer points are divisible by or the polynomial is not squarefree as a polynomial. We also show that generally the upper squarefree density satisfies .

3D large-scale forward modeling of gravitational fields using triangular spherical prisms with polynomial densities in depth

To take the sphericity of the Earth into account, tesseroids are often utilized as grid elements in large-scale gravitational forward modeling. However, such elements in a latitude–longitude mesh suffer from degenerating into poorly shaped triangles near poles. Moreover, tesseroids have limited flexibility in describing laterally variable density distributions with irregular boundaries and also face difficulties in achieving completely equivalent division over a spherical surface that may be desired in a gravity inversion. We develop a new method based on triangular spherical prisms (TSPs) for 3D gravitational modeling in spherical coordinates. A TSP is defined by two spherical surfaces of triangular shape, with one of which being the radial projection of the other. Due to the spherical triangular shapes of the upper and lower surfaces, TSPs enjoy more advantages over tesseroids in describing mass density with different lateral resolutions. In addition, such an element also allows subdivisions with nearly equal weights in spherical coordinates. To calculate the gravitational effects of a TSP, we assume the density in each element to be polynomial along radial direction so as to accommodate a complex density environment. Then, we solve the Newton’s volume integral using a mixed Gaussian quadrature method, in which the surface integral over the spherical triangle is calculated using a triangle-based Gaussian quadrature rule via a radial projection that transforms the spherical triangles into linear ones. A 2D adaptive discretization strategy and an extension technique are also combined to improve the accuracy at observation points near the mass sources. The numerical experiments based on spherical shell models show that the proposed method achieves good accuracy from near surface to a satellite height in the case of TSPs with various dimensions and density variations. In comparison with the classical tesseroid-based method, the proposed algorithm enjoys better accuracy and much higher flexibility for density models with laterally irregular shapes. It shows that to achieve the same accuracy, the number of elements required by the proposed method is much less than that of the tesseroid-based method, which substantially speeds up the calculation by more than 2 orders. The application to the tessellated LITHO1.0 model further demonstrates its capability and practicability in realistic situations. The new method offers an attractive tool for gravity forward and inverse problems where the irregular grids are involved.

Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation

This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.

On density of polynomials in algebra of holomorphic functions of exponential type on linear Lie group

The question of the density of the algebra of polynomials (regular functions) in the algebra of holomorphic functions of exponential type on a complex Lie group arose in the study of duality for Hopf algebras of holomorphic functions. It was shown by the author in [J . Lie Theory, 29:4, 1045 - 1070 (2019)] that the answer is affirmative in the connected linear case. However, the argument is quite involved and here we present a short proof. It contains two ingredients. The first is the existences of a finite - dimensional faithful holomorphic representation with closed range. To prove it, we use an approach developed by Ðjoković. The second is a lower bound for the norm of a one - parameter matrix subgroup, which is based on some elementary linear algebra consideration. The rest of the proof is close to the original one and uses a decomposition of the group into a semidirect product of a simply connected solvable and linearly complex reductive factors.

Squarefree density of polynomials

Squarefree density of polynomials

Density of Complex and Quaternionic Polyanalytic Polynomials in Polyanalytic Fock Spaces

In this paper we consider the polyanalytic Fock spaces both in the complex and in the quaternionic case. In this latter case, the polyanalytic functions are considered in the slice regular case, and we shall treat Fock spaces of the first and of the second kind. In all these spaces we prove quantitative results in the approximation by polyanalytic polynomials. The quantitative approximation results are given in terms of higher order Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{p}$$\\end{document}-moduli of smoothness.

On the Jones polynomial modulo primes

AbstractWe derive an upper bound on the density of Jones polynomials of knots modulo a prime number $p$ , within a sufficiently large degree range: $4/p^7$ . As an application, we classify knot Jones polynomials modulo two of span up to eight.

Geometric sieve over number fields for higher moments

The geometric sieve for densities is a very convenient tool proposed by Poonen and Stoll (and independently by Ekedahl) to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a geometric sieve that allows the computation of all higher moments corresponding to the density, over a general number field K. This work advances the understanding of geometric sieve for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the geometric sieve for densities over number fields; on the other hand, it extends the recent result on a geometric sieve for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.

Isomorphic Copies of ell ^infty in the Weighted Hardy Spaces on the Unit Disc

It is still unclear whether the density of analytic polynomials in an H-admissible space is sufficient to the minimality of the space? This question has a purely foundational background, relating fundamental concepts from the theory of H^p spaces. We hypothesize that there is no general relationship between the density of analytic polynomials and the R-admissibility of an H-admissible space. We solve this problem by finding suitable counterexamples of Hardy spaces built upon some weighted Lebesgue spaces. In particular, we provide a direct construction of weights from Szegő class, which guarantees the existence of isomorphic copies of the space of bounded sequences in weighted Hardy spaces on the unit disc.

On the density of multivariate polynomials with varying weights

In this paper we consider multivariate approximation by weighted polynomials of the form w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) , where p n p_n is a multivariate polynomial of degree at most n n , w w is a given nonnegative weight with nonempty zero set, and γ n ↑ ∞ \gamma _n\uparrow \infty . We study the question if every continuous function vanishing on the zero set of w w is a uniform limit of weighted polynomials w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) . It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that γ n = o ( n ) . \gamma _n=o(n).

Evaluation of gravitational curvatures for a tesseroid and spherical shell with arbitrary-order polynomial density

In recent years, there are research trends from constant to variable density and low-order to high-order gravitational potential gradients in gravity field modeling. Under the research circumstances, this paper focuses on the variable density model for gravitational curvatures (or gravity curvatures, third-order derivatives of gravitational potential) of a tesseroid and spherical shell in the spatial domain of gravity field modeling. In this contribution, the general formula of the gravitational curvatures of a tesseroid with arbitrary-order polynomial density is derived. The general expressions for gravitational effects up to the gravitational curvatures of a spherical shell with arbitrary-order polynomial density are derived when the computation point is located above, inside, and below the spherical shell. When the computation point is located above the spherical shell, the general expressions for the mass of a spherical shell and the relation between the radial gravitational effects up to arbitrary-order and the mass of a spherical shell with arbitrary-order polynomial density are derived. The influence of the computation point’s height and latitude on gravitational curvatures with the polynomial density up to fourth-order is numerically investigated using tesseroids to discretize a spherical shell. Numerical results reveal that the near-zone problem exists for the fourth-order polynomial density of the gravitational curvatures, i.e., relative errors in log _{10} scale of gravitational curvatures are large than 0 below the height of about 50 km by a grid size of 15'times 15'. The polar-singularity problem does not occur for the gravitational curvatures with polynomial density up to fourth-order because of the Cartesian integral kernels of the tesseroid. The density variation can be revealed in the absolute errors as the superposition effects of Laplace parameters of gravitational curvatures other than the relative errors. The derived expressions are examples of the high-order gravitational potential gradients of the mass body with variable density in the spatial domain, which will provide the theoretical basis for future applications of gravity field modeling in geodesy and geophysics.

Kantorovich type topologies on spaces of measures and convergence of barycenters

We study two topologies $ \tau_{KR} $ and $ \tau_K $ on the space of measures on a completely regular space generated by Kantorovich–Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich–Rubinshtein topology $ \tau_{KR} $ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures, some $ s $-convex measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures.

Markov Moment Problems on Special Closed Subsets of ℝn

First, this paper provides characterizing the existence and uniqueness of the linear operator solution 𝑇 for large classes of full Markov moment problems on closed subsets 𝐹 of ℝ𝑛. One uses approximation by special nonnegative polynomials. The case when 𝐹 is compact is studied. Then the cases when 𝐹 = ℝ𝑛 and 𝐹 = ℝ+n are under attention. Here, the main findings consist in proving and applying the density of special polynomials, which are sums of squares, in the positive cone of Lv1(ℝ𝑛), and respectively of Lv1(ℝ+n), for a large class of measures 𝜈. One solves the important difficulty created by the fact that on ℝ𝑛, 𝑛 ≥ 2, there exist nonnegative polynomials which are not expressible in terms of sums of squares. This is the second aim of the paper. On the other hand, two types of symmetry are outlined. Both these symmetry properties appear naturally from the thematic mentioned above. This is the third aim of the paper. They lead to new statements, illustrated in corollaries, and supported by a few examples.

Density of polyanalytic polynomials in complex and quaternionic polyanalytic weighted Bergman spaces

We introduce the concepts of complex polyanalytic weighted Bergman spaces and of quaternionic polyanalytic weighted Bergman spaces of first and second kind. In these spaces we then prove qualitative and quantitative results in approximation by polyanalytic polynomials. The quantitative approximation results are given in terms of higher order $L^{p}$-moduli of smoothness and in terms of the best approximation quantity.

Fast calculation of gravitational effects using tesseroids with a polynomial density of arbitrary degree in depth

Fast and accurate calculation of gravitational effects on a regional or global scale with complex density environment is a critical issue in gravitational forward modelling. Most existing significant developments with tessroid-based modelling are limited to homogeneous density models or polynomial ones of a limited order. Moreover, the total gravitational effects of tesseroids are often calculated by pure summation in these methods, which makes the calculation extremely time-consuming. A new efficient and accurate method based on tesseroids with a polynomial density up to an arbitrary order in depth is developed for 3D large-scale gravitational forward modelling. The method divides the source region into a number of tesseroids, and the density in each tesseroid is assumed to be a polynomial function of arbitrary degree. To guarantee the computational accuracy and efficiency, two key points are involved: (1) the volume Newton’s integral is decomposed into a one-dimensional integral with a polynomial density in the radial direction, for which a simple analytical recursive formula is derived for efficient calculation, and a surface integral over the horizontal directions evaluated by the Gauss–Legendre quadrature (GLQ) combined with a 2D adaptive discretization strategy; (2) a fast and flexible discrete convolution algorithm based on 1D fast Fourier transform (FFT) and a general Toepritz form of weight coefficient matrices is adopted in the longitudinal dimension to speed up the computation of the cumulative contributions from all tesseroids. Numerical examples show that the gravitational fields predicted by the new method have a good agreement with the corresponding analytical solutions for spherical shell models with both polynomial and non-polynomial density variations in depth. Compared with the 3D GLQ methods, the new algorithm is computationally more accurate and efficient. The calculation time is significantly reduced by 3 orders of magnitude as compared with the traditional 3D GLQ methods. Application of the new algorithm in the global crustal CRUST1.0 model further verifies its reliability and practicability in real cases. The proposed method will provide a powerful numerical tool for large-scale gravity modelling and also an efficient forward engine for inversion and continuation problems.

Length statistics of random multicurves on closed hyperbolic surfaces

A multicurve can be decomposed into components. The ratios of the length of each component to the total length give a hint of the shape of the multicurve. In this paper, we determine the distribution of these ratios of a random multicurve with a given topological type on a closed hyperbolic surface, using the methods of Margulis’ thesis and Mirzakhani’s equidistribution theorem for horospheres. This distribution admits a polynomial density whose coefficients can be expressed explicitly in terms of intersection numbers of psi-classes on the Deligne–Mumford compactification of the moduli space of complex curves, and in particular it does not depend on the hyperbolic metric. This result generalizes prior work of M. Mirzakhani in the case of random pants decompositions. Results very close to ours were obtained independently and simultaneously by F. Arana-Herrera.

Approximation in the Mean on Rational Curves

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue \(L^2\)-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan are obtained on rational curves.

Nontransitivity of tuples of random variables with polynomial density and its effects in Bayesian models

The problem of nontransitivity of stochastic precedence relation is considered. In particular, nontransitivity is studied in the case of three and four random variables with densities given by cubic polynomials on the unit interval. Lower bounds for maximal probabilities and the corresponding expressions for the densities and distribution functions are obtained. Moreover, the effect of stochastic nontransitivity in application to Bayesian queueing models is studied. Namely, we compare the behavior of the traffic intensities of three and four single-server queuing systems with random service parameters and random parameters of input flows under the assumption that the latter are subject to a stochastic precedence relation.

The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots in Qp${\mathbb {Q}}_p$

We determine the probability that a random polynomial of degree n $n$ over Z p ${\mathbb {Z}}_p$ has exactly r $r$ roots in Q p ${\mathbb {Q}}_p$ , and show that it is given by a rational function of p $p$ that is invariant under replacing p $p$ by 1 / p $1/p$ .

Squarefree values of polynomial discriminants I

We determine the density of monic integer polynomials of given degree $$n>1$$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials f(x), such that f(x) is irreducible and $${\mathbb Z}[x]/(f(x))$$ is the ring of integers in its fraction field, is positive, and is in fact given by $$\zeta (2)^{-1}$$ . It also follows from our methods that there are $$\gg X^{1/2+1/n}$$ monogenic number fields of degree n having associated Galois group $$S_n$$ and absolute discriminant less than X, and we conjecture that the exponent in this lower bound is optimal.