Related papers: Equidistribution of zeros of polynomials
In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure mu on…
We address the distribution properties of points of small height on proper toric varieties and applications to the related Bogomolov property. We introduce the notion of monocritical toric metrized divisor and we prove that equidistribution…
Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion…
Let $f(Z)=Z^n-a_{1}Z^{n-1}+\cdots+(-1)^{n-1}a_{n-1}Z+(-1)^na_n$ be a monic polynomial with coefficients in a ring~$R$ with identity, not necessarily commutative. We study the ideal $I_f$ of $R[X_1,\dots,X_n]$ generated by…
We prove an effective equidistribution theorem for semisimple closed orbits on compact adelic quotients. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. The…
We study the small scale distribution of the eigenfunctions of a point scatterer (the Laplacian perturbed by a delta potential) on two- and three-dimensional flat tori. In two dimensions, we establish small scale equidistribution for the…
In this paper, we obtain several new classes of irreducible polynomials having integer coefficients whose zeros lie inside an open disk around the origin or outside a closed annular region in the complex plane. Such irreducible polynomials…
A certain real number, depending on two neighbouring sides of a quadrilateral and the diagonal meeting these two sides at their common point, is shown to be invariant under affinity. As an application we demonstrate a nice formula for the…
We prove that Misiurewicz parameters with prescribed combinatorics and hyperbolic parameters with (d - 1) distinct attracting cycles with given multipliers are equidistributed with respect to the bifurcation measure in the moduli space of…
An example of trigonometric polynomials with extremely small uniform norm is given. This example demonstrates the potential limits for extension of Sidon's inequality for lacunary polynomials in a certain direction.
We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric…
We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near…
A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our…
We use the theory of unipotent flows to prove, in a general setting, the equidistribution of Hecke points. We follow the approach of Burger-Sarnak, and use a theorem of Mozes-Shah.
The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are…
It is known that the image in $\mathbb{R}^{2}/\mathbb{Z}^{2}$ of a circle of radius $\rho$ in the plane becomes equidistributed as $\rho\to\infty$. We consider the following sparse version of this phenomenon. Starting from a sequence of…
We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima…
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…
In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…
In this note we consider the distribution of values of weighted sums of the von Mangoldt arithmetical function. By using a formula for the distribution of values of trigonometric polynomials, we are able to present evidence supporting the…