Related papers: On the gaps between zeros of trigonometric polynom…

We describe the closures of locally divergent orbitsunder the action of tori on Hilbert modular spaces of rank r = 2. In particular, we prove that if D is a maximal R-split torus acting on a real Hilbert modular space then every locally…

The main result of this paper is a proof that for any integrable function $f$ on the torus, any sequence of its orthogonal projections $(\widetilde{P}_n f)$ onto periodic spline spaces with arbitrary knots $\widetilde{\Delta}_n$ and…

For every fixed dimension $d$ and sufficiently large $n$, we determine the maximum possible diameter of a strongly connected $d$-dimensional simplicial complex on $n$ vertices. This improves on a sequence of previous results and settles a…

Let k be an algebraically closed field of characteristic p > 0. Let H be a subgroup of GL(n,k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d…

In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials.…

Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result…

Given a probability measure $\mu$ with infinite support on the unit circle $\partial\mathbb{D}=\{z:|z|=1\}$, we consider a sequence of paraorthogonal polynomials $\h_n(z,\lambda)$ vanishing at $z=\lambda$ where $\lambda \in \T$ is fixed. We…

The trigonometric monomial $\cos(\left\langle k, x \right\rangle)$ on $\mathbb{T}^d$, a harmonic polynomial $p: \mathbb{S}^{d-1} \rightarrow \mathbb{R}$ of degree $k$ and a Laplacian eigenfunction $-\Delta f = k^2 f$ have root in each ball…

We prove that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors.

The classical Szego polynomial approximation theorem states that the polynomials are dense in the space $L^2(\rho)$, where $\rho$ is a measure on the unit circle, if and only if the logarithmic integral of the measure $\rho$ diverges. In…

First a formula for the number of zeros of the orthogonal polynomial in the intervals is presented. Then a criteria about the appearance of a zero in a gap is given. Finally a necessary and sufficient condition is derived such that the…

We prove that in each dimension $d$ there is a constant $w^\infty(d)\in \mathbb{N}$ such that for every $n\in \mathbb{N}$ all but finitely many $d$-polytopes with $n$ lattice points have width at most $w^\infty(d)$. We call $w^\infty(d)$…

We give a uniform approximation of the characteristic function of the boundary of a centrally symmetric n-dimensional compact and convex set by homogeneous polynomials of even degree $d$ fulfilling $|g_d-1|\leq E/d^{1/2-\beta}$, for every…

In this paper we study a number of conjectures on the behavior of the value distribution of eigenfunctions. On the two dimensional torus we observe that the symmetry conjecture holds in the strongest possible sense. On the other hand we…

Let T be a topology on the finite set Xn. We consider the open polynomial associated with the topology T. Its coefficients are the cardinality of open sets of size j=0,...,n. J. Brown [4] asked when this polynomial has only real zeros. We…

We present a full geometric characterization of the $1$-dimensional (semialgebraic) images $S$ of either $n$-dimensional closed balls $\overline{\mathcal B}_n\subset{\mathbb R}^n$ or $n$-dimensional spheres ${\mathbb S}^n\subset{\mathbb…

We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of…

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…

In this article, we consider piecewise smooth differential equations $Z_{X_-X_+}$, where $X_-$ and $X_+$ are linear vector fields in dimension 3, having the torus as discontinuity manifold. We consider that $Z_{X_-X_+}$ is an inelastic…

The main goal of this article is to provide a proof of the Pederson-Roy-Szpirglas theorem about counting common real zeros of real polynomial equations by using basic results from Linear algebra and Commutative algebra. The main tools are…