Related papers: Zero Sets of Univariate Polynomials
We prove that any lower unitriangular and totally nonnegative matrix gives rise to a family of polynomials with only real zeros. This has consequences for problems in several areas of mathematics. We use it to develop a general theory for…
We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…
Let $(K,v)$ be a henselian valued field. Let $\mathbb{P}^{dless}\subset K[x]$ be the set of monic, irreducible polynomials which are defectless and have degree greater than one. For a certain equivalence relation $\,\approx\,$ on…
Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi_0,\xi_1,\ldots,\xi_n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its…
An $n$-independent set in two dimensions is a set of nodes admitting (not necessarily unique) bivariate interpolation with polynomials of total degree at most $n.$ For an arbitrary $n$-independent node set $\mathcal X$ we are interested…
The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of $\mathbb Z_{\ge 0}$ occur as the set of catenary degrees of a…
This paper discusses the location of zeros of polynomials in a polynomial sequence $\{P_n(z)\}$ generated by a three-term recurrence relation of the form $P_n(z)+ B(z)P_{n-1}(z) +A(z) P_{n-k}(z)=0$ with $k>2$ and the standard initial…
We consider the problem of bounding away from 0 the minimum value m taken by a polynomial P of Z[X_1,...,X_k] over the standard simplex, assuming that m>0. Recent algorithmic developments in real algebraic geometry enable us to obtain a…
The Sendov conjecture asserts that if all the zeros of a polynomial p lie in the closed unit disk then there must be a zero of p ' within unit distance of each zero. In this paper we give a partial result when p has simple zeros.
The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently this tool has been…
A method of constructing an entire function with given zeros and estimates of growth is suggested. It gives a possibility to describe zero sets of certain classes of entire functions of one and several variables in terms of growth of volume…
We will show that, consistently, every uncountable set can be continuously mapped onto a non measure zero set, while there exists an uncountable set whose all continuous images into a Polish space are meager.
Let S(n,0) be the set of monic complex polynomials of degree $n\ge 2$ having all their zeros in the closed unit disk and vanishing at 0. For $p\in S(n,0)$ denote by $|p|_{0}$ the distance from the origin to the zero set of $p'$. We…
For a nonempty compact set D of R we determine the maximal possible dimension of a subspace X of polynomial functions of degree at most m which possesses a positive bases (where positivity is understood on D). The exact value of this…
Given a symmetric variety Y defined over the rationals and a non-zero polynomial with integer coefficients, we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r-free for a Zariski…
A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for…
We give results on zeros of a polynomial of $\zeta(s),\zeta'(s),\ldots,\zeta^{(k)}(s)$. First, we give a zero free region and prove that there exist zeros corresponding to the trivial zeros of the Riemann zeta function. Next, we estimate…
In this paper we shall show that there exists L_0 such that for each even integer L >= L_0 there exists $c_1 \in \rz$ for which the Julia set of $z --> z^L + c_1$ has positive Lebesgue measure. This solves an old problem. Editor's note: In…
We obtain some results on the asymptotic behaviour of Geometric polynomials in both the complex plane minus $[-1,0]$ and the interval $(-1,0)$. We also find the distance of consecutive zeros of these polynomials in the bulk of the interval…
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…