Related papers: A Generalization of the B\^{o}cher-Grace Theorem

An \emph{s-graph} is a graph with two kinds of edges: \emph{subdivisible} edges and \emph{real} edges. A \emph{realisation} of an s-graph $B$ is any graph obtained by subdividing subdivisible edges of $B$ into paths of arbitrary length (at…

We show how rational points on certain varieties parametrize phenomena arising in the Galois theory of iterates of quadratic polynomials. As an example, we characterize completely the set of quadratic polynomials $x^2+c$ whose third iterate…

We consider Bochner-Riesz means on weighted $L^p$ spaces, at the critical index $\lambda(p)=d(\frac 1p-\frac 12)-\frac 12$. For every $A_1$-weight we obtain an extension of Vargas' weak type $(1,1)$ inequality in some range of $p>1$. To…

We generalise the Siegel-Voloch theorem about S-integral points on elliptic curves as follows: let K/F denote a global function field over a finite field F of characteristic p>3, let S denote a finite set of places of K and let E/K denote a…

Given an arbitrary complex-valued infinite matrix A and a positive integer n we introduce a naturally associated polynomial basis B_A of C[x0...xn]. We discuss some properties of the locus of common zeros of all polynomials in B_A having a…

A new version of the Graeffe algorithm for finding all the roots of univariate complex polynomials is proposed. It is obtained from the classical algorithm by a process analogous to renormalization of dynamical systems. This iteration is…

Computing the critical points of a polynomial function $q\in\mathbb Q[X_1,\ldots,X_n]$ restricted to the vanishing locus $V\subset\mathbb R^n$ of polynomials $f_1,\ldots, f_p\in\mathbb Q[X_1,\ldots, X_n]$ is of first importance in several…

The article proved the upper bound of leading coefficient of characteristic polynomial of graded ideal in a ring of generalized polynomials.

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval $I \subset \mathbb{R}$ can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical…

A generic orthotope is an orthogonal polytope whose tangent cones are described by read-once Boolean functions. The purpose of this note is to develop a theory ofEhrhart polynomials for integral generic orthotopes. The most remarkable part…

Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the…

In this paper we study the operation of cutting off edges of a simple $3$-polytope $P$ along the graph $\Gamma$. We give the criterion when the resulting polytope is simple and when it is flag. As a corollary we prove the analog of…

In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart…

We consider a Br\'ezis-Nirenberg type critical growth $p$-Laplacian problem involving a parameter $\mu > 0$ in a smooth bounded domain $\Omega$. We prove the existence of multiple nontrivial solutions if either $\mu$ or the volume of…

We show that there exists a simple generalization of Kazakov's multicritical one-matrix model, which interpolates between the various multicritical points of the model. The associated multicritical potential takes the form of a power series…

The classical Hermite-Biehler theorem describes possible zero sets of complex linear combinations of two real polynomials whose zeros strictly interlace. We provide the full characterization of zero sets for the case when this interlacing…

We consider a graph polynomial \xi(G;x,y,z) introduced by Averbouch, Godlin, and Makowsky (2007). This graph polynomial simultaneously generalizes the Tutte polynomial as well as a bivariate chromatic polynomial defined by Dohmen, Poenitz…

A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem,…

Let $p_n$ be the characteristic polynomial of an $n \times n$ random matrix drawn from one of the compact classical matrix groups. We show that the critical points of $p_n$ converge to the uniform distribution on the unit circle as $n$…