arXiv subjects

Julius Borcea

Publications and source records attributed to Julius Borcea.

26 records · Page 2Linked to original sources

arXiv2006

Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations

A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials generated by discrete planar charge configurations. In the case of $n$ positive charges we prove that the equilibrium points satisfy $\binom{n}{2}$ weighted majorization relations and are uniquely determined by $n-1$ such relations. It is further shown that the Hausdorff geometry of the equilibrium points and the charged particles is controlled by the weighted standard deviation of the latter. By using finite-rank perturbations of compact normal Hilbert space operators we establish similar relations for infinite charge distributions. We also discuss a hierarchy of weighted de Bruijn-Springer relations and inertia laws, the existence of zeros of Borel series with positive $l^1$-coefficients, and an operator version of the Clunie-Eremenko-Rossi conjecture.

math.CV↗

arXiv2006

Maximal and linearly inextensible polynomials

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 determine all 0-maximal polynomials of degree $n$, that is, all polynomials $p\in S(n,0)$ such that $|p|_{0}\ge |q|_{0}$ for any $q\in S(n,0)$. Using a second order variational method we then show that although some of these polynomials are linearly inextensible, they are not locally maximal for Sendov's conjecture.

math.CV↗

arXiv2006

Spectral order and isotonic differential operators of Laguerre-Polya type

The spectral order on $\bR^n$ induces a natural partial ordering on the manifold $\calH_{n}$ of monic hyperbolic polynomials of degree $n$. We show that all differential operators of Laguerre-Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space $\li$ of real bounded sequences. As a consequence, we deduce that the monoid $\calA'$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre-Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\calA'$-orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.

math.CA↗

arXiv2005

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function $f(z)$ by a sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by $f(z)$. Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ in the complement $\mathbb{CP}^1\setminus \D_{f}$, where $\D_{f}$ is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family $\{r_{n}(z)\}_{n\in \mathbb{N}}$. As an application we settle the so-called 3-conjecture of Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.

math.CA↗

arXiv2004

Classifying real polynomial pencils

Let $\bP^n$ be the space of all homogeneous polynomials of degree $n$ in two variables with real coefficients. The standard discriminant $\D_{n+1}\subset \bP^n$ is Whitney stratified according to the number and the multiplicities of multiple real zeros. A real polynomial pencil, that is, a line $L\subset \bP^n$ is called generic if it intersects $\D_{n+1}$ transversally. Nongeneric pencils form the Grassmann discriminant $\D_{2,n+1}\subset \gtn$, where $\gtn$ is the Grassmannian of lines in $\bP^n$. We enumerate the connected components of the set $\widetilde \gtn=\gtn\setminus \D_{2,n+1}$ of all generic lines in $\bP^n$ and relate this topic to the Hawaii conjecture and the classical theorems of Obreschkoff and Hermite-Biehler.

math.AG↗

arXiv2003

Hyperbolic polynomials and spectral order

The spectral order on $\bR$ induces a partial ordering on the manifold $\calH_{n}$ of monic hyperbolic polynomials of degree $n$. We show that the semigroup $\tilde{\calS}$ generated by differential operators of the form $(1-\la \frac{d}{dx})e^{\la \frac{d}{dx}}$, $\la \in \bR$, acts on the poset $\calH_{n}$ in an order-preserving fashion. We also show that polynomials in $\calH_{n}$ are global minima of their respective $\tilde{\calS}$-orbits and we conjecture that a similar result holds even for complex polynomials. Finally, we show that only those pencils of polynomials in $\calH_{n}$ which are of logarithmic derivative type satisfy a certain local minimum property for the spectral order.

math.CA↗

arXiv2002

Dualities and vertex operator algebras of affine type

We notice that for any positive integer $k$, the set of $(1,2)$-specialized characters of level $k$ standard $A_{1}^{(1)}$-modules is the same as the set of rescaled graded dimensions of the subspaces of level $2k+1$ standard $A_{2}^{(2)}$-modules that are vacuum spaces for the action of the principal Heisenberg subalgebra of $A_{2}^{(2)}$. We conjecture the existence of a semisimple category induced by the "equal level" representations of some algebraic structure which would naturally explain this duality-like property, and we study potential such structures in the context of generalized vertex operator algebras.

math.QA↗

arXiv2001

Annihilating Fields of Standard Modules for Affine Lie Algebras

Given an affine Kac-Moody Lie algebra $\tilde{\mathfrak{g}}[σ]$ of arbitrary type, we determine certain minimal sets of annihilating fields of standard $\tilde{\mathfrak{g}}[σ]$-modules. We then use these sets in order to obtain a characterization of standard $\tilde{\mathfrak{g}}[σ]$-modules in terms of irreducible loop $\tilde{\mathfrak{g}}[σ]$-modules, which proves to be a useful tool for combinatorial constructions of bases for standard $\tilde{\mathfrak{g}}[σ]$-modules.

math.QA↗