Related papers: A Haagerup inequality through the use of orthogona…

Together with Speicher, in 2007 the first author proved the strong Haagerup inequality for operator norms of homogeneous holomorphic polynomials in freely independent $\mathscr{R}$-diagonal elements (including in particular circular random…

With any integer convex polytope $P\subset\midR^n$ we associate a multivariate hypergeometric polynomial whose set of exponents is $\midZ^{n}\cap P.$ This polynomial is defined uniquely up to a constant multiple and satisfies a holonomic…

A mean step in Haagerup's proof for the optimal constants in Khintchine's inequality is to show integral inequalities of type $\int(g^s-f^s)\mathrm{d}\mu\geq 0$. F.L. Nazarov and A.N. Podkorytov made Haagerup's proof much more clearer for…

It follows from the Garloff-Wagner Theorem that the set of stable polynomials of degree $n$, denoted by $\mathcal{H}_n$, i.e., those whose zeros all lie in the open left complex half-plane, with the Hadamard product $*$, forms an abelian…

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…

A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre…

We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\it geometric degree of the…

In this paper we explore inequalities between symmetric homogeneous polynomials of degree four of three real variables and three nonnegative real variables. The main theorems describe the cases in which the smallest possible coefficient is…

In this paper we discuss the natural candidate for the one dimensional free Poincar\'e inequality. Two main strong points sustain this candidacy. One is the random matrix heuristic and the other the relations with the other free functional…

We use free probability techniques to compute borders of spectra of non hermitian operators in finite von Neumann algebras which arise as `free sums' of `simple' operators. To this end, the resolvent is analyzed with the aid of the Haagerup…

We estimate the operator norm of radial non-negative functions on hyperbolic groups. As a consequence, we show that several forms of Haagerup's inequality are optimal.

We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, H_d denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then…

Let H(N) denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in H(N) whose largest coefficients are as small as possible and also for…

This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…

Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z)$, $Z \in C^{n}$. We call such a polynomial $p$ {\bf H-Stable} if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This…

The main aim of this work is to apply the matrix approach of ortho\-gonal polynomials associated with infinite Hermitian definite positive matrices in relation with an important question regarding the location of zeros of Sobolev orthogonal…

We exhibit several bounds for operator norms of the sum of $\epsilon$-free semicircular random variables introduced in the paper of Speicher and Wysocza\'{n}ski. In particular, using the first and second largest eigenvalues of the adjacency…

Led by the key example of the Korteweg-de Vries equation, we study pairs of Hamiltonian operators which are non-homogeneous and are given by the sum of a first-order operator and an ultralocal structure. We present a complete classification…

Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|\alpha|=m} a_{\alpha} z^{\alpha}$ in $n$ complex variables, $$\big( \sum_{|\alpha|=m} |a_{\alpha}|^{r}…

Let $g$ and $h$ be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials \begin{equation*} P_n^{g,h}(x):= \frac{x}{h(n)} \sum_{k=1}^{n} g(k) \,…