Positive polynomial matrices play a fundamental role in systems and control theory: they represent e.g. spectral density functions of stochastic processes and show up in spectral factorizations, robust control and filter design problems. Positive polynomials obviously form a convex set and were recently studied in the area of convex optimization [1, 5]. It was shown in [2, 5] that positive poly...

This paper explores the geometric structure of spectrahedral cone, called symmetry adapted positive semidefinite (PSD) and Gram spectrahedron a symmetric polynomial. In particular, we determine dimension PSD describe its extremal rays, discuss matrix representations. We also consider spectrahedra for specific families polynomials including binary polynomials, quadratics, ternary quartics sextic...

Abstract. It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel measure on the real line. In this paper we extend this result and show that every system of polynomials satisfying some (2N + 1)-term recurrence re...

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are g...

Let G be a perfectly oriented planar graph. Postnikov’s boundary measurement construction provides a rational map from the set of positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov’s map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weight...

We study the strong asymptotics of orthogonal polynomials with respect to a measure of the type dμ/2π + ∑∞ j=1Ajδ(z− zk), where μ is a positive measure on the unit circle Γ satisfying the Szegö condition and {zj}j=1 are fixed points outside Γ. The masses {Aj}j=1 are positive numbers such that ∑∞ j=1Aj < +∞. Our main result is the explicit strong asymptotic formulas for the corresponding orthogo...

Generalized polynomials are defined as products of polynomials raised to positive real powers. The generalized degree can be defined in a natural way. A number of classical inequalities holding for polynomials can be extended for generalized polynomials utilizing the generalized degree in place of the ordinary one. Remez established a sharp upper bound for the maximum modulus on [— 1,1] of alge...

We consider polynomials over the binary field, F 2. A polynomial f of degree m is called primitive if k = 2 m − 1 is the smallest positive integer such that f divides x k + 1. We consider polynomials over the binary field, F 2. A polynomial f of degree m is called primitive if k = 2 m − 1 is the smallest positive integer such that f divides x k + 1.

Let pk(x) = x +■■■ , k e N0 , be the polynomials orthogonal on [-1, +1] with respect to the positive measure da . We give sufficient conditions on the real numbers p , j = 0, ... , m , such that the linear combination of orthogonal polynomials YfLo^jPn-j has n simple zeros in (—1,-1-1) and that the interpolatory quadrature formula whose nodes are the zeros of Yfj=oßjPn-j has positive weights.

The quantum integer [n]q is the polynomial 1+q+q+ · · ·+q. Two sequences of polynomials U = {un(q)}∞n=1 and V = {vn(q)} ∞ n=1 define a linear addition rule ⊕ on a sequence F = {fn(q)}∞n=1 by fm(q) ⊕ fn(q) = un(q)fm(q)+vm(q)fn(q). This is called a quantum addition rule if [m]q⊕[n]q = [m + n]q for all positive integers m and n. In this paper all linear quantum addition rules are determined, and a...