Related papers: Aspects of stable polynomials
We give a formula and an estimation for the number of irreducible polynomials in two (or more) variables over a finite field.
We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…
We present some inequalities that provide different sufficient conditions for an univariate monic polynomial to be Hurwitz unstable. These are motivated by difficult control problems where direct application of the Li\'enard-Chipart…
The relationship between a stable multivariable polynomial $p(z)$ and the Fourier coefficients of its spectral density function $1/|p(z)|^2$, is further investigated. In this paper we focus on the radial asymptotics of the Fourier…
In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
Explicit examples of {\bf positive} crystalline measures and Fourier quasicrystals are constructed using pairs of stable of polynomials, answering several open questions in the area.
We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on…
For a polynomial in several variables depending on some parameters, we discuss some results to the effect that for almost all values of the parameters the polynomial is irreducible. In particular we recast in this perspective some results…
The concept of stability, originally introduced for polynomials, will be extended to apply to the class of entire functions. This generalization will be called Hurwitz stablility and the class of Hurwitz stable functions will serve as the…
This paper derives two stabilizability theorems for a basic class of discrete-time nonlinear systems with multiple unknown parameters. First, we claim that a discrete-time multi-parameter system is stabilizable if its nonlinear growth rate…
Using a sums of squares formula for two variable polynomials with no zeros on the bidisk, we are able to give a new proof of a representation for distinguished varieties. For distinguished varieties with no singularities on the two-torus,…
An infinite family of Boolean polynomials which correspond to the discrete average maps, defined in [2], is constructed and their algebraic and combinatorial properties are investigated. They turn out to be balanced, and some recurrence…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
A polynomial $f(x)$ over a field $K$ is called stable if all of its iterates are irreducible over $K$. In this paper we study the stability of trinomials over finite fields. Specially, we show that if $f(x)$ is a trinomial of even degree…
Consider polynomial sequences that satisfy a first-order differential recurrence. We prove that if the recurrence is of a special form, then the Tur\'an expressions for the sequence are weakly Hurwitz stable (non-zero in the open right…
In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…
Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the so-called ideal property: if a polynomial P belongs to a class Q, then the composition u o P o v of P with…