Related papers: Random complex fewnomials, I

We consider natural Hamiltonian systems of $n>1$ degrees of freedom with polynomial homogeneous potentials of degree $k$. We show that under a genericity assumption, for a fixed $k$, at most only a finite number of such systems is…

Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/epsilon. The bound depends on an integral…

In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure mu on…

Given an odd integer polynomial f(x) of a degree k >=3, we construct a non-negative valued, normed trigonometric polynomial with the spectrum in the set of integer values of f(x) not greater than n, and a small free coefficient…

We study integer coefficient polynomials of fixed degree and maximum height $H$, that are irreducible by Dumas's criterion. We call such polynomials Dumas polynomials. We derive upper bounds on the number of Dumas polynomials, as $H$…

This paper contains some estimates for the {\it integral-uniform} norm and the uniform norm of a wide class of random polynomials. The family of integral-uniform norms introduced by Kasin and Tzafriri is a natural generalization of the…

In this article we study the limiting empirical measure of zeros of higher derivatives for sequences of random polynomials. We show that these measures agree with the limiting empirical measure of zeros of corresponding random polynomials.…

It is known that random monic integral polynomials of bounded degree $d$ and integral coefficients distributed uniformly and independently in $[-H,H]$ are irreducible over $\mathbb{Z}$ with probability tending to $1$ as $H\to \infty$. In…

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for…

The paper presents linear integral predictors for continuous time high-frequency signals with a a finite spectrum gap. The predictors are based on approximation of a complex valued periodic exponential (complex sinusoid) by rational…

Inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of random monomial ideals. We prove theorems about the…

In this paper, we give error bounds on the number of monic irreducible polynomials $a_0+a_1x+\dots+a_{n-1}x^{n-1}+x^n$ over a finite field $\mathbb{F}_q$ of degree $n$ with $(a_0, a_1, \dots, a_{n-1}, 1)$ lying in a fixed affine algebraic…

Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external…

We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$.…

We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…

For a joint probability density function f(x) of a random vector X the mixed partial derivatives of log f(x) can be interpreted as limiting cumulants in an infinitesimally small open neighborhood around x. Moreover, setting them to zero…

For a finite set of natural numbers $D$ consider a complex polynomial of the form $f(z) = \sum_{d \in D} c_d z^d$. Let $\rho_+(f)$ and $\rho_-(f)$ be the fractions of the unit circle that $f$ sends to the right($\operatorname{Re} f(z) > 0$)…

The probability for two monic polynomials of a positive degree n with coefficients in the finite field F_q to be relatively prime turns out to be identical with the probability for an n x n Hankel matrix over F_q to be nonsingular.…

We compute averages of products and ratios of characteristic polynomials associated with Orthogonal, Unitary, and Symplectic Ensembles of Random Matrix Theory. The pfaffian/determinantal formulas for these averages are obtained, and the…

We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric…