Related papers: On the zeros of polyanalytic polynomials

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most $d$. For this, we interpret the basis polynomials as vector…

Let $X$ be a (real or complex) infinite dimensional linear space. We establish conditions on a homogeneous polynomial $P$ on $X$ so that, if $W$ is any finite dimensional subspace of $X$ on which $P$ vanishes, then $P$ vanishes on an…

Let $P_s \in \mathcal{D}_s[X_0,X_1, \ldots,X_l]$ be a polynomial whose coefficients are the ring of all general Dirichlet series which converge absolutely in the half-plane $\Re (s) > 1/2$. In the present paper, we show that the function…

We give an example of a polynomial of degree 4 in 5 variables that is the sum of squares of 8 polynomials and cannot be decomposed as the sum of 7 squares. This improves the current existing lower bound of 7 polynomials for the Pythagoras…

An $n$-independent set in two dimensions is a set of nodes admitting (not necessarily unique) bivariate interpolation with polynomials of total degree at most $n.$ For an arbitrary $n$-independent node set $\mathcal X$ we are interested…

The Gauss--Lucas theorem states that any convex set $K\subset\mathbb{C}$ which contains all $n$ zeros of a degree $n$ polynomial $p\in\mathbb{C}[z]$ must also contain all $n-1$ critical points of $p$. In this paper we explore the following…

Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of…

In this paper, we obtain several new classes of irreducible polynomials having integer coefficients whose zeros lie inside an open disk around the origin or outside a closed annular region in the complex plane. Such irreducible polynomials…

This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed…

Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the…

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-\Delta^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these…

In this paper we obtained the formula for the number of irreducible polynomials with degree $n$ over finite fields of characteristic two with given trace and subtrace. This formula is a generalization of the result of Cattell et al.(2003)…

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…

Applying the Picard-Fuchs equation to the discontinuous differential system, we obtain the upper bounds of the number of zeros for Abelian integrals of four kinds of quadratic differential systems when it is perturbed inside all…

We give results on zeros of a polynomial of $\zeta(s),\zeta'(s),\ldots,\zeta^{(k)}(s)$. First, we give a zero free region and prove that there exist zeros corresponding to the trivial zeros of the Riemann zeta function. Next, we estimate…

We present new bounds for the numerical radius of bounded linear operators and $2\times 2$ operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new…

For any integer n >= 2 and any nonnegative integers r,s with r+2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r…

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\bf P}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. We generalize this theorem to…

Let $N_n(a, b)$ denote the number of real zeros of Gaussian elliptic polynomials of degree $n$ on the interval $(a, b)$, where $a$ and $b$ may vary with $n$. We obtain a precise formula for the variance of $N_n(a, b)$ and utilize this…

The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way, and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials we associate…