Related papers: Zeros of differential polynomials in real meromorp…

This paper studies the poles of the real Archimedean zeta function for a weighted homogeneous polynomial $f \in \mathbb{R}[x, y]$ with an isolated singularity at the origin. By applying a weighted blow-up, we derive the meromorphic…

In this paper, we investigate meromorphic solutions in $\mathbb{C}^m$ of the nonlinear differential equation \[\displaystyle f^n\partial_u(f)g^n\partial_u(g)=1,\] where $\partial_u(f)=\sum_{j=1}^mu_j\partial_j(f)$ and $\sum_{j=1}^m u_j\neq…

We prove Polya's conjecture of 1943: For a real entire function of order greater than 2, with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity with n. We use the saddle point method and…

We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou…

Let $f$ be a transcendental meromorphic function, defined in the complex plane $\mathbb{C}$. In this paper, we give a quantitative estimations of the characteristic function $T(r,f)$ in terms of the counting function of a homogeneous…

This paper is devoted to establish sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extend some results for entire functions to meromorphic functions. Actually, we shall mainly…

The interplay among the time-evolution of the coefficients and the zeros of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently this tool has been…

We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these functions. We prove that the…

Let $s_0,s_1,\dots,s_{m-1}$ be complex numbers and $r_0,\dots,r_{m-1}$ rational integers in the range $0\le r_j\le m-1$. Our first goal is to prove that if an entire function $f$ of sufficiently small exponential type satisfies…

The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By…

In this paper, we relate the set of asymptotic critical values of a polynomial function $f$ with the set of discontinuity of two functions, the multivalued function which associate to each value $t$ the set of tangent directions at infinity…

In this paper, we investigate the zero distributions of $q$-shift difference-differential polynomials of meromorphic functions with zero-order that extends and generalizes the classical Hayman results of the zeros of differential…

In this paper, we establish transcendental entire function $A(z)$ and polynomial $B(z)$ such that the differential equation $f''+A(z)f'+B(z)f=0$, has all non-trivial solution of infinite order. We use the notion of \emph{critical rays} of…

We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case…

Let h be a complex meromorphic function decomposed in two different ways P(f) and Q(g), where f, g are meromorphic functions and P, Q are rational functions. We follow an approach due to C.-C. Yang, P. Li and K. H. Ha who handle similar…

Let $\Phi$ be an irreducible (possibly noncrystallographic) root system of rank $l$ of type $P$. For the corresponding cluster complex $\Delta(P)$, which is known as pure $(l-1)$-dimensional simplicial complex, we define the generating…

Let U be the open unit disc in C. Given a continuous function g: bU --> C-{0} denote by W(g) the winding number of g around the origin. We prove that a continuous function f: bU --> C extends meromorphically through U if and only if there…

The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to…