Related papers: Differentiation Evens Out Zero Spacings

In this paper, we investigate zeros of difference polynomials of the form $f(z)^nH(z, f)-s(z)$, where $f(z)$ is a meromorphic function, $H(z, f)$ is a difference polynomial of $f(z)$ and $s(z)$ is a small function. We first obtain some…

We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\neq k, k\neq i$. $D_2$…

The complex zeros of the Riemannn zeta-function are identical to the zeros of the Riemann xi-function, $\xi(s)$. Thus, if the Riemann Hypothesis is true for the zeta-function, it is true for $\xi(s)$. Since $\xi(s)$ is entire, the zeros of…

We consider sparse polynomials in $N$ variables over a finite field, and ask whether they vanish on a set $S^N$, where $S$ is a set of nonzero elements of the field. We see that if for a polynomial $f$, there is $\mathbf{c}\in S^N$ with $f…

Suppose $C \subset \mathbb{C}$ is compact. Let $q_k$ be a sequence of polynomials of degree $n_k \to \infty$, such that the locus of roots of all the polynomials is bounded, and the number of roots of $q_k$ in any closed set $L$ not meeting…

If f is a real entire function and ff" has only real zeros then f belongs to the Laguerre-Polya class, the closure of the set of real polynomials with real zeros. This result completes a long line of development originating from a…

We consider the problem of characterizing all functions $f$ defined on the set of integers modulo $n$ with the property that an average of some $n$th roots of unity determined by $f$ is always an algebraic integer. Examples of such…

Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function $ f $ when sharing a set $ \mathcal{S}=\{\alpha(z),…

Consider a random polynomial $P_n$ of degree $n$ whose roots are independent random variables sampled according to some probability distribution $\mu_0$ on the complex plane $\mathbb C$. It is natural to conjecture that, for a fixed $t\in…

It has been conjectured that the statistical properties of zeros of the Riemann zeta function near $z = 1/2 + \ui E$ tend, as $E \to \infty$, to the distribution of eigenvalues of large random matrices from the Unitary Ensemble. At finite…

Given an entire function $f$ of finite order $\rho$, let $L(z,f)=\sum_{j=0}^{m}b_{j}(z)f^{(k_{j})}(z+c_{j})$ be a linear delay-differential polynomial of $f$ with small coefficients in the sense of $O(r^{\lambda+\varepsilon})+S(r,f)$,…

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0,…

If all the zeros of $n$th degree polynomials $f(z)$ and $g(z) = \sum_{k=0}^{n}\lambda_k\binom{n}{k}z^k$ respectively lie in the cricular regions $|z|\leq r$ and $|z| \leq s|z-\sigma|$, $s>0$, then it was proved by Marden \cite[p. 86]{mm}…

We study explicit continued fraction expansions for certain series. Some of these expansions have symmetry that generalizes some remarkable examples discovered independently by Kmosek and Shallit. Furthermore, we prove the following…

In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc,…

In this paper, we investigate the uniqueness problem of difference polynomials $f^{n}(z)P(f(z))L_c(f)$ and $g^{n}(z)P(g(z))L_c(g)$, where $L_c(f)=f(z+c)+c_0f(z)$, $P(z)$ is a polynomial with constant coefficients of degree $m$ sharing a…

Let $\mu$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $\mu$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for…

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le…

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…

For a given entire function $f(z)=\sum_{j=0}^{\infty}a_{j}z^{j}$, we study the zero distribution of $f_{r}(z)=\sum_{j\equiv r\pmod m}a_{j}z^{j}$ where $m\in\mathbb{N}$ and $0\le r<m$. We find conditions under which the zeros of $f_{r}(z)$…