Related papers: Differentiation Evens Out Zero Spacings

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…

Let $f(x) \in \mathbb{C}[x]$ of degree $n$. We attach to $f$ a $\mathbb{C}$-vector space $W(f)$ which consists of complex polynomials $p(x)$ of degree at most $n - 2$ such that $f(x)$ divides $f"(x)p(x) - f'(x) p'(x)$. The space $W(f)$…

We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional differential operator of the form $z^{a}…

Let $\phi(x)=\sum \alpha_n x^n$ be a formal power series with real coefficients, and let $D$ denote differentiation. It is shown that "for every real polynomial $f$ there is a positive integer $m_0$ such that $\phi(D)^mf$ has only real…

Let F be a family of functions meromorphic in a domain D. If {|f|/(1+|f|^3):f in F} is locally uniformly bounded away from zero, then F is normal.

Given a central division algebra $D$ of degree $d$ over a field $F$, we associate to any standard polynomial $\phi(z)=z^n+c_{n-1} z^{n-1}+\dots+c_0$ over $D$ a "companion polynomial" $\Phi(z)$ of degree $n d$ with coefficients in $F$ whose…

Let $P(z)=z^{n}+a_{n-2}z^{n-2}+\cdots+a_0$ be a nonconstant polynomial and $S(z)$ be a nonzero rational function and denote $h(z)=S(z)e^{P(z)}$. Let $\theta\in(0,\pi/2n)$ be a constant and $\varepsilon>0$ be a small constant. It is shown…

Let $f$ and $F$ be two polynomials satisfying $F(x)=u(x)f(x)+v(x)f'(x)$. We characterize the relation between the location and multiplicity of the real zeros of $f$ and $F$, which generalizes and unifies many known results, including the…

The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are…

Let $f \in S_k(\Gamma_0(N))$ be a newform, and let $r_f^{\pm}(X)$ denote its corresponding even and odd period polynomials. For sufficiently large level and weight, we show that the zeros of $r_f^{\pm}(X)$ all lie on the circle $|X| =…

Let $F$ be a function from $\mathbb{F}_{p^n}$ to itself and $\delta$ a positive integer. $F$ is called zero-difference $\delta$-balanced if the equation $F(x+a)-F(x)=0$ has exactly $\delta$ solutions for all non-zero $a\in\mathbb{F}_{p^n}$.…

We consider the problem of computing the partition function $\sum_x e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\Bbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$. In the case of a quadratic polynomial $f$,…

Let f(x) be a differentiable function on the real line R, and let P be a point not on the graph of f(x). Define the illumination index of P to be the number of distinct tangents to the graph of f which pass thru P. We prove that if f '' is…

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions…

In this paper we investigate the approximation of continuous functions on the Wasserstein space by smooth functions, with smoothness meant in the sense of Lions differentiability. In particular, in the case of a Lipschitz function we are…

We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices,…

It is known that if $f\colon {\mathbb R}^2 \to {\mathbb R}$ is a polynomial in each variable, then $f$ is a polynomial. We present generalizations of this fact, when ${\mathbb R}^2$ is replaced by $G\times H$, where $G$ and $H$ are…

Many interesting questions in arithmetic dynamics revolve, in one way or another, around the (local and/or global) reducibility behavior of iterates of a polynomial. We show that for very general families of integer polynomials $f$ (and,…

A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be…

We study the distribution of the zeros of functions of the form $f(s)=h(s) \pm h(2a-s)$, where $h(s)$ is a meromorphic function, real on the real line, $a$ a real number. One of our results establishes sufficient conditions under which all…