Related papers: Divisor Divisibility Sequences on Tori

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the…

Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function,…

A divisibility relation on ultrafilters on the set $\mathbb{N}$ of natural numbers is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also…

Let $\Lambda = \{\lambda_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{\lambda_{k} \in \Lambda : | \lambda_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an…

Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate…

A linear code $C$ over $\mathbb{F}_q$ is called $\Delta$-divisible if the Hamming weights $\operatorname{wt}(c)$ of all codewords $c \in C$ are divisible by $\Delta$. The possible effective lengths of $q^r$-divisible codes have been…

Let $f=(f_1,\ldots,f_n)$ be a system of $n$ complex homogeneous polynomials in $n$ variables of degree $d$. We call $\lambda\in\mathbb{C}$ an eigenvalue of $f$ if there exists $v\in\mathbb{C}^n\backslash\{0\}$ with $f(v)=\lambda v$,…

We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random…

A natural first step in the classification of all `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ lies in understanding the commutant of the modular matrices $S$ and $T$. We begin this paper extending the work of…

We obtain estimates on the number $|\mathcal{A}_{\boldsymbol{\lambda}}|$ of elements on a linear family $\mathcal{A}$ of monic polynomials of $\mathbb{F}_q[T]$ of degree $n$ having factorization pattern…

To study a Dirichlet polynomial $f(s)=\frac{a_{m}}{m^{s}}+\cdots +\frac{a_{n}}{n^{s}}$ by regarding it as a multivariate polynomial via the canonical map $\phi$ sending $p_i^{-s}$ to an indeterminate $X_i$, with $p_i$ the $i$th prime…

A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side…

Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $\Delta(f_n)$ denote the discriminant of $f_n$, that is $\Delta(f_n) =…

We introduce the notion of a pre-sequence of matrix orthogonal polynomials to mean a sequence {F_n} of matrix orthogonal functions with respect to a weight function W, satisfying a three term recursion relation and such that det(F_0) is not…

A divisibility relation on ultrafilters is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also belongs to $\cal G$. After describing the…

For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},..., x_n^{\pm1}] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of…

Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $\varphi(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient…

Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under…

For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f) \geq 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*}…

A Fej\'er-Dirichlet lift is developed that turns divisor information at the integers into entire interpolants with explicit Dirichlet-series factorizations. For absolutely summable weights the lift interpolates $(a*1)(n)$ at each integer…