Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

On the projective normality of double coverings over a rational surface

Abstract: We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $\pi$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+r\pi^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.

Integer division circuit and method of performing integer division in hardware

Abstract: An integer division circuit includes a first bit-shifting circuit configured to shift the bits of a dividend to produce a normalized dividend. The circuit also includes a second bit-shifting circuit configured to shift the bits of a divisor to produce a normalized divisor. A divider tree circuit is configured to divide the normalized dividend by the normalized divisor to produce a normalized quotient. The circuit further includes a third bit-shifting circuit configured to shift the bits of the normalized quotient to produce a quotient.

Integral points of bounded degree on the projective line and in dynamical orbits

Abstract: Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $\varphi(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $\varphi^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.

The behavior of ds-divisors of positive integers

Abstract: We study the behavior of DS-divisors of positive integers. Here "DS" stands for "divisor-squared." For an integer n, a positive divisor q of n is called a DS-divisor if q 2 j n − q. Such a pair (n,q) is called a DS-pair. Using a table generated for DS-pairs, we examine the existence and the numbers of positive DS-divisors of prime powers, products of two prime powers, and other cases represented by primary factorization. We also investigate patterns and structures of DS-divisors derived from our observations of the table. In addition, we study relationships between the numbers of DS-divisors and the values of Euler function. This research is related to the Primality Test problem of positive integers.

Divisors of Modular Parameterizations of Elliptic Curves

Abstract: The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$ there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is the conductor of $E$. These maps may be expressed in terms of pairs of modular functions $X(z)$ and $Y(z)$ where $X(z)$ and $Y(z)$ satisfy the Weierstrass equation for $E$ as well as a certain differential equation. Using these two relations, a recursive algorithm can be used to calculate the $q$ - expansions of these parametrizations at any cusp. %These functions are algebraic over $\mathbb{Q}(j(z))$ and satisfy modular polynomials where each of the coefficient functions are rational functions in $j(z)$. Using these functions, we determine the divisor of the parametrization and the preimage of rational points on $E$. We give a sufficient condition for when these preimages correspond to CM points on $X_0(N)$. We also examine a connection between the algebras generated by these functions for related elliptic curves, and describe sufficient conditions to determine congruences in the $q$-expansions of these objects.