Topic
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.
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13 Dec 2008
TL;DR: This paper demonstrates a method to successfully attack on RSA cryptosystem algorithm factorization of large integers using matrices and shows that the method can be generalized.
Abstract: Factorization of large integers gives a method to successfully attack on RSA cryptosystem algorithm. Williams p+1 gives us such algorithm to factorize the integer n; if there exists a prime divisor p, such that p+1 will have only a small prime divisors. In this paper we demonstrate this algorithm using matrices and show that the method can be generalized.
TL;DR: In this article, the Betti numbers of the Hilbert scheme of points of a smooth divisor on a non-singular surface were computed relative to a line in the Euclidean space.
Abstract: Let $D$ be a smooth divisor on a non singular surface $S$. We compute the Betti numbers of the Hilbert scheme of points of $S$ relative to $D$. In the case of $\mathbb{P}^2$ and a line in it, we give an explicit set of generators and relations for the corresponding cohomology groups.
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TL;DR: In this article, it was shown that if an abelian variety admits an embedding into the Jacobian of a smooth projective curve, then the embedding induces an injective push-forward homomorphism at the level of Chow groups.
Abstract: In this text we prove that if an abelian variety $A$ admits of an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $\Th_A$ to be the divisor $\Th_C\cap A$, where $\Th_C$ denotes the theta divisor of $J(C)$, then the embedding of $\Th_A$ into $A$ induces an injective push-forward homomorphism at the level of Chow groups We show that this is the case for every principally polarized abelian varieties
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TL;DR: In this article, the Riemann-Roch inequality holds for smooth tropical toric surfaces, where the value of a divisor $D$ on a tropical variety $X$ is estimated by terms of global sections and computed more easily than $h^{0}(X,D)
Abstract: For a divisor $D$ on a tropical variety $X$, we define two amounts in order to estimate the value of $h^{0}(X,D)$, which are described by terms of global sections and computed more easily than $h^{0}(X,D)$. As an application of its estimation, we show that a Riemann-Roch inequality holds for smooth tropical toric surfaces.
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TL;DR: In this paper, the authors considered the problem of divisibility of a given element of the exterior product of a free module over a commutative unital ring and gave sufficient and necessary conditions for such divisability.
Abstract: Let $M$ be a free module of rank $m$ over a commutative unital ring $R$ and let $N$ be its free submodule. We consider the problem of when a given element of the exterior product $\Lambda^pM$ is divisible, in a sense, over elements of the exterior product $\Lambda^r N$, where $r\le p$. Precisely, we give conditions under which a given $\eta\in\Lambda^pM$ can be expressed as a finite sum of elements of $\Lambda^r N$ multiplied (via the exterior product) by elements of $\Lambda^{p-r} M$. Necessary and sufficient conditions for such divisibility take a simple form, provided that the submodule is embedded in $M$ with singularities having the depth larger then $p-r+1$. In the special case where $r=rank N$ the divisibility property means that $\eta=\Omega\wedge\gamma$ where $\Omega$ is the product $\omega_1\wedge\cdots\wedge\omega_r$ of elements of a basis of $N$ and $\gamma$ is an element of $\Lambda^{p-r}M$.
More detailed statements of these results are then used to state criteria for existence and uniqueness of algebraic residua when the "divisor" is defined by elements $f_1,\dots,f_k\in R$. Special cases are multidimensional logarithmic residua in complex analysis.