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.
Papers published on a yearly basis
Papers
More filters
TL;DR: In this paper, the authors established a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp ( n, Z ) which satisfy X 2 + I = 0 (resp. X 2+ X + X + I) and the set for hermitian forms of rank n over Z [√−1] (resp.
1 citations
01 Jan 2013
TL;DR: In this paper, the coefficients for the Weierstrass}(x) and } 00 (x)(x = 1, �, �+1 2 )-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ((23)).
Abstract: In this paper, we find the coefficients for the Weierstrass}(x) and } 00 (x)(x = 1 , � , �+1 2 )-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ((23)). Finally, we reprove congruences for the functions µ(n) and �(n) in Hahn's article (11, Theorems 6.1 and 6.2).
1 citations
Journal Article•
TL;DR: In this article, the irreducible components of an algebraically closed field of characteristic zero are associated with a pair of numerical data, where Ni and vi 1 are the multiplicities of Ei in the divisor of respectively f o h and h*(dx A dy) on X.
Abstract: Introduction. Let k be an algebraically closed field of characteristic zero and f(x, y) E k[x, y]. Let (X, h) be an embedded resolution of f = 0 in the affine plane A2, constructed by successive blowing-ups, and denote by Ei, i E I, the irreducible components of h-'(f '{0}). We associate to each Ei, i E I, a pair of numerical data (N,, vi), where Ni and vi 1 are the multiplicities of Ei in the divisor of respectively f o h and h*(dx A dy) on X. Fix one exceptional curve E with numerical data (N, v) and say E intersects k times another irreducible component. Denote these components by E1, . . ., Ek. Then we have
1 citations
Journal Article•
TL;DR: In this article, the problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms.
Abstract: The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the art by focusing on prime degree polynomials; let n be the degree. If (n−1) has a ‘large’ r-smooth divisor s, then we find a nontrivial factor of f(x) in deterministic poly(n^r, log q) time, assuming GRH and that s=Ω(√(n/2^r)). Thus, for r=O(1) our algorithm is polynomial time. Further, for r=Ω(loglog n) there are infinitely many prime degrees n for which our algorithm is applicable and better than the best known, assuming GRH. Our methods build on the algebraic-combinatorial framework of m-schemes initiated by Ivanyos, Karpinski and Saxena (ISSAC 2009). We show that the m-scheme on n points, implicitly appearing in our factoring algorithm, has an exceptional structure, leading us to the improved time complexity. Our structure theorem proves the existence of small intersection numbers in any association scheme that has many relations, and roughly equal valencies and indistinguishing numbers.
1 citations
TL;DR: There is a permutation $\sigma\in S_{n-1}$ such that all the elements $sa_{\sigma(s)}\ (s=1,\ldots, n-1)$ are nonzero if and only if the group G is the cyclic group Z/n Z.
Abstract: Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}$ be elements of $G$. We show that there is a permutation $\sigma\in S_{n-1}$ such that all the elements $sa_{\sigma(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only if $$\left|\left\{1\leqslant s
1 citations