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.

Lattices over Polynomial Rings and Applications to Function Fields

Abstract: This paper deals with lattices $(L,\Vert~\Vert)$ over polynomial rings, where $L$ is a finitely generated module over $k[t]$, the polynomial ring over the field $k$ in the indeterminate $t$, and $\Vert~\Vert$ is a discrete real-valued length function on $L\otimes_{k[t]}k(t)$. A reduced basis of $(L,\Vert~\Vert)$ is a basis of $L$ whose vectors attain the successive minima of $(L,\Vert~\Vert)$. We develop an algorithm which transforms any basis of $L$ into a reduced basis of $(L,\Vert~\Vert)$. By identifying a divisor $D$ of an algebraic function field with a lattice $(L,\Vert~\Vert)$ over a polynomial ring, this reduction algorithm can be addressed to the computation of the Riemann-Roch space of $D$ and the successive minima of $(L,\Vert~\Vert)$, without the use of any series expansion.

Combinatorial identities and Titchmarsh's divisor problem for multiplicative functions

Abstract: Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|

An improved sieve of Eratosthenes

Abstract: We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and essentially linear time. These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and essentially linear time. We can also apply our sieve to any subinterval of [1,N] of length O(N^{1/3} (log N)^{2/3}) in time essentially linear on the length of the interval. Before, such a thing was possible only for subintervals of [1,N] of length >> sqrt{N}. Just as in (Galway, 2000), our approach is related to Diophantine approximation, and also has close ties to Voronoi's work on the Dirichlet divisor problem. The advantage of the method here resides in the fact that, because the method we will give is based on the sieve of Eratosthenes, we will also be able to use it to factor integers, and not just to produce lists of consecutive primes.

A mirror theorem for multi-root stacks and applications.

Abstract: Given a smooth projective variety $X$ with a simple normal crossing divisor $D:=D_1+D_2+...+D_n$, where $D_i\subset X$ are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $X_{D,\vec r}$ by constructing an $I$-function, a slice of Givental's Lagrangian cone for Gromov--Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of $X_{D,\vec r}$ stabilize for sufficiently large $\vec r$. (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau--Ginzburg potentials using orbifold invariants of $X_{D,\vec r}$.

Finite Groups with Four Conjugacy Class Sizes

Abstract: We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and that the set of class sizes is exactly {1, m, n, mk}, where m, n > 1 are coprime numbers and k > 1 is a divisor of n.