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.

A lower bound for the variance of generalized divisor functions in arithmetic progressions

Abstract: We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $\alpha$-fold divisor function, for any complex number $\alpha ot\in \{1\}\cup-\mathbb{N}$, even when considering a sequence of parameters $\alpha$ close in a proper way to $1$. Our work builds on that of Harper and Soundararajan, who handled the particular case of $k$-fold divisor functions $d_k(n)$, with $k\in\mathbb{N}_{\geq 2}$.

Single-cycle modulus operation

Abstract: Several different approaches to performing the modulus operation are presented. In one, a method of performing the modulus operation upon a dividend and a divisor within a limited range is discussed. The method involves storing a reference value, receiving a dividend value, and calculating a number of derived inputs. Each of the derived inputs corresponds to the dividend value minus the reference value, and is then further modified by a multiple of the divisor. Using the divisor to select between these derived inputs provides the answer.

On Jacobian group arithmetic for typical divisors on curves

Abstract: In a previous joint article with Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of “typical” divisor classes on \(C_{3,4}\) curves, improving on similar results by other authors. At that time, we could only state that a general divisor was typical, and hence unlikely to be encountered if one implemented these algorithms over a very large finite field. This article pins down an explicit characterization of these typical divisors, for an arbitrary smooth projective curve of genus \(g \ge 1\) having at least one rational point. We give general algorithms for Jacobian group arithmetic with these typical divisors, and prove not only that the algorithms are correct if various divisors are typical, but also that the success of our algorithms provides a guarantee that the resulting output is correct and that the resulting input and/or output divisors are also typical. These results apply in particular to our earlier algorithms for \(C_{3,4}\) curves. As a byproduct, we obtain a further speedup of approximately 15% on our previous algorithms for \(C_{3,4}\) curves.

Polynomial Matrices, Splitting Subspaces and Krylov Subspaces over Finite Fields

Abstract: Let $T$ be a linear operator on an $\mathbb{F}_q$-vector space $V$ of dimension $n$. For any divisor $m$ of $n$, an $m$-dimensional subspace $W$ of $V$ is $T$-splitting if $$ V =W\oplus TW\oplus \cdots \oplus T^{d-1}W, $$ where $d=n/m$. Let $\sigma(m,d;T)$ denote the number of $m$-dimensional $T$-splitting subspaces. Determining $\sigma(m,d;T)$ for an arbitrary operator $T$ is an open problem. This problem is closely related to another open problem on Krylov spaces. We discuss this connection and give explicit formulae for $\sigma(m,d;T)$ in the case where the invariant factors of $T$ satisfy certain degree conditions. A connection with another enumeration problem on polynomial matrices is also discussed.