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.

Efficient reduction of large divisors on hyperelliptic curves

Abstract: We present an algorithm for reducing a divisor on a hyperelliptic curve of arbitrary genus over any finite field. Our method is an adaptation of a procedure for reducing ideals in quadratic number fields due to Jacobson, Sawilla and Williams, and shares common elements with both the Cantor and the NUCOMP algorithms for divisor arithmetic. Our technique is especially suitable for the rapid reduction of a divisor with very large Mumford coefficients, obtained for example through an efficient tupling technique. Results of numerical experiments are presented, showing that our algorithm is superior to the standard reduction algorithm in many cases.

The shifted convolution of generalized divisor functions

Abstract: We prove an asymptotic formula for the shifted convolution of the divisor functions \(d_k(n)\) and \(d(n)\) with \(k \geq 3\), which is uniform in the shift parameter and which has a power saving error term. Along the way, we also consider a few other variants of the shifted convolution problem.

Connectivity of Circulant Graphs

Abstract: In this paper, an explicit expression is derived for the connectivity of a connected circulant graph whose connectivity is less than its point degree, that is k(G) = :m is a proper divisor of n and.

Lang-Vojta Conjecture over function fields for surfaces dominating $\mathbb{G}_m^2$

Abstract: We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $\mathbb{G}_m^2$. This extends results of Corvaja and Zannier, who proved the conjecture in the split case, and results of Corvaja and Zannier and the second author that were obtained in the case of the complement of a degree four and three component divisor in $\mathbb{P}^2$. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

Strongly real classes in finite unitary groups of odd characteristic

Abstract: We classify all strongly real conjugacy classes of the finite unitary group $\U(n, F_q)$ when $q$ is odd. In particular, we show that $g \in \U(n, F_q)$ is strongly real if and only if $g$ is an element of some embedded orthogonal group $O^{\pm}(n, F_q)$. Equivalently, $g$ is strongly real in $\U(n, F_q)$ if and only if $g$ is real and every elementary divisor of $g$ of the form $(t \pm 1)^{2m}$ has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group $\Sp(2n, F_q)$, $q$ odd, and a generating function for the number of strongly real classes in $\U(n, F_q)$, $q$ odd, and we also give partial results on strongly real classes in $\U(n, F_q)$ when $q$ is even.