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.

On some analogues of Titchmarsh divisor problem

Abstract: In [15] Titchmarsh posed and solved under the generalized Riemann Hypothesis, the problem of an asymptotic behavior of the number of the solutions of the equation 1 = p — n 1 n 2 for a prime p ≦ x and natural numbers n 1 and n 2 . When we put then the above problem is to get an asymptotic law for the sum

On deficient-perfect numbers

Abstract: For a positive integer \(n\), let \(\sigma(n)\) denote the sum of the positive divisors of \(n\). Let \(d\) be a proper divisor of \(n\), we call \(n\) a deficient-perfect number if \(\sigma(n) = 2n - d\). In this paper, we show that there is no odd deficient-perfect number with three distinct prime divisors. DOI: 10.1017/S0004972714000082

Explicit Rationality of Some Special Fano Fourfolds

Abstract: Recent results of Hassett, Kuznetsov and others pointed out countably many divisors \( \mathcal{C}_{d} \) in the open subset of \( \mathbb{P}^{55}= \mathbb{P}(\textit{H}^{o}(\mathcal{O}_{\mathbb{P}^{5}}(3))) \) parametrizing all cubic 4-folds and lead to the conjecture that the cubics corresponding to these divisors should be precisely the rational ones. Rationality has been proved by Fano for the first divisor \( \mathcal{C}_{14} \), in [RS19a] for the divisors \( \mathcal{C}_{26} \) and \( \mathcal{C}_{38} \), and in [RS19b] for \( \mathcal{C}_{42} \). In this note we describe explicit birational maps from a general cubic fourfold in \( \mathcal{C}_{14} \), in \( \mathcal{C}_{26} \) and in \( \mathcal{C}_{38} \) to P4, providing concrete geometric realizations of the more abstract constructions in [RS19a] and of the theoretical framework developed in [RS19b]. We also exhibit an explicit relationship between the divisor C14 and a certain divisor in the open subset of \( \mathbb{P}^{39}= \mathbb{P}(\textit{H}^{o}(\mathcal{O}_{Y}(2))) \) parametrizing smooth quadratic sections of a del Pezzo fivefold \( \textit{Y}=\mathbb{G}(1,4)\cap\mathbb{P}^{8}\subset\mathbb{P}^{8} \), the so-called Gushel–Mukai fourfolds.

Hopf-Galois Structures Arising From Groups with Unique Subgroup of Order p

Abstract: For $\Gamma$ a group of order $mp$ for $p$ prime where $gcd(p,m)=1$, we consider those regular subgroups $N\leq Perm(\Gamma)$ normalized by $\lambda(\Gamma)$, the left regular representation of $\Gamma$. These subgroups are in one-to-one correspondence with the Hopf-Galois structures on separable field extensions $L/K$ with $\Gamma=Gal(L/K)$. This is a follow up to the author's earlier work where, by assuming $p>m$, one has that all such $N$ lie within the normalizer of the $p$-Sylow subgroup of $\lambda(\Gamma)$. Here we show that one only need assume that all groups of a given order $mp$ have a unique $p$-Sylow subgroup, and that $p$ not be a divisor of the automorphism groups of any group of order $m$. As such, we extend the applicability of the program for computing these regular subgroups $N$ and concordantly the corresponding Hopf-Galois structures on separable extensions of degree $mp$.

Combinatorially Determined Zeroes of Bernstein--Sato Ideals for Tame and Free Arrangements

Abstract: For a central, not necessarily reduced, hyperplane arrangement $f$ equipped with any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing $f$, we consider a more general type of Bernstein--Sato ideal consisting of the polynomials $B(S) \in \mathbb{C}[s_{1}, \dots, s_{r}]$ satisfying the functional equation $B(S) f^{\prime} f_{1}^{s_{1}} \cdots f_{r}^{s_{r}} \in \text{A}_{n}(\mathbb{C})[s_{1}, \dots, s_{r}] f_{1}^{s_{1} + 1} \cdots f_{r}^{s_{r} + 1}$ Generalizing techniques due to Maisonobe, we compute the zero locus of the standard Bernstein--Sato ideal in the sense of Budur (ie $f^{\prime} = 1)$ for any factorization of a free and reduced $f$ and for certain factorizations of a non-reduced $f$ We also compute the roots of the Bernstein--Sato polynomial for any power of a free and reduced arrangement If $f$ is tame, we give a combinatorial formula for the roots lying in $[-1,0)$ For $f^{\prime} eq 1$ and any factorization of a line arrangement, we compute the zero locus of this ideal For free and reduced arrangements of larger rank, we compute the zero locus provided $\text{deg}(f^{\prime}) \leq 4$ and give good estimates otherwise Along the way we generalize a duality formula for $\mathscr{D}_{X,\mathfrak{x}}[S]f^{\prime}f_{1}^{s_{1}} \cdots f_{r}^{s_{r}}$ that was first proved by Narvaez-Macarro for $f$ reduced, $f^{\prime} = 1$, and $r = 1$ As an application, we investigate the minimum number of hyperplanes one must add to a tame $f$ so that the resulting arrangement is free This notion of freeing a divisor has been explicitly studied by Mond and Schulze, albeit not for hyperplane arrangements We show that small roots of the Bernstein--Sato polynomial of $f$ can force lower bounds for this number