Related papers: $S$-Unit Equations and the Asymptotic Fermat Conje…

Here we give brief account of hermitian symplectic spaces, showing that they are intimately connected to symmetric as well as self-adjoint extensions of a symmetric operator. Furthermore we find an explicit parameterisation of the Lagrange…

The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded…

In this paper we introduce the notion of Shimura's period symbols over function fields in positive characteristic and establish their fundamental properties. We further formulate and prove a function field analogue of Shimura's conjecture…

In this work we explore the construction of abelian extensions of number fields with exactly one complex place using multivariate analytic functions in the spirit of Hilbert's 12th problem. To this end we study the special values of the…

Let $F$ be a totally real number field. Dasgupta conjectured an explicit $p$-adic analytic formula for the Gross-Stark units of $F$. In a later paper, Dasgupta-Spiess conjectured a cohomological formula for the principal minors and the…

This paper investigates asymptotic properties of multifractal products of random fields. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces $L_q.$ New results on the rate of…

Different viewpoints on the asymptotic expansion of Feynman diagrams are reviewed. The relations between the field theoretic and diagrammatic approaches are sketched. The focus is on problems with large masses or large external momenta.…

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the…

We study certain unitary CM fields whose Galois closure has Galois group $\operatorname{PSL}_2(\mathbb{F}_q) \times \mathbb{Z}/2\mathbb{Z}$. After investigating the CM types of these fields, we turn towards Colmez's conjectural formula on…

We study the asymptotics conjecture of Malle for dihedral groups $D_\ell$ of order $2\ell$, where $\ell$ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class…

Using a simple Asymptotically Strong N=1 Susy SU(2) Gauge theory coupled to a {\bf{5}}-plet Chiral superfield we demonstrate the plausibility of the `` truly minimal '' Asymptotically Strong Grand Unification scenario proposed by us…

In this paper, we introduce the notion of unit reducibility for number fields, that is, number fields in which all positive unary forms attain their nonzero minimum at a unit. Furthermore, we investigate the link between unit reducibility…

We give conjectures on the "asymptotic" behaviour of the Hilbert series of (quotients by) generic ideals in the exterior algebra, as the number of variables tend to infinity. Our conjectures are supported by extensive computer calculations.

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units.…

Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^\delta < |A| < |F|^{1-\delta}$ for some $\delta > 0$, then we prove the estimate $|A+A| + |A.A| \geq c(\delta) |A|^{1+\eps}$ for some $\eps = \eps(\delta) > 0$.…

Kummer's conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer's ratio under three scenarios: no Siegel zero, presence of Siegel zero and…

In this article, we provide a comprehensive analysis of the asymptotic behavior of Bell numbers, enhancing and unifying various results previously dispersed in the literature. We establish several explicit lower and upper bounds. The main…

We study the sums $$ S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor\right) $$ when $f$ is supported on $r$th powers with $r\geq 2$. This restriction allows us to give nontrivial estimates for one of the error terms in…

In this article, we present streamlined proofs of results of Ankeny, Artin, and Chowla concerning the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{p})$ for primes $p\equiv 1 \bmod{4}$ while providing a generalization of…

We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field…