Related papers: Sum of two squares in cyclic quartic fields

A \emph{magic square} is an $n \times n$ array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for $n=4$, as a function of either the magic sum or an…

Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series $\sum_K|\Disc(K)|^{-s}$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel…

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, \delta \in \mathcal{O}_K^\times$ satisfying $\varepsilon + \delta = n$. We…

Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using…

We apply the circle method to obtain an asymptotic formula for the number of integral points on a certain sliced cubic hypersurface related to the Segre cubic. Unusually, the major and minor arc integrals in this application are both…

The aim of this note is to show that the generalized supertrace, constructed in another paper of the author, inducing an isomorphism between the Hochschild homology of a superalgebra and that of the superalgebra of square supermatrices of a…

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. This paper mainly researches the polynomial representation of double cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^2\mathbb{F}_{q}$ with $v^3=v$. Firstly, we give the…

We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version…

Sectors at centre of affine quadrics with point symmetry are investigated over arbitrary fields of characteristic different from two. As an application we demonstrate nice formulas for the area and the volume of such planar and spatial…

In paper this paper it is considered the summation problem for trigonometric integrals with quadratic phase. This problem considered in the papers \cite{Chub},\cite{Chax},\cite{Jabbar} in particular cases. Our results generalized the…

We give an estimate of exponential sums over singular binary quintic forms in a characteristic-free form, based on the Waring decomposition of binary forms. This extends the method on our preceding result on the space of binary quartics to…

The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form.

This paper presents a method for computing two-dimensional constant mean curvature surfaces. The method in question uses the variational aspect of the problem to implement an efficient algorithm. In principle it is a flow like method in…

We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues…

Let us consider a cyclic extension of a function field defined over a finite field. For a character (non-trivial) of this extension, we calculate, as a linear combinations of products of Jacobi sums, the coefficients of the polynomial given…

By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x\not\equiv y…

Spherical field theory is a new non-perturbative method for studying quantum field theories. It uses the spherical partial wave expansion to reduce a general d-dimensional Euclidean field theory into a set of coupled one-dimensional…

We obtain a new classification of the finite metacyclic group in terms of group invariants. We present an algorithm to compute these invariants, and hence to decide if two given finite metacyclic groups are isomorphic, and another algorithm…

A curious number is a palindromic number whose base ten representation has the form $a \ldots a b \ldots b a \ldots a$. In this paper, we determine all curious numbers that are perfect squares. Our proof involves reducing the search for…

In Euclidean geometry, the Pythagorean theorem is presented as an equation involving three squares. This paper explores how analogous expressions may be identified in spherical and hyperbolic geometries.