Related papers: On classic $n$-universal quadratic forms over dyad…

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form.

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 study quadratic forms over totally real number fields by using an associated ring of quaternions. We examine some properties of residue class rings of these quaternions and use geometry of numbers to prove that certain ideals of the ring…

A positive definite even Hermitian lattice is called \emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $\Q{-m}$ for…

For quadratic spaces which represent 1 there is a characterization of hermitian compositions in the language of algebras-with-involutions using the even Clifford algebra. We extend this notion to define a generalized composition based on…

We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be 3. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar…

Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. We construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical…

An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we provide a characterization of almost universal ternary quadratic…

In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.

Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we show that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if $\gcd(2x,y)=1$ and $y>C$ where…

We prove that two general ternary forms are simultaneously identifiable only in the classical cases of two quadratic and a cubic and a quadratic form. We translate the problem into the study of a certain linear system on a projective bundle…

We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. One can find more details about the content of present paper in Extended Abstract.

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity…

We provide a concise approach to generalized dilaton theories with and without torsion and coupling to Yang-Mills fields. Transformations on the space of fields are used to trivialize the field equations locally. In this way their solution…

In this paper, we study the representations of integral quadratic polynomials. Particularly, it is shown that there are only finitely many equivalence classes of positive ternary universal integral quadratic polynomials, and that there are…

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic…

In this paper, we present a complete classification of all imaginary $n$-quadratic fields of class number 1.

The paper presents a classification of quadratic extension algebras, also known as algebras of degree 2, as well as several characterizations of quaternion algebras over a field (of characteristic not 2). The presentation is not restricted…

In this paper, we investigate the conditions under which a diagonal quadratic form $\sum_{i=1}^{m}a_i X_i^2$ represents every $n \times n$ integral matrix, where $a_i$ ($1 \leq i \leq m$) are integers. For $n=2$, we give a necessary and…

We study two-dimensional classically integrable field theory with independent boundary condition on each end, and obtain three possible generating functions for integrals of motion when this model is an ultralocal one. Classically…