Related papers: Universal quadratic forms and indecomposables in n…

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit upper bound for the rank of universal quadratic forms over a given number field $K$, under the assumption…

The problem of writing a totally positive element as a sum of squares has a long history in mathematics, going back to Bachet and Lagrange. While for some specific rings (like integers or polynomials over the rationals), there are known…

Let $ n $ be an integer and $ n\ge 2 $. A classic integral quadratic form over local fields is called classic $ n $-universal if it represents all $n$-ary classic integral quadratic forms. We determine the equivalent conditions and minimal…

This paper describes the centers of the universal enveloping algebras and the invariant rings of the standard filiform Lie algebras over fields of characteristic zero and also over large enough prime characteristic. We determine explicit…

We study the distribution of principal ideals generated by irreducible elements in an algebraic number field.

We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of…

In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.

In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to…

We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible…

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\mathbb Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of…

In this article, we introduce a new object, a virtual quadratic space, and its group of isometries. They are presented as natural generalizations of quadratic spaces and orthogonal groups. It is then shown that by replacing quadratic spaces…

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…

We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras…

Faithful representations of regular $\ast$-rings and modular complemented lattices with involution within orthosymmetric sesquilinear spaces are studied within the framework of Universal Algebra. In particular, the correspondence between…

We study totally positive, additively indecomposable integers in a real quadratic field $\mathbb Q(\sqrt D)$. We estimate the size of the norm of an indecomposable integer by expressing it as a power series in $u_i^{-1}$, where $\sqrt D$…

We prove that arithmetic is interpretable in any indecomposable polynomial ring (in any set of variables), and in addition we provide an alternative uniform proof of undecidability for all members in this class of rings.

Quaternions, split quaternions, and hybrid numbers are very well-known number systems. These number systems are used to make geometry in Euclidean and Lorentz spaces. These number systems can be obtained with the help of a quadratic form.…

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the…

We address some questions concerning indecomposable polynomials and their spectrum. How does the spectrum behave via reduction or specialization, or via a more general ring morphism? Are the indecomposability properties equivalent over a…

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than…