Related papers: Quadratic forms with a strong regularity property …

A Kaehler metric $g$ with integral Kaehler form is said to be partially regular if the partial Bergman kernel associated to mg is a positive constant for all integer m sufficiently large. The aim of this paper is to prove that for all n\geq…

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$…

Let $G$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p\geq 0$ and let $\phi$ be a $p$-restricted irreducible representation of $G$. Let $T$ be a maximal torus of $G$ and $s\in T$. We say…

The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every…

Let $f$ be a positive definite integral ternary quadratic form and let $r(k,f)$ be the number of representations of an integer $k$ by $f$. In this article we study the number of representations of squares by $f$. We say the genus of $f$,…

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive…

The goal of this note is to provide an analysis of the positive integers that are represented everywhere locally, but not globally, by each of the 29 spinor regular positive definite integral ternary quadratic forms that are not regular.

A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…

Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present…

Kaplansky conjectured that if two positive-definite real ternary quadratic forms have perfectly identical representations over $\mathbb{Z}$, they are constant multiples of regular forms, or is included in either of two families parametrized…

In 1995, Reznick showed an important variant of the obvious fact that any positive semidefinite (real) quadratic form is a sum of squares of linear forms: If a form (of arbitrary even degree) is positive definite then it becomes a sum of…

For all positive integers $k$ and $N$ we prove that there are infinitely many totally real multiquadratic fields $K$ of degree $2^k$ over $\mathbb Q$ such that each universal quadratic form over $K$ has at least $N$ variables.

We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…

In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…

Given an endomorphism u of a finite-dimensional vector space over an arbitrary field K, we give necessary and sufficient conditions for the existence of a regular quadratic form (resp. a symplectic form) for which u is orthogonal (resp.…

A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. Semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadatric…

For a (positive definite and integral) quadratic form $f$, a quadratic form is said to be {\it an isolation of $f$ from its proper subforms} if it represents all proper subforms of $f$, but not $f$ itself. It was proved that the minimal…

We obtain an explicit formula for the quadratic embedding constant (QEC) of a strongly regular graph $\mathrm{srg}(n,k,\lambda,\mu)$ with $\mu\ge1$. By using QEC we give a necessary and sufficient condition for a strongly regular graph to…

Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite…