相关论文: Geometric elements and classification of quadrics …

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of…

We give an upper bound for the number of rational points of height at most $B$, lying on a surface defined by a quadratic form $Q$. The bound shows an explicit dependence on $Q$. It is optimal with respect to $B$, and is also optimal for…

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…

We investigate factorizability of a quadratic split quaternion polynomial. In addition to inequality conditions for existence of such factorization, we provide lucid geometric interpretations in the projective space over the split…

We present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms. Beginning from basic results developed by Riehm, our solution to this problem hinges on the classification of…

We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors. We demonstrate the effectiveness of the approach by proving of a number of integral identities with vector integrands.

We show that the shape of a complex cubic field lies on the geodesic of the modular surface defined by the field's trace-zero form. We also prove a general such statement for all orders in \'etale Q-algebras. Applying a method of Manjul…

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg…

In this paper, we examine a ciphertext on the basis of using geometrical objects. Each symbol normative alphabet is determined as a point on the plane. We consider possible ways for presentation of these points.

Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it…

In this paper, the concept of the metric matrix is introduced to establish a concise and unified formulation for the inner product in barycentric coordinates. Building on this framework, we explore the intrinsic algebraic identities of…

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine Coxeter groups W(D3) and W(B3)=Aut(D3). The rank-3 Coxeter-Weyl groups describing the…

It is well known that every non-degenerate quadratic form admits a decomposition into an orthogonal sum of its anisotropic part and a hyperbolic form. This decomposition is unique up to isometry. In this paper we present an algorithm for…

A unifying overview of the ways to parameterize the linear group GL(4.C) and its subgroups is given. As parameters for this group there are taken 16 coefficients G = G(A,B,A_{k}, B_{k}, F_{kl}) in resolving matrix G in terms of 16 basic…

Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost…

The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are…

Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in $\mathbb{R}^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in…

Barycentric coordinates provide solutions to the problem of expressing an element of a compact convex set as a convex combination of a finite number of extreme points of the set. They have been studied widely within the geometric…

We investigate the Hasse principle for complete intersections cut out by a quadric and cubic hypersurface defined over the rational numbers.

Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over the finite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$. The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ with respect…