Related papers: Binary normal regular Hermitian lattices over imag…

We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials consatitute orthogonal families of vectors in ambient quaternionic $L^2$-spaces. Using these…

Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's…

An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form…

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…

In this paper we describe an algorithm for classifying orbits of vectors in Lorentzian lattices. The main point of this is that isomorphism classes of positive definite lattices in some genus often correspond to orbits of vectors in some…

In this paper, we introduce and study the Dirichlet series enumerating (proper) equivalence classes of full rank subforms/sublattices of a given quadratic form/lattice, focusing on the positive definite binary case. We obtain formulas…

For a positive integer $m$, a (positive definite integral) quadratic form is called primitively $m$-universal if it primitively represents all quadratic forms of rank $m$. It was proved in arXiv:2202.13573 that there are exactly $107$…

We give an overview of universal quadratic forms and lattices, focusing on the recent developments over the rings of integers in totally real number fields. In particular, we discuss indecomposable algebraic integers as one of the main…

Properties of Hermitian forms are used to investigate several natural questions from CR Geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the…

Let \mathbb{F}_q^{n+l} denote the (n+l)-dimensional singular linear space over a finite field \mathbb{F}_q. For a fixed integer m\leq\min\{n,l\}, denote by \mathcal{L}^m_o(\mathbb{F}_q^{n+l}) the set of all subspaces of type (t,t_1), where…

Hermitian cubic norm structures were recently introduced in order to study the class of skew-dimension one structurable algebras (which are typically only defined over fields of characteristic different from $2$ and $3$) over arbitrary…

For every positive integer k, it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in k arithmetic progressions. For k=1,…

Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two…

For a given odd integer $n>1$, we provide some families of imaginary quadratic number fields of the form $\mathbb{Q}(\sqrt{x^2-t^n})$ whose ideal class group has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$.

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and…

In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic…

We collect a number of striking recent results in a study of dimers on infinite regular bipartite lattices and also on regular bipartite graphs. We clearly separate rigorously proven results from conjectures. A primary goal is to show…

In this note we study an analogy between a positive definite quadratic form for elliptic curves over finite fields and a positive definite quadratic form for elliptic curves over the rational number field. A question is posed of which an…

We study a general class of PT-symmetric tridiagonal Hamiltonians with purely imaginary interaction terms in the quasi-hermitian representation of quantum mechanics. Our general Hamiltonian encompasses many previously studied lattice models…

For given real or complex $m \times n$ data matrices $X$, $Y$, we investigate when there is a matrix $A$ such that $AX = Y$, and $A$ is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex…