Related papers: Hermite reduction and a Waring's problem for integ…

For each positive integer $n$, let $g_{\mathbb Z}(n)$ be the smallest integer such that if an integral quadratic form in $n$ variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of…

We improve the bound of the $g$-invariant of the ring of integers of a totally real number field, where the $g$-invariant $g(r)$ is the smallest number of squares of linear forms in $r$ variables that is required to represent all the…

The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on…

The Korkine-Zolotareff (KZ) reduction is a widely used lattice reduction strategy in communications and cryptography. The Hermite constant, which is a vital constant of lattice, has many applications, such as bounding the length of the…

We employ the slice spectral sequence, the motivic Steenrod algebra, and Voevodsky's solutions of the Milnor and Bloch-Kato conjectures to calculate the hermitian $K$-groups of rings of integers in number fields. Moreover, we relate the…

The Korkine--Zolotareff (KZ) reduction, and its generalisations, are widely used lattice reduction strategies in communications and cryptography. The KZ constant and Schnorr's constant were defined by Schnorr in 1987. The KZ constant can be…

In this paper, we study the unary Hermitian lattices over imaginary quadratic fields. Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of…

We present a variation of the modular algorithm for computing the Hermite normal form of an $\mathcal O_K$-module presented by Cohen, where $\mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996)…

We introduce the projective Hermite constant for positive definite binary hermitian forms associated with an imaginary quadratic number field $K$. It is a lower bound for the classical Hermite constant, and these two constants coincide when…

For each positive integer $n$, let $g_\Delta(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd…

Let $E=\mathbb{Q}\big(\sqrt{-d}\big)$ be an imaginary quadratic field for a square-free positive integer $d$, and let $\mathcal{O}$ be its ring of integers. For each positive integer $m$, let $I_m$ be the free Hermitian lattice over…

For a symmetric $R$-space $K/L=G/P$ the standard intertwining operators provide a canonical $G$-invariant pairing between sections of line bundles over $G/P$ and its opposite $G/\overline{P}$. Twisting this pairing with an involution of $G$…

We compute low-dimensional K-groups of certain rings associated with the study of the Hermite ring conjecture. This includes a monoid ring whose low-dimensional K-groups were recently computed by Krishna and Sarwar in the case where the…

The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity relating the number of $\mathrm{GL}_2 \mathbb{Z}$-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants $D$, $-27D$; it…

We study the number of real zeros of finite combinations of $K+1$ consecutive normalized Hermite polynomials of the form $$ q_n(x)=\sum_{j=0}^K\gamma_j\tilde H_{n-j}(x),\quad n\ge K, $$ where $\gamma_j$, $j=0,\dots ,K$, are real numbers…

We present a theory of reduction of binary quadratic forms with coefficients in Z[lambda], where lambda is the minimal translation in a Hecke group. We generalize from the modular group Gamma(1) = SL(2,Z) to the Hecke groups and make…

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 Hurwitz--Radon number originates in the composition problem of quadratic forms and is related to the maximum number of pointwise linearly independent vector fields on spheres. Kannaka--Tojo [arXiv:2602.04544] reformulated the…

Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the…

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree $n$ binary form. A survey of a…