Related papers: Sums of Integral Squares In Certain Complex Bi-qua…

Let K = Q(\omega) with \omega^3 = m be a pure cubic number field. We show that the elements\alpha \in K^\times whose squares have the form a - \omega form a group isomorphic to the group of rational points on the elliptic curve E_m: y^2=…

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all…

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…

Let S be a fixed symmetric finite subset of SL_d(O_K) that generates a Zariski dense subgroup of SL_d(O_K) when we consider it as an algebraic group over Q by restriction of scalars. We prove that the Cayley graphs of SL_d(O_K/I) with…

In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We…

Sums-of-squares formulas over the integers have been studied extensively using their equivalence to consistently signed intercalate matrices. This representation, combined with combinatorial arguments, has been used to produce…

Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f=\sum_{i=1}^r f_i^2$ where $\mathrm{deg} f_i=d/2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having…

Let $S_{g}$ denote the genus $g$ closed orientable surface. For $k\in \mathbb{N}$, a $k$-system is a collection of pairwise non-homotopic simple closed curves such that no two intersect more than $k$ times. Juvan-Malni\v{c}-Mohar…

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…

A positive square-free integer is called a \textit{congruent number} if it arises as the area of a right triangle with rational side lengths. Let $ n = p_1p_2 \cdots p_t q $ be a square-free integer, where each $ p_i \equiv 1 \pmod{8} $ and…

We prove that (under the assumption of the generalized Riemann hypothesis) a totally real multiquadratic number field $K$ has a positive density of primes $p \in \mathbb{Z}$ for which the image of the unit group $(\mathcal{O}_K)^{\times})$…

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by…

In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such a matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most…

Let $\mathfrak o$ be the ring of integers of a totally real number field. If $f$ is a quadratic form over $\mathfrak o$ and $g$ is another quadratic form over $\mathfrak o$ which represents all proper subforms of $f$, does $g$ represent…

This paper gives an algorithm to determine whether a number in a biquadratic field is a sum of two squares, based on local-global principle of isotropy of quadratic forms.

Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers…

We demonstrate the existence of $K$-multimagic squares of order $N$ consisting of distinct integers whenever $N>2 K(K+1)$. This improves upon our earlier result in which we only required $N+1$ distinct integers. Additionally, we present a…

We study the problem of how many different sums of squares decompositions a general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and the…

We prove a result about the non-existence of certain sums-of-squares formulas over a field. This generalizes an old theorem which used topological K-theory to obtain obstruction conditions when the field is the real numbers. Our result…

We associate with any simplicial complex $\K$ and any integer $m$ a system of linear equations and inequalities. If $\K$ has a simplicial embedding in $\R^m$ then the system has an integer solution. This result extends the work of I. Novik…