Related papers: Separating singular moduli and the primitive eleme…

Let $k$ be a field and consider the path algebra $kQ$ of the quiver $Q$. A pair of indecomposable $kQ$-modules $(Y,X)$ is called an orthogonal exceptional pair if the modules are exceptional and…

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and $\mathbb{K}[x,y]$ the polynomial ring. The group $\text{SL}_{2}\left(\mathbb{K}[x,y]\right)$ of all matrices with determinant equal to $1$ over $\mathbb{K}[x,y]$…

Let $q$ be an even prime power and $m\geq2$ an integer. By $\mathbb{F}_q$, we denote the finite field of order $q$ and by $\mathbb{F}_{q^m}$ its extension degree $m$. In this paper we investigate the existence of a primitive normal pair…

In this article, we study the multiparameter second quantum Weyl algebra at roots of unity. In this setting, the algebra is a polynomial identity (PI) algebra, and the dimension of its simple modules is bounded above by its PI degree. We…

It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $\Sigma$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). A corollary…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the…

In this note we establish existence of solutions of singular boundary value problem $-(p(x)y^{\prime }(x))^{\prime}=q(x)f(x,y,py')$ for $0< x\leq b$ and $y'(0)=0$, $\alpha_{1}y(b)+\beta_{1}p(b)y^{\prime}(b)=\gamma_{1}$ with $p(0)=0$ and…

In this article, we establish a sufficient condition for the existence of a primitive element $\alpha \in {\mathbb{F}_{q^n}}$ such that the element $\alpha+\alpha^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and…

Let $r$ be a positive divisor of $q-1$ and $f(x,y)$ a rational function of degree sum $d$ over $\mathbb{F}_q$ with some restrictions, where the degree sum of a rational function $f(x,y) = f_1(x,y)/f_2(x,y)$ is the sum of the degrees of…

We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter…

Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(\alpha)$. For any unit $\epsilon$ of $K$ such that ${\mathbf Q}(\alpha\epsilon)=K$, we consider…

We determine couples of singular moduli which have rational products

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…

A. Mukhopadhyay, M. R. Murty and K. Srinivas (http://arxiv.org/abs/0808.0418) have recently studied various arithmetic properties of the discriminant $\Delta_n(a,b)$ of the trinomial $f_{n,a,b}(t) = t^n + at + b$, where $n \ge 5$ is a fixed…

Let Q denote the field of rational numbers. Let F \subseteq R is a euclidean field. We prove that: (1) if x,y \in F^n (n>1) and |x-y| is constructible by means of ruler and compass then there exists a finite set S(x,y) \subseteq F^n…

Let $x$ and $n$ be positive integers. We prove a non-trivial lower bound for $x$, dependant only on $\omega_n$, the number of distinct prime factors of $x^n-1$. By considering the divisibility of $\varphi \mid x^n-1$ for $\varphi \mid n$,…

In this paper we generalize the results of Sharma, Awasthi and Gupta (see \cite{SAG}). We work over a field of any characteristic with $q = p^k$ elements and we give a sufficient condition for the existence of a primitive element $\alpha…

For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb…

We consider and completely solve the parametrized family of Thue equations \begin{eqnarray*}X(X-Y)(X+Y)(X-\lambda Y)+Y^4=\xi,\end{eqnarray*} where the solutions $x,y$ come from the ring $\mathbb{C}[T]$, the parameter…

Let $\mathbb{F}_q$ be the finite field of $q$ elements, and let $k\mid q-1$ be a positive integer. Let $f(x)=ax^2+bx+c$ be a quadratic polynomial in $\mathbb{F}_q[x]$ with $b^2-4ac\ne0$. In this paper, we show that if…