Related papers: Algebraic Characterizations of Angle Multisections…

Let $\mathbb{D}$ be a division ring, and let ${\mathbb{D}}^{m\times n}$ be the set of $m\times n$ matrices over $\mathbb{D}$. Two matrices $A,B\in {\mathbb{D}}^{m\times n}$ are adjacent if ${\rm rank}(A-B)=1$. By the adjacency,…

Let $t\ge 1$ be a given integer. Let ${\cal F}$ be a family of subsets of $[m]=\{1,2,\ldots,m\}$. Assume that for every pair of disjoint sets $S,T\subset [m]$ with $|S|=|T|=k$, there do not exist $2t$ sets in ${\cal F}$ where $t$ subsets of…

Let $F$ be a finite set of monomials of the same degree $d\geq 2$ in a polynomial ring $R=k[x_1,...,x_n]$ over an arbitrary field $k$. We give some necessary and/or sufficient conditions for the birationality of the ring extension…

For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a…

We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary…

We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a…

For a field $\mathbb{F}$, let $R(n, m)$ be the ring of invariant polynomials for the action of $\mathrm{SL}(n, \mathbb{F}) \times \mathrm{SL}(n, \mathbb{F})$ on tuples of matrices -- $(A, C)\in\mathrm{SL}(n, \mathbb{F}) \times…

We enumerate factorisations of the complete bipartite graph into spanning semiregular graphs in several cases, including when the degrees of all the factors except one or two are small. The resulting asymptotic behaviour is seen to…

Let ${\mathcal B}=\{b_i \}_{i=1}^\infty$ be a fixed sequence of pairwise distinct elements of a number field $k$. Given the integers $2\leq s \leq r$, assuming a quantitative version of Vojta's conjecture on the bounded degree algebraic…

We suggest a method of solving the problem of existence of a triangle with prescribed two bisectors and one third element which can be taken as one of the angles, the sides, the heights or the medians, or the third bisector.

Let $T(n,m)$ be the set of all plane labelled bipartite trees with $n$ white vertices and $m$ black. If the number $n+m$ of vertices is even, then the set $T(n,m)$ is a union of two disjoint subsets --- subset od "even" trees and subset of…

We address the question of existence of sections of fibrations in two settings. First, we show that a bundle with base a finite 2-complex admits a section if and only if the inclusion of the fiber is $\pi_1$-injective and the associated…

Let $K=\Q(\theta)$ be an algebraic number field with $\theta$ a root of an irreducible quadrinomial $f(x) = x^6+ax^m+bx+c\in\Z[x] $ with $m\in\{2,3,4,5\}$. In the present paper, we give some explicit conditions involving only $a,~b,~c$ and…

The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…

In this paper we consider the H\'enon problem in a ball. We prove the existence of (at least) one branch of nonradial solutions that bifurcate from the radial ones and that this branch is unbounded.

In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(\theta^n)$ with $\theta<e$ then $f$ is a polynomial. In this paper, we investigate the…

The eponymous theorem of P.L. Wantzel presents a necessary and sufficient criterion for angle trisectability in terms of the third Chebyshev polynomial $T_3$, thus making it easy to prove that there exist non-trisectable angles. We…

In this paper, motivated by a problem of Scott and a conjecture of Lee, Loh and Sudakov we consider bisections of directed graphs. We prove that every directed graph with $m$ arcs and minimum semidegree at least $d$ admits a bisection in…

Let $A$ be a finite-dimensional division algebra containing a base field $k$ in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if $A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that: (1) In…

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let $\mathsf{G}$ be a group,…