Related papers: On presentations of integer polynomial points of s…

Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a finite field extension of $K$ defined by a monic irreducible polynomial $f(x)\in R[x]$. We give an easy version of Dedekind's criterion which computationally improves…

Let A be a basic connected finite dimensional algebra over a field k and let Q be the ordinary quiver of A. To any presentation of A with Q and admissible relations, R. Martinez-Villa and J. A. de La Pena have associated a group called the…

Let $G$ be an infinite simple group of finite Morley rank and of Pr\"{u}fer $2$-rank $1$ which admits a supertight automorphism $\alpha$ such that the fixed-point subgroup $C_G(\alpha^n)$ is pseudofinite for all integers $n > 0$. We prove…

Let $G$ be a finite group and let $k$ be a sufficiently large finite field. Let $R(G)$ denote the character ring of $G$ (i.e. the Grothendieck ring of the category of ${\mathbb{C}}G$-modules). We study the structure and the representations…

Let $R$ be a finite-dimensional algebra over an algebraically closed field $F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite…

We classify the finite-dimensional rational representations $V$ of the exceptional algebraic groups $G$ with $\mathfrak g={\sf Lie}(G)$ such that the symmetric invariants of the semi-direct product $\mathfrak g\ltimes V$, where $V$ is an…

All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…

Let K be an algebraically closed field. For a finitely generated graded K algebra R, let cmdef R := dim R - depth R denote the Cohen-Macaulay-defect of R. Let G be a linear algebraic group over K that is reductive but not linearly…

Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic ${\rm char}(k)=p\geq 3$. In generalization of the familiar notion from the modular representation theory of finite groups, we define the…

It is known from work by H. Abels and P. Abramenko that for a classical Fq-group G of rank n the arithemetic lattice G(Fq[t]) of Fq[t]-rational points is of type Fn-1 provided that q is large enough. We show that the statement is true…

We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the…

Let X(F,G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in Mathematica, SageMath, and in Python, that takes a…

We give an example of a finitely presented simple group containing a finitely generated subgroup which is not finitely presented.

Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ is…

All nonabelian finite simple groups of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, have presentations with at most $80 $ relations and bit-length $O(\log n +\log q)$. Moreover, $A_n$…

A group of matrices $G$ with entries in a number field $K$ is defined to be numerical if $G$ has a finite index subgroup of matrices whose entries are algebraic integers. It is shown that an irreducible or completely reducible subgroup of…

Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$,…

Let $K$ be an unramified $p$-adic local field and let $W$ be the ring of integers of $K$. Let $(X,S)/W$ be a smooth proper scheme together with a normal crossings divisor. We show that there are only finitely many log crystalline $\mathbb…

Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with…