Related papers: Algebraic G-functions associated to matrices over …

Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F<x_{1,g_1},...,x_{r,g_r}> be the corresponding free G-graded algebra where F is a field of…

We construct indecomposable cycles in the motivic cohomology group $H^3_{{\mathcal M}}(A,{\mathbb Q}(2))$ where $A$ is an Abelian surface over a number field or the function field of a base. When $A$ is the self product of the universal…

Let V be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field F. Suppose that v_{i,j} only depends on i-j and is 0 for |i-j| large. Then V^n is defined for all n, and one has a "generating…

According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and ${\bf P}^s$…

As an instance of a linear action of a Hopf algebra on a free associative algebra, we consider finite group gradings of a free algebra induced by gradings on the space spanned by the free generators. The homogeneous component corresponding…

We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group G and a nonabelian subgroup A of G, we describe G as a…

Let p be a prime. A p-adic functional on a torsion-free abelian group G is a group homomorphism from G to the p-adic integers. The group of all such p-adic functionals is viewed as a p-adic dual group of G, and is studied from the point of…

Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,\sigma)$, where $X$ is a variety over $k$ and $\sigma$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of…

Let p be a prime and G be a torsion-free abelian group. A homomorphism from G to the p-adic integers is called a p-adic functional on G. If G has finite rank, then G can be represented as an inductive limit of an inductive sequence of free…

We study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a very special case of complete free commutative Rota-Baxter algebras. This allows us to use free…

In this paper we develop computational tools to study the higher algebraic $K$-theory of Green functors. We construct a spectral sequence converging to the algebraic $\mathbb{G}$-theory of any $G$-Green functor, for $G$ a cyclic $p$-group.…

For an integer $M\geq 2$ and a finite group $G$, an element $\alpha\in G$ is called an $M$-th power if it satisfies $A^M=\alpha$ for some $A\in G$. In this article, we will deal with the case when $G$ is finite symplectic or orthogonal…

Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of…

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. In a previous article, we gave a necessary and sufficient condition for X to be free of given rank d…

A subgroup of a group $G$ is called algebraic if it can be expressed as a finite union of solution sets to systems of equations. We prove that a non-elementary subgroup $H$ of an acylindrically hyperbolic group $G$ is algebraic if and only…

Siegel defined in 1929 two classes of power series, the E-functions and G-functions, which generalize the Diophantine properties of the exponential and logarithmic functions respectively. In 1949, he asked whether any E-function can be…

We define a generalization $\mathfrak{G}$ of the Grassmann algebra $G$ which is well-behaved over arbitrary commutative rings $C$, even when $2$ is not invertible. In particular, this enables us to define a notion of superalgebras that does…

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the…

Let $G$ be a finite group of order $n$, and $Z_G=\mathbb{Z}\langle\zeta_{i,g}\mid g\in G,\ i=1,2,\dots,n\rangle$ be the free generic algebra, with canonical action of $G$ according to $(\zeta_{i,g})^x=\zeta_{i,x^{-1}g}$. It is proved that…

We consider a rank one group $G = \langle A,B \rangle $ which acts cubically on a module $V$, this means $[V,A,A,A] =0$ but $[V,G,G,G] \ne 0$. We have to distinguish whether the group $A_0 :=C_A([V,A]) \cap C_A(V/C_V(A))$ is trivial or not.…