Related papers: Exponentiation in power series fields

We explain how the field of logarithmic-exponential series constructed in \cite{DMM1} and \cite {DMM2} embeds as an exponential field in any field of exponential-logarithmic series constructed in \cite{KK1}, \cite {K} and \cite {KS}. On the…

We show that every non-trivial ordered abelian group $G$ is augmentable by infinite elements, i.e., we have $G\preccurlyeq H\oplus G$ for some non-trivial ordered abelian group $H$. As an application, we show that when $k$ is a field of…

A classical tool in the study of real closed fields are the fields $K((G))$ of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian…

In math.AC/9608214 it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural…

Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map ${\mathop{exp}}_!:X\mathbb K[[X]]\longrightarrow 1+X\mathbb K[[X]]$ for the associated Lie group…

Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{G}_a$ denote the additive group of $k$. Let $n \geq 1$ and let ${\rm Mat}(n, k[T])^E$ denote the set of all exponential matrices of ${\rm Mat}(n,…

We continue our exploration of various approaches to integration of representations from a Lie algebra $\mbox{Lie} (G)$ to an algebraic group $G$ in positive characteristic. In the present paper we concentrate on an approach exploiting…

We investigate valued fields which admit a valuation basis. Given a countable ordered abelian group G and a real closed, or algebraically closed field F, we give a sufficient condition for a valued subfield of the field of generalized power…

Let $F$ be an archimedean field, $G$ a divisible ordered abelian group and $h$ a group exponential on $G$. A triple $(F,G,h)$ is realised in a non-archimedean exponential field $(K,\exp)$ if the residue field of $K$ under the natural…

A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian…

Let $\mathbb F$ be a local field and $G$ be a linear algebraic group defined over $\mathbb F$. For $k\in\mathbb N$, let $g\to g^k$ be the $k$-th power map $P_k$ on $G(\mathbb F)$. The purpose of this article is two-fold. First, we study the…

Let $G$ be a connected exponential Lie group and $R$ be the solvable radical of $G$. We describe a condition on $G/R$ under which one can then conclude that $R$ is an exponential Lie group. The condition holds in particular when $G$ is a…

We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential…

In [26], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered…

A classical result in descriptive complexity theory states that Datalog expresses exactly the class of polynomially computable queries on ordered databases. In this paper we extend this result to the case of higher-order Datalog. In…

We establish a correspondence between automorphisms and derivations on certain algebras of generalised power series. In particular, we describe a Lie algebra of derivations on a field $k(\!(G)\!)$ of generalised power series, exploiting our…

Let $H$ be a linear algebraic group over an algebraically closed field of characteristic $p>0$. We prove that any "exponential map" for $H$ induces a bijection between the variety of $r$-tuples of commuting $[p]$-nilpotent elements in…

Let $G$ be a group containing a nilpotent normal subgroup $N$ with central series $\{N_j\}$, such that each $N_j/N_{j+1}$ is a $\mathbb{F}$-vector space over a field $\mathbb{F}$ and the action of $G$ on $N_j/N_{j+1}$ induced by the…

We consider the valued field $\mathds{K}:=\mathbb{R}((\Gamma))$ of formal series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma>$). We investigate how to endow $\mathds{K}$ with a logarithm $l$,…

Let $G$ be an acylindrically hyperbolic group and $E$ an exponential equation over $G$. We show that if $E$ is solvable in $G$, then there exists a solution whose components, corresponding to loxodromic elements, can be linearly estimated…