arXiv subjects

Vincent Bagayoko

Publications and source records attributed to Vincent Bagayoko.

12 recordsLinked to original sources

arXiv2026

A formal Lie correspondence

We establish an equivalence between categories of 'formally nilpotent' Lie algebras and exponential groups in characteristic zero. It extends the equivalences of Mal'cev, Lazard, Quillen and Warfield, and applies to groups under composition of generalized formal series or automorphisms of algebras of generalized formal series. We obtain first-order transfer results from finite dimensional nilpotent objects to formally nilpotent ones. We give applications to solving equations over groups, to the theory of nilpotent exponential groups as per Miasnikov-Remeslennikov, and to definability problems in certain groups of formal series.

math.RA↗

arXiv2025

Automorphisms and derivations on algebras endowed with formal infinite sums

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 knowledge of its group of valuation preserving automorphisms. The correspondence is given by the formal Taylor expansion of the exponential. In order to define the exponential map, we develop an appropriate notion of summability of infinite families in algebras. We show that there is a large class of algebras in which the exponential induces the above correspondence.

math.RA↗

arXiv2025

Groups with infinite linearly ordered products

We introduce a formalism of infinite, linearly ordered products in general groups. Using this, we define infinite compositions in certain groups of formal power series such as transseries. We show that such groups can sometimes be represented as infinite, linearly ordered, semidirect products of ordered Abelian groups.

math.GR↗

arXiv2025

Ordered groups of formal series, and a conjugacy problem

Given an ordered field $\mathbb{T}$ of formal series over an ordered field $\mathbf{R}$ equipped with a composition law $\circ \colon \mathbb{T} \times \mathbb{T}^{>\mathbb{R}} \longrightarrow \mathbb{T}$, we give conditions for $(\mathbb{T}^{>\mathbb{R}},\circ)$ to be a group. We show that classical fields of transseries and hyperseries satisfy these conditions. We then give further conditions on $\mathbb{T}$ under which $(\mathbb{T}^{>\mathbb{R}},\circ,<)$ is a linearly ordered group with exactly three conjugacy classes, and solve the open problem of existence of such a group.

math.LO↗

arXiv2025

Taylor expansions over generalised power series

We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an exponential function, and apply this to the case of standard fields of transseries, such as $\log$-$\exp$ transseries and $ω$-series.

math.LO↗

arXiv2025

Formal conjugacy and asymptotic differential algebra

We study conjugacy of formal derivations on fields of generalised power series in characteristic 0. Casting the problem of Poincaré resonance in terms of asymptotic differential algebra, we give conditions for conjugacy of parabolic flat log-exp transseries, flat grid-based transseries, logarithmic transseries, power series with exponents and coefficients in an ordered field, and formal Puiseux series.

math.DS↗

arXiv2025

Equations over valued groups

We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using simple valuation theoretic arguments.

math.GR↗

arXiv2024

Hyperseries subfields of surreal numbers

We study subfields of surreal numbers, called hyperseries fields, that are suited to be equipped with derivations and composition laws. We show how to define embeddings on hyperseries fields that commute with transfinite sums and all hyperexponential and hyperlogarithmic functions.

math.LO↗

arXiv2024

Sign sequences of log-atomic numbers

Log-atomic numbers are surreal numbers whose iterated logarithms are monomials, and consequently have a trivial expansion as transseries. Presenting surreal numbers as sign sequences, we give the sign sequence formula for log-atomic numbers. To that efect, we relate log-atomic numbers to fixed-points of certain surreal functions.

math.LO↗

arXiv2023

The hyperserial field of surreal numbers

For any ordinal $α> 0$, we show how to define a hyperexponential $E_{ω^α}$ and a hyperlogarithm $L_{ω^α}$ on the class $\mathbf{No}^{>, \succ}$ of positive infinitely large surreal numbers. Such functions are archetypes of extremely fast and slowly growing functions at infinity. We also show that the surreal numbers form a so-called hyperserial field for our definition.

math.LO↗

arXiv2023

Surreal numbers as hyperseries

Surreal numbers form the ultimate extension of the field of real numbers with infinitely large and small quantities and in particular with all ordinal numbers. Hyperseries can be regarded as the ultimate formal device for representing regular growth rates at infinity. In this paper, we show that any surreal number can naturally be regarded as the value of a hyperseries at the first infinite ordinal $ω$. This yields a remarkable correspondence between two types of infinities: numbers and growth rates.

math.LO↗

arXiv2023

Surreal substructures

Conway's field No of surreal numbers comes both with a natural total order and an additional "simplicity relation" which is also a partial order. Considering No as a doubly ordered structure for these two orderings, an isomorphic copy of No into itself is called a surreal substructure. It turns out that many natural subclasses of No are actually of this type. In this paper, we study various constructions that give rise to surreal substructures and analyze important examples in greater detail.

math.LO↗