Mickaël Matusinski

Hardy type derivations on fields of exponential logarithmic series

Abstract We consider the valued field K : = R ( ( Γ ) ) of formal series (with real coefficients and monomials in a totally ordered multiplicative group Γ). We investigate how to endow K with a logarithm l, which satisfies some natural properties such as commuting with infinite products of monomials. We studied derivations on K (Kuhlmann and Matusinski, in press [KM10] ). Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyze sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In Kuhlmann (2000) [Kuh00] , the first author described the exponential closure K EL of ( K , l ) . Here we show how to extend such a log-compatible derivation on K to K EL .

Hardy type derivations on generalised series fields

Abstract We consider the valued field K : = R ( ( Γ ) ) of generalised series (with real coefficients and monomials in a totally ordered multiplicative group Γ). We investigate how to endow K with a series derivation, that is a derivation that satisfies some natural properties such as commuting with infinite sums (strong linearity) and (an infinite version of) Leibniz rule. We characterise when such a derivation is of Hardy type, that is, when it behaves like differentiation of germs of real valued functions in a Hardy field. We provide a necessary and sufficient condition for a series derivation of Hardy type to be surjective.

The Exponential-Logarithmic Equivalence Classes of Surreal Numbers

In his monograph, H. Gonshor showed that Conway’s real closed field of surreal numbers carries an exponential and logarithmic map. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the additive and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length of these exponential classes.

Surreal numbers with derivation, Hardy fields and transseries: a survey

The present survey article has two aims: - To provide an intuitive and accessible introduction to the theory of the field of surreal numbers with exponential and logarithmic functions. - To give an overview of some of the recent achievements. In particular, the field of surreal numbers carries a derivation which turns it into a universal domain for Hardy fields.

On the algebraicity of Puiseux series

We deal with the algebraicity of a formal Puiseux series in terms of the properties of its coefficients. We show that the algebraicity of a Puiseux series for given bounded degrees is determined by a finite number of explicit universal polynomial formulas. Conversely, given a vanishing polynomial, there is a closed-form formula for the coefficients of the series in terms of the coefficients of the polynomial and of an initial part of the series.

On generalized series fields and exponential-logarithmic series fields with derivations

In geometric terms, given a singular foliation of the plane, a dicritical divisor is (whenever it exists) an irreducible component of the exceptional divisor which is transverse to the foliation. Abhyankar gave recently a definition of the dicritical divisors which generalize and algebraicize the geometrical definition in the local case and the polynomial case. Following his work, we give a geometrical interpretation of these dicritical divisors and new proofs of their existence.Developments in valuation theory, specially the study of algebraically closed valued fields, have used the model theory of C-minimal structures in different places (specially the work of Hrushovski-Kazdhan in [HK] and Haskell-Hrushovski-Macpherson in [HHM]). We intend with this text both to divulgate a basic comprehension of C-minimality for those mathematicians interested in valuation theory having a basic knowledge in model theory and to provide a slightly different presentation of the cell decomposition theorem proved by Haskell and Macpherson in [HM94]. Studying algebraic structures from a model-theoretic point of view can be described as studying the category of definable sets of algebraic structures: objects correspond to definable sets (i.e., solution sets of a particular first order formula) and morphisms correspond to definable functions (i.e., functions for which the graph is a definable set). A model theoretic perspective allows different ways of generalizing properties one can extract from algebraic structures. For instance, quantifier elimination for real closed fields (R,≤,+, ·, 0, 1) implies that definable subsets of R are exactly the semi-algebraic sets but also induced the fruitful notion of o-minimality: an ordered structure (M,≤, ...) is o-minimal if every definable subset of M is a finite union of points and intervals. In the same spirit, quantifier elimination for theories of valued structures like algebraically closed valued fields or the p-adic fields induce different notions of minimality, C-minimality being one of them. The aim of the text is to provide the reader with a basic comprehension of C-minimality (hopefully giving her tools to easy the reading of articles like [HK, HHM]), and to expose a proof of a deep theorem proved by Haskell and Macpherson in [HM94]: the cell decomposition theorem for dense C-minimal structures. We do not present new results and most of the article follows the same scheme as [HM94] though most of the proofs (and some definitions) have been simplified (and in some cases corrected). Section 1 contains a brief introduction to C-minimality together with definitions, examples and basic properties. In section 2 we define what cells are and study definable functions. Finally the cell decomposition theorem is proved in section 3 together with some results about dimension. Notation will be standard with the following remarks. Capital L is restricted for firstorder languages (with all possible subscripts and superscripts like L′, L0, etc.). For a set A, L(A) is the expansion of L with a new constant for every element in A. We say a formula φ has parameters from A if it is an L(A)-formula. Given an L-structure M , A ⊆ Mn and a formula φ(x) with |x| = n (the length of the tuple), we denote by φ(A) the set {a ∈ A : M |= φ(a)}. We say A is definable if there is an L(M)-formula φ(x)We study Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. For zero dimensional valuations we prove that whenever the ring is complete and the semigroup of values taken by the valuation is finitely generated (which implies that the valuation is Abhyankar) the valuation can be uniformized in an embedded way by a birational map which is monomial with respect to a suitable system of generators of the maximal ideal. We prove that conversely if a valuation is Abhyankar after a birational modification and localization at the point picked by the valuation one obtains a ring whose semigroup of values is finitely generated. Combining the two results and using the good behavior of Abhyankar valuations with respect to composition and completion gives local uniformization for all Abhyankar valuations of excellent equicharacteristic local domains with an algebraically closed residue field. Some general results on Abhyankar valuations are by-products of the method of proof.In this paper, we give an introduction of the phenomenon of lifting with respect to residually transcendental extensions, the notion of distinguished pairs and complete distinguished chains which lead to the study of certain invariants associated to irreducible polynomials over valued fields. We give an overview of various results regarding these concepts and their applications.Given such a series a = ∑ γ aγt γ ∈ K and α ∈ Γ we can truncate the series at t to give the series a|α := ∑ γ<α aγt γ ∈ K. We call a|α a truncation of a; if in addition a 6= a|α (equivalently, α 6 β for some β ∈ supp a), then a|α is said to be a proper truncation of a. Thus all truncations of a are proper iff supp a is cofinal in Γ. The only proper truncation of ct with nonzero c ∈ k is 0. A subset of K is said to be closed under truncation (or just truncation closed) if it contains all truncations of all its elements. The property of being truncation closed turns out to be stable under various operations. I learned about this from Ressayre [10] (where k = R) and confess to being very surprised by this phenomenon. My modest goal here is to bring together in one place the various stability results of this kind, with proofs and without unnecessary restrictions. For example, the next theorem (from Section 2) gives the stability of truncation closedness under certain arithmetic extension procedures.where aij ∈ K. Additive polynomials over valued fields in positive characteristic play an important role in understanding many algebraic and model theoretic properties of maximal fields of positive characteristic, see [7] for a thorough examination of the issue. A subset S of a valued field (K, v) has the optimal approximation property if for all a ∈ K, the set {v(s − a) : s ∈ S} has a maximal element. By the image of a polynomial f(x1, . . . , xn) over K we mean the set {f(a1, . . . , an) : a1, . . . , an ∈ K}.The text is based on notes from a class entitled {\em Model Theory of Berkovich Spaces}, given at the Hebrew University in the fall term of 2009, and retains the flavor of class notes. It includes an exposition of material from \cite{hhmcrelle}, \cite{hhm} and \cite{HL}, regarding definable types in the model completion of the theory of valued fields, and the classification of imaginary sorts. The latter is given a new proof, based on definable types rather than invariant types, and on the notion of {\em generic reparametrization}. I also try to bring out the relation to the geometry of \cite{HL} - stably dominated definable types as the model theoretic incarnation of a Berkovich point.Suppose

Existence des diviseurs dicritiques, d’après S.S. Abhyankar

F

A note on Schanuel’s conjectures for exponential logarithmic power series fields

is a field with a nontrivial valuation

Exponential fields and Conway's omega-map

v

Sur l'alg\'ebricit\'e des s\'eries de Puiseux

and valuation ring