Related papers: A Note on Schanuel's Conjectures for Exponential L…
We provide a characterisation of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to…
We axiomatize a class of existentially closed exponential fields equipped with an $E$-derivation. We apply our results to the field of real numbers endowed with $exp(x)$ the classical exponential function defined by its power series…
In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equations satisfied by…
In this paper we use tools from set theory and the uncountable categoricity of Zilber's pseudo-exponential field to show that Zilber's field is isomorphic to the complex field with (standard) exponentiation and hence Schanuel's conjecture…
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several…
We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the…
It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem, whether a given D-finite power series is algebraic or…
Given $b=-A\pm i$ with $A$ being a positive integer, we can represent any complex number as a power series in $b$ with coefficients in $\mathcal A=\{0,1,\ldots, A^2\}$. We prove that, for any real $\tau\geq 2$ and any non-empty proper…
We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially…
We introduce and discuss a variant of Schanuel conjecture in the framework of the Carlitz exponential function over Tate algebras and allied functions. Another purpose of the present paper is to widen the horizons of possible investigations…
It is proved that for all but a finite set of the square-free integers $d$ the value of transcendental function $\exp~(2\pi i ~x+\log\log y)$ is an algebraic number for the algebraic arguments $x$ and $y$ lying in a real quadratic field of…
We extend the Ax-Schanuel theorem recently proven for Shimura varieties by Mok-Pila-Tsimerman to all varieties supporting a pure polarized integral variation of Hodge structures. The essential new ingredient is a volume bound on Griffiths…
Given a logarithmic $1$-form on the snc locus of a log canonical surface pair $(X, D)$ over a perfect field of characteristic $p \ge 7$, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also…
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 $l$,…
We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter…
Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the set $U:=\{…
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…
We adapt the construction of the field of logarithmic-exponential transseries of van den Dries, Macintyre, and Marker to build an ordered differential field of sublogarithmic-transexponential series. We use this structure to build a…
I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property. A corollary is that there are at most…
It is proved that the associative differential graded algebra of (polynomial) polyvector fields on a vector space (may be infinite- dimensional) is quasi-isomorphic to the corresponding cohomological Hochschild complex of (polynomial)…