10 research outputs found
Around Zilber's quasiminimality conjecture
This is an extended abstract for a survey talk given in Oberwolfach on 1st
December 2022, slightly updated in June 2023. I survey some work around the
notion of quasiminimality and some of the progress towards Zilber's conjecture
from the last 25 years.Comment: 6 page
Finitely presented exponential fields
The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that finitely generated strong extensions are finitely presented, and these extensions are classified. An algebraic construction is given of Zilber's pseudo-exponential fields. As applications of the general results and methods of the paper, it is shown that Zilber's fields are not model-complete, answering a question of Macintyre, and a precise statement is given explaining how Schanuel's conjecture answers all transcendence questions about exponentials and logarithms. Connections with the Kontsevich-Zagier, Grothendieck, and Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested
Model Theory of Holomorphic Functions
This thesis is concerned with a conjecture of Zilber: that the complex field expanded with the exponential function should be `quasi-minimal'; that is, all its definable subsets should be countable or have countable complement. Our purpose is to study the geometry of this structure and other expansions by holomorphic functions of the complex field without having first to settle any number-theoretic problems, by treating all countable sets on an equal footing.
We present axioms, modelled on those for a Zariski geometry, defining a non-first-order class of ``quasi-Zariski'' structures endowed with a dimension theory and a topology in which all countable sets are of dimension zero. We derive a quantifier elimination theorem, implying that members of the class are quasi-minimal.
We look for analytic structures in this class. To an expansion of the complex field by entire holomorphic functions we associate a sheaf of analytic germs which is closed under application of the implicit function theorem. We prove that is also closed under partial differentiation and that it admits Weierstrass preparation. The sheaf defines a subclass of the analytic sets which we call -analytic. We develop analytic geometry for this class proving a Nullstellensatz and other classical properties. We isolate a condition on the asymptotes of the varieties of certain functions in . If this condition is satisfied then the -analytic sets induce a quasi-Zariski structure under countable union. In the motivating case of the complex exponential we prove a low-dimensional case of the condition, towards the original conjecture
The theory of Liouville functions
SIGLEAvailable from INIST (FR), Document Supply Service, under shelf-number : RP 16128 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc
The Theory of Liouville Functions
A Liouville function is an analytic function H: C â\u86\u92 C with a Taylor series � â\u88\u9e n=1 xn /an such the anâ\u80\u99s form a â\u80\u9cvery fast growing â\u80\u9d sequence of integers. In this paper we exhibit the complete first-order theory of the complex field expanded with H