Title:

AxSchanuel type inequalities in differentially closed fields

In this thesis we study AxSchanuel type inequalities for abstract differential equations. A motivating example is the exponential differential equation. The AxSchanuel theorem states positivity of a predimension defined on its solutions. The notion of a predimension was introduced by Hrushovski in his work from the 1990s where he uses an amalgamationwithpredimension technique to refute Zilber's Trichotomy Conjecture. In the differential setting one can carry out a similar construction with the predimension given by AxSchanuel. In this way one constructs a limit structure whose theory turns out to be precisely the firstorder theory of the exponential differential equation (this analysis is due to Kirby (for semiabelian varieties) and Crampin, and it is based on Zilber's work on pseudoexponentiation). One says in this case that the inequality is adequate. Thus, by an AxSchanuel type inequality we mean a predimension inequality for a differential equation. Our main question is to understand for which differential equations one can find an adequate predimension inequality. We show that this can be done for linear differential equations with constant coefficients by generalising the AxSchanuel theorem. Further, the question turns out to be closely related to the problem of recovering the differential structure in reducts of differentially closed fields where we keep the field structure (which is quite an interesting problem in its own right). So we explore that question and establish some criteria for recovering the derivation of the field. We also show (under some assumptions) that when the derivation is definable in a reduct then the latter cannot satisfy a nontrivial adequate predimension inequality. Another example of a predimension inequality is the analogue of AxSchanuel for the differential equation of the modular jfunction due to Pila and Tsimerman. We carry out a Hrushovski construction with that predimension and give an axiomatisation of the firstorder theory of the strong Fraïssé limit. It will be the theory of the differential equation of j under the assumption of adequacy of the predimension. We also show that if a similar predimension inequality (not necessarily adequate) is known for a differential equation then the fibres of the latter have interesting model theoretic properties such as strong minimality and geometric triviality. This, in particular, gives a new proof for a theorem of Freitag and Scanlon stating that the differential equation of j defines a trivial strongly minimal set.
