Adequate predimension inequalities in differential fields

Annals of Pure and Applied Logic 173 (1):103030 (2022)
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Abstract

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the j-function (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterović and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the j-function, that is, the Ax-Schanuel inequality for the j-function is adequate.

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References found in this work

A new strongly minimal set.Ehud Hrushovski - 1993 - Annals of Pure and Applied Logic 62 (2):147-166.
A Note on the Axioms for Zilber’s Pseudo-Exponential Fields.Jonathan Kirby - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):509-520.
Algebraically closed field with pseudo-exponentiation.B. Zilber - 2005 - Annals of Pure and Applied Logic 132 (1):67-95.
Model theory of special subvarieties and Schanuel-type conjectures.Boris Zilber - 2016 - Annals of Pure and Applied Logic 167 (10):1000-1028.

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