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Preprint Number 1393

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1393. Vahagn Aslanyan
Predimension Inequalities in Differential Fields
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Submission date: 13 March 2018

Abstract:

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. 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 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 its turn is analogous to Zilber's pseudo-exponentiation) though there are many significant differences.

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Full text arXiv 1803.04753: pdf, ps.


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