Adequate Predimension Inequalities in Differential Fields

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 jfunction (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 based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences.

A differential approach to the Ax-Schanuel, I

“…We have a canonical decomposition of the tangent bundle T P = T (P/Y ) ⊕ H ∇ as the direct sum of its vertical and ∇-horizontal subbundles. 1 Let D ⊂ T Y be a subset of the tangent bundle of a smooth manifold Y such that for each y ∈ Y, the fiber above y, Dy ⊆ TyY is a d-dimensional subspace. Further, suppose that for any point y there is an Zariski open neighborhood U of y such that for any z ∈ U , we have independent vector fields X1(z), .…”

“…Remark 5.7. We will use the following model theoretic conventions (1) The notation υ will be used both for a tuple and an element.…”

“…Over the past several years, in a series of works Aslanyan, and later Aslanyan, Kirby and Eterović [1,2,5,6,4] develop the connection between Ax-Schanuel type transcendence statements and the existential closedness of certain reducts of differentially closed fields related to equations satisfied by the j-function. This series of work builds on the earlier program of Kirby, Zilber and others mainly around the exponential function, see e.g.…”

“…Intermediate differential algebraic results from Ax's work are utilized in the program studying existential closedness results around the exponential function. A different approach was required in [1,2,5,6,4] for studying the j-function, in part since the intermediate results of the functional transcendence results of [31] take place in the o-minimal rather than abstract differential setting. For instance, the motivation for a differential algebraic proof of Ax-Schanuel results and this issue is pointed out specifically following Theorem 1.3 of [5] and in Section 4.4 of [4].…”