Adequate predimension inequalities in differential fields

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, here give a precise definition quite general context. We also discuss the connection of problem definability derivations reducts differentially closed fields. The Ax-Schanuel exponential equation (proved Ax) its analogue j -function (established Pila Tsimerman) are our main examples predimensions. carry out Hrushovski construction with latter obtain natural candidate first-order theory -function. It is analogous Kirby's axiomatisation (which turn based on axioms Zilber's pseudo-exponentiation), although there many significant differences. joint work Sebastian Eterovi? Jonathan Kirby have recently proven that obtained indeed -function, is, adequate.