From the Introduction: The object of diophantine geometry is the study of the sets of ratio- nal points of algebraic varieties. The classical case is the "arithmetic case" when our varieties are defined over a number field. One soon discoveres however that most arithmetic diophantine questions have interesting "geo- metric analogues" in which number fields are replaced by function fields over algebraically closed fields of characteristic zero. The aim of the present book is to describe a new approach to diophantine geometry over function fields; it is based 011 proving analogues of diophantine results in "differential algebraic geometry". In some significant cases, these analogues imply in their turn the corresponding "geometric analogues"..
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