Algebraic theory of differential inequalities

We obtain differential-algebraic analogues of some basic theorems of real algebra and semialgebraic geometry. Proofs are based on: a differential version of the real spectrum of a differential ring containing Q; an Artin-Schreier theory for such rings; the model theory of ordered differential fields. Results include: an algebraic characterization of the differential inequalities which are consequences of a given finite set of algebraic differential equations and inequalities; a differential counterpart of the Hormander-Lojasiewicz inequality.