Completeness in partial differential algebraic geometry

This paper is part of the model theory of fields of characteristic 0, equipped with m   commuting derivation operators (DCF0,mDCF0,m). It continues to partial differential fields work begun by Wai-Yan Pong, who treated the case m=1m=1. We study the concept of completeness in differential algebraic geometry, applying methods of model theory and differential algebra. Our central tool in applying the valuative criterion developed in differential algebra by E.R. Kolchin, Peter Bloom, and Sally Morrison is a fundamental theorem in classical elimination theory due to the model theorist Lou van den Dries. We use this valuative criterion to give a new family of complete differential algebraic varieties. In addition to completeness, we prove some embedding theorems for differential algebraic varieties of arbitrary differential transcendence degree. As a special case, we show that every differential algebraic subvariety of the projective line which has Lascar rank less than ωmωm can be embedded in the affine line.