The model theory of differential fields with finitely many commuting derivations

In this paper we set oat the basic model theory of differential fields of c haracteristic 0, which have finitely many commuting derivations. We give ax ioms for the theory of differentially closed differential fields with rn de rivations and show that this theory is omega-stable. model complete, and qu antifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be om ega(m) + 1.