Nonstandard methods for bounds in differential polynomial rings

Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt [L. van den Dries, K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984) 77–91] are here extended to differential polynomial rings over differential fields. Among the standard consequences of this work are: a partial answer to the primality problem, the equivalence of this problem with several others related to the Ritt problem, and the existence of bounds for characteristic sets of minimal prime differential ideals and for the differential Nullstellensatz.