  Using the functional interpretation from proof theory, we analyze
nonconstructive proofs of several central theorems about polynomial and
differential polynomial rings. We extract effective bounds, some of which are
new to the literature, from the resulting proofs. In the process we discuss the
constructive content of Noetherian rings and the Nullstellensatz in both the
classical and differential settings. Sufficient background is given to
understand the proof-theoretic and differential-algebraic framework of the main
results.
