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The theory of difference-differential fields of characteristic zero has amodel-companion denoted by $\it DCFA$. Previously we proved a weak version ofZilber's dichotomy for $\it DCFA$. In this paper we use arc spaces techniquesas developed by Moosa, Pillay and Scanlon to suppress the extra hypothesisneeded before and prove the full Zilber's dichotomy for $\it DCFA$, we alsostate how these techniques generalise to partial differential fields with anautomorphism.

Zilber's Dichotomy for Differentially Closed Fields with an Automorphism