Could you tell me how we could prove that the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ is undecidable?

Which undecidable positive existential theory can we reduce to the above one?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?

How can we prove the (un)decidability?