When we know that the positive existential theory of a ring $R[x]$ in a language $L$ is undecidable, does it follow that the positive existential theory of $R[x,y]$ in the same language $L$ is also undecidable?
Or can it be that the positive existential theory of $R[x,y]$ is decidable in $L$ ?
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EDIT:
I am looking at the specific case $\mathbb{C}[x]$ and $\mathbb{C}[x, e^{\mu x} \mid \mu \in \mathbb{C}]$ and the language $L=\{+, \cdot , \frac{d}{dx}, 0, 1, x\}$.
The positive existential theory of $\mathbb{C}[x]$ in $L$ is undecidable.
Does it follow that the positive existential theory of $\mathbb{C}[x, e^{\mu x} \mid \mu \in \mathbb{C}]$ in $L$ is undecidable, since $\mathbb{C}[x] \subset \mathbb{C}[x, e^{\mu x} \mid \mu \in \mathbb{C}]$ ?
Or do we have to reduce the positive existential theory of $\mathbb{C}[x]$ into the positive existential theory of $\mathbb{C}[x, e^{\mu x} \mid \mu \in \mathbb{C}]$ ?