Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D_T$ as follows:

Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether $$ T \vdash \exists \bar{x}\, p(\bar{x}) = 0, $$ i.e., whether $T$ shows that $p$ has a root.

By the MDRP theorem, we know that $D_T$ is undecidable for $T = \operatorname{Th}(\mathbb{N})$.

Using the same theorem, we can also show that for any computably enumerable $T$ containing Robinson arithmetic there is a $p$ such that $p$ has a root in $\mathbb{N}$, but $T$ does not show this, so $D_T$ is not equivalent to solving diophantic equations.

My question is for which smaller $T$ the problem $D_T$ is still undecidable, especially

• $T$ = Peano arithmetic?
• $T$ = Robinson arithmetic?
• Some computably enumerable $T$?
• Any computably enumerable $T$ containing, say, P.A. or R.A.? (this is my intuition, but I haven't been able to show it)

Given a theory $T \subseteq \operatorname{Th}(\mathbb{N})$, define the decision problem $D_T$ as follows:

Given a polynomial $p$ with integer coefficients and variables $\bar{x}$, decide whether $$ T \vdash \forall \bar{x}\, p(\bar{x}) \neq 0, $$ i.e., whether $T$ shows that $p$ does not have a root.

By the MDRP theorem, we know that $D_T$ is undecidable for $T = \operatorname{Th}(\mathbb{N})$.

Using the same theorem, we can also show that for any computably enumerable $T$ there is a $p$ such that $p$ has no root in $\mathbb{N}$, but $T$ does not show this, so $D_T$ is not equivalent to solving diophantic equations.

My question is for which smaller $T$ the problem $D_T$ is still undecidable, especially

• $T$ = Peano arithmetic?
• $T$ = Robinson arithmetic?
• Some computably enumerable $T$?
• Any computably enumerable $T$ containing, say, P.A. or R.A.? (this is my intuition, but I haven't been able to show it)

Edit: Had "... has a root" before, which is kinda trivial (thanks to @NoahS for pointing this out)