Let $P$ be a prime ideal in $S=\mathbb{C}[x_1,\ldots , x_n],$ and write $[n] = \{ 1, \ldots , n \}.$ The *algebraic matroid* of $P$ can be defined according to circuit axioms as follows: $C\subset [n]$ is a circuit if $P \cap \mathbb{C} [x_i \mid i \in C]$ is principal, and we call a generator of this ideal a circuit polynomial. The *circuit ideal* $P_{\mathcal{C}}\subset S$ is generated by all circuit polynomials.

**Question** For which $P$ do we have $\sqrt{P_{\mathcal{C}}}=P$?

For context, I include the following facts:

- If $P$ is generated by monomials, the answer is trivially always.
- If $P$ is generated by binomials, the answer is always, though seemingly less trivial. This follows from results in the article "Binomial Ideals" by Eisenbud and Sturmfels.
- If $P$ is homogeneous, the circuit polynomials need not be
*scheme-theoretic generators*for $P$ (even in the binomial case.)