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Set-theoretic generation by circuit polynomials

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: $Csubset [n]$ 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:

  1. If $P$ is generated by monomials, the answer is trivially always.
  2. 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.
  3. If $P$ is homogeneous, the circuit polynomials need not be scheme-theoretic generators for $P$ (even in the binomial case.)

MathOverflow Asked on January 5, 2022

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