# Set-theoretic generation by circuit polynomials

MathOverflow Asked on January 5, 2022

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.)

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