The Circuit Ideal of a Vector Configuration

Abstract

The circuit ideal, $\ica$, of a configuration $\A = \{\a_1, ..., \a_n\} \subset \Z^d$ is the ideal generated by the binomials ${\x}^{\cc^+} - {\x}^{\cc^-} \in \k[x_1, ..., x_n]$ as $\cc = \cc^+ - \cc^- \in \Z^n$ varies over the circuits of $\A$. This ideal is contained in the toric ideal, $\ia$, of $\A$ which has numerous applications and is nontrivial to compute. Since circuits can be computed using linear algebra and the two ideals often coincide, it is worthwhile to understand when equality occurs. In this paper we study $\ica$ in relation to $\ia$ from various algebraic and combinatorial perspectives. We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of $\ica$ and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that $\ia$ is the unique minimal prime of $\ica$ and that the embedded primes of $\ica$ are indexed by certain faces of the cone spanned by $\A$. We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to $\ia$ and $\ica$. The Gr\"obner fan of $\ica$ is shown to refine that of $\ia$ when the codimension of the ideals is at most two.